Image Processing Toolbox For Use with MATLAB ® Computation Visualization Programming User’s Guide Version 3 How to Contact The MathWorks: www.mathworks.com comp.soft-sys.matlab Web Newsgroup [email protected] Technical support Product enhancement suggestions Bug reports Documentation error reports Order status, license renewals, passcodes Sales, pricing, and general information 508-647-7000 Phone 508-647-7001 Fax The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 Mail [email protected] [email protected] [email protected] [email protected] [email protected] For contact information about worldwide offices, see the MathWorks Web site. Image Processing Toolbox User’s Guide COPYRIGHT 1993 - 2002 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc. FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by or for the federal government of the United States. By accepting delivery of the Program, the government hereby agrees that this software qualifies as "commercial" computer software within the meaning of FAR Part 12.212, DFARS Part 227.7202-1, DFARS Part 227.7202-3, DFARS Part 252.227-7013, and DFARS Part 252.227-7014. The terms and conditions of The MathWorks, Inc. Software License Agreement shall pertain to the government’s use and disclosure of the Program and Documentation, and shall supersede any conflicting contractual terms or conditions. If this license fails to meet the government’s minimum needs or is inconsistent in any respect with federal procurement law, the government agrees to return the Program and Documentation, unused, to MathWorks. MATLAB, Simulink, Stateflow, Handle Graphics, and Real-Time Workshop are registered trademarks, and TargetBox is a trademark of The MathWorks, Inc. Other product or brand names are trademarks or registered trademarks of their respective holders. Printing History: August 1993 May 1997 January 1998 January 1999 September 2000 April 2001 June 2001 July 2002 First printing Second printing Online only Online only Online only Third printing Online only Online only Version 1 Version 2 Revised for Version 2.1 Revised for Version 2.2 (Release 11) Revised for Version 2.2.2 (Release 12) Revised for Version 3.0 Revised for Version 3.1 (Release 12.1) Revised for Version 3.2 (Release 13) Contents Preface What Is the Image Processing Toolbox? . . . . . . . . . . . . . . . . . . xii Related Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Configuration Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv About the Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Structure of the Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . xv Image Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii MATLAB Newsgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Typographical Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Image Processing Demos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Getting Started 1 Example 1 — Some Basic Topics . . . . . . . . . . . . . . . . . . . . . . . . 1. Read and Display an Image . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Check the Image in Memory . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Perform Histogram Equalization . . . . . . . . . . . . . . . . . . . . . . 4. Write the Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Check the Contents of the Newly Written File . . . . . . . . . . . 1-2 1-2 1-3 1-4 1-7 1-7 Example 2 — Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . 1-9 1. Read and Display an Image . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9 2. Use Morphological Opening to Estimate the Background . . 1-9 3. Display the Background Approximation as a Surface . . . . . 1-10 i 4. Subtract the Background Image from the Original Image . 5. Adjust the Image Contrast . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Apply Thresholding to the Image . . . . . . . . . . . . . . . . . . . . . 7. Determine the Number of Objects in the Image . . . . . . . . . 8. Examine the Label Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Measure Object Properties in the Image . . . . . . . . . . . . . . . 10. Compute Statistical Properties of Objects in the Image . . 1-12 1-13 1-14 1-15 1-16 1-18 1-21 Where to Go from Here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-23 Online Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-23 Toolbox Demos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-23 Introduction 2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 Images in MATLAB and the Image Processing Toolbox . . . 2-4 Storage Classes in the Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4 Image Types in the Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5 Indexed Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5 Intensity Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7 Binary Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8 RGB Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8 Multiframe Image Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11 Summary of Image Types and Numeric Classes . . . . . . . . . . . 2-12 Converting Image Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12 Working with Image Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading a Graphics Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . Writing a Graphics Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . Querying a Graphics File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Converting Image Storage Classes . . . . . . . . . . . . . . . . . . . . . . Converting Graphics File Formats . . . . . . . . . . . . . . . . . . . . . . 2-15 2-16 2-17 2-18 2-19 2-20 Image Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21 ii Contents Summary of Image Arithmetic Functions . . . . . . . . . . . . . . . . Image Arithmetic Truncation Rules . . . . . . . . . . . . . . . . . . . . . Adding Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subtracting Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplying Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dividing Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nesting Calls to Image Arithmetic Functions . . . . . . . . . . . . . 2-22 2-22 2-23 2-24 2-25 2-27 2-27 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-28 Pixel Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-28 Spatial Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-29 Displaying and Printing Images 3 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2 Displaying Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3 Displaying Indexed Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3 Displaying Intensity Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4 Displaying Binary Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6 Displaying RGB Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10 Displaying Images Directly from Disk . . . . . . . . . . . . . . . . . . . 3-11 Special Display Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . Adding a Colorbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displaying Multiframe Images . . . . . . . . . . . . . . . . . . . . . . . . . Displaying Multiple Images . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12 3-12 3-13 3-17 Setting Toolbox Display Preferences . . . . . . . . . . . . . . . . . . . 3-23 Toolbox Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23 Using the truesize Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-24 Zooming in on a Region of an Image . . . . . . . . . . . . . . . . . . . 3-26 Zooming In or Out with the Zoom Buttons . . . . . . . . . . . . . . . . 3-26 Zooming In or Out from the Command Line . . . . . . . . . . . . . . 3-26 iii Texture Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-28 Printing Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-29 Troubleshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-30 Spatial Transformations 4 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3 Image Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4 Image Resizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specifying the Size of the Output Image . . . . . . . . . . . . . . . . . . Specifying the Interpolation Method . . . . . . . . . . . . . . . . . . . . . Using Filters to Prevent Aliasing . . . . . . . . . . . . . . . . . . . . . . . . 4-5 4-5 4-6 4-6 Image Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8 Specifying the Interpolation Method . . . . . . . . . . . . . . . . . . . . . 4-8 Specifying the Size of the Output Image . . . . . . . . . . . . . . . . . . 4-9 Image Cropping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10 Performing General Spatial Transformations . . . . . . . . . . . Specifying the Transformation Type . . . . . . . . . . . . . . . . . . . . . Performing the Transformation . . . . . . . . . . . . . . . . . . . . . . . . Advanced Spatial Transformation Techniques . . . . . . . . . . . . 4-12 4-12 4-14 4-15 Image Registration 5 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 iv Contents Registering an Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4 Point Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4 Example: Registering to a Digital Orthophoto . . . . . . . . . . . . . . 5-6 Types of Supported Transformations . . . . . . . . . . . . . . . . . . 5-13 Selecting Control Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using the Control Point Selection Tool . . . . . . . . . . . . . . . . . . . Starting the Control Point Selection Tool . . . . . . . . . . . . . . . . . Viewing the Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specifying Matching Control Point Pairs . . . . . . . . . . . . . . . . . Saving Control Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15 5-15 5-16 5-18 5-22 5-30 Using Correlation to Improve Control Points . . . . . . . . . . . 5-33 Neighborhood and Block Operations 6 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2 Block Processing Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3 Types of Block Processing Operations . . . . . . . . . . . . . . . . . . . . 6-3 Sliding Neighborhood Operations . . . . . . . . . . . . . . . . . . . . . . 6-5 Padding Borders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6 Linear and Nonlinear Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6 Distinct Block Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9 Overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10 Column Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12 Sliding Neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12 Distinct Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13 v Linear Filtering and Filter Design 7 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2 Linear Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6 Filtering Using imfilter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7 Using Predefined Filter Types . . . . . . . . . . . . . . . . . . . . . . . . . 7-14 Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency Transformation Method . . . . . . . . . . . . . . . . . . . . . Frequency Sampling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . Windowing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creating the Desired Frequency Response Matrix . . . . . . . . . Computing the Frequency Response of a Filter . . . . . . . . . . . . 7-16 7-16 7-17 7-18 7-19 7-20 7-21 Transforms 8 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3 Definition of Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 8-3 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11 Discrete Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-16 The DCT Transform Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-17 The DCT and Image Compression . . . . . . . . . . . . . . . . . . . . . . 8-18 Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-20 Using the Radon Transform to Detect Lines . . . . . . . . . . . . . . 8-24 The Inverse Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . 8-26 vi Contents Morphological Operations 9 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2 Dilation and Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-4 Understanding Dilation and Erosion . . . . . . . . . . . . . . . . . . . . . 9-4 Structuring Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-7 Dilating an Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-11 Eroding an Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-12 Combining Dilation and Erosion . . . . . . . . . . . . . . . . . . . . . . . . 9-14 Dilation- and Erosion-Based Functions . . . . . . . . . . . . . . . . . . 9-16 Morphological Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . Marker and Mask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pixel Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flood-Fill Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finding Peaks and Valleys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19 9-19 9-23 9-26 9-29 Distance Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-38 Example: Marker-Controlled Watershed Segmentation . . Step 1: Read in Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Step 2: Create the Structuring Element . . . . . . . . . . . . . . . . . . Step 3: Enhance the Image Contrast . . . . . . . . . . . . . . . . . . . . Step 4: Exaggerate the Gaps Between Objects . . . . . . . . . . . . Step 5: Convert Objects of Interest . . . . . . . . . . . . . . . . . . . . . . Step 6: Detect Intensity Valleys . . . . . . . . . . . . . . . . . . . . . . . . Step 7: Watershed Segmentation . . . . . . . . . . . . . . . . . . . . . . . Step 8: Extract Features from Label Matrix . . . . . . . . . . . . . . 9-41 9-41 9-42 9-42 9-43 9-44 9-45 9-46 9-47 Objects, Regions, and Feature Measurement . . . . . . . . . . . Connected-Component Labeling . . . . . . . . . . . . . . . . . . . . . . . . Selecting Objects in a Binary Image . . . . . . . . . . . . . . . . . . . . . Finding the Area of Binary Images . . . . . . . . . . . . . . . . . . . . . . Finding the Euler Number of a Binary Image . . . . . . . . . . . . . 9-49 9-49 9-51 9-51 9-52 Lookup Table Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-53 vii Analyzing and Enhancing Images 10 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2 Pixel Values and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pixel Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intensity Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Image Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Image Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Region Property Measurement . . . . . . . . . . . . . . . . . . . . . . . . . 10-3 10-3 10-4 10-8 10-8 10-9 10-9 Image Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-10 Edge Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-10 Quadtree Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-11 Image Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-14 Intensity Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-14 Noise Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-20 Region-Based Processing 11 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2 Specifying a Region of Interest . . . . . . . . . . . . . . . . . . . . . . . . 11-3 Selecting a Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-3 Other Selection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-4 Filtering a Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-6 Filling a Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8 viii Contents Image Deblurring 12 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2 Understanding Deblurring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-3 Causes of Blurring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-3 Deblurring Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-3 Using the Deblurring Functions . . . . . . . . . . . . . . . . . . . . . . . 12-5 Deblurring with the Wiener Filter . . . . . . . . . . . . . . . . . . . . . . 12-6 Deblurring with a Regularized Filter . . . . . . . . . . . . . . . . . . . . 12-8 Deblurring with the Lucy-Richardson Algorithm . . . . . . . . . . 12-9 Deblurring with the Blind Deconvolution Algorithm . . . . . . 12-14 Creating Your Own Deblurring Functions . . . . . . . . . . . . . . . 12-19 Avoiding Ringing in Deblurred Images . . . . . . . . . . . . . . . . 12-20 Color 13 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-2 Working with Different Screen Bit Depths . . . . . . . . . . . . . 13-3 Determining Your Systems Screen Bit Depth . . . . . . . . . . . . . 13-3 Choosing a Screen Bit Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-4 Reducing the Number of Colors in an Image . . . . . . . . . . . . 13-6 Using rgb2ind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-7 Reducing Colors in an Indexed Image . . . . . . . . . . . . . . . . . . 13-12 Dithering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-13 Converting to Other Color Spaces . . . . . . . . . . . . . . . . . . . . NTSC Color Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . YCbCr Color Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HSV Color Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-15 13-15 13-16 13-16 ix Function Reference 14 Functions — By Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-2 Image Input, Output, and Display . . . . . . . . . . . . . . . . . . . . . . 14-2 Spatial Transformation and Registration . . . . . . . . . . . . . . . . . 14-5 Image Analysis and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 14-6 Image Enhancement and Restoration . . . . . . . . . . . . . . . . . . . 14-7 Linear Filtering and Transforms . . . . . . . . . . . . . . . . . . . . . . . 14-8 Morphological Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-10 Region-Based, Neighborhood, and Block Processing . . . . . . . 14-12 Colormap and Color Space Functions . . . . . . . . . . . . . . . . . . . 14-13 Miscellaneous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-14 Functions — Alphabetical List . . . . . . . . . . . . . . . . . . . . . . . 14-16 x Contents Preface This section introduces you to the Image Processing Toolbox and describes conventions used by the documentation. What Is the Image Processing Toolbox? Introduces the Image Processing Toolbox and its (p. xii) capabilities Related Products (p. xiii) Highlights other MathWorks products that are related to image processing Configuration Notes (p. xiv) Provides some information about installing and configuring the image processing toolbox About the Documentation (p. xv) Details the structure of the Image Processing Toolbox documentation and credits the sources of the images used in the documentation Typographical Conventions (p. xix) Lists typographical conventions used in the documentation Image Processing Demos (p. xx) Lists all the demos included with the Image Processing Toolbox Preface What Is the Image Processing Toolbox? The Image Processing Toolbox is a collection of functions that extend the capability of the MATLAB® numeric computing environment. The toolbox supports a wide range of image processing operations, including: • Spatial image transformations • Morphological operations • Neighborhood and block operations • Linear filtering and filter design • Transforms • Image analysis and enhancement • Image registration • Deblurring • Region of interest operations Many of the toolbox functions are MATLAB M-files, a series of MATLAB statements that implement specialized image processing algorithms. You can view the MATLAB code for these functions using the statement type function_name You can extend the capabilities of the Image Processing Toolbox by writing your own M-files, or by using the toolbox in combination with other toolboxes, such as the Signal Processing Toolbox and the Wavelet Toolbox. For a list of the new features in Version 3.0, see the Release Notes documentation. xii Related Products Related Products The MathWorks provides several products that are especially relevant to the kinds of tasks you can perform with Image Processing Toolbox. For more information about any of these products, see either • The online documentation for that product if it is installed or if you are reading the documentation from the CD • The MathWorks Web site, at http://www.mathworks.com; see the “products” section The toolboxes listed below all include functions that extend MATLAB. The blocksets all include blocks that extend Simulink. . Product Description DSP Blockset Design and simulate DSP systems Mapping Toolbox Analyze and visualize geographically based information MATLAB The Language of Technical Computing Signal Processing Toolbox Perform signal processing, analysis, and algorithm development Wavelet Toolbox Analyze, compress, and denoise signals and images using wavelet techniques xiii Preface Configuration Notes To determine if the Image Processing Toolbox is installed on your system, type this command at the MATLAB prompt. ver When you enter this command, MATLAB displays information about the version of MATLAB you are running, including a list of all toolboxes installed on your system and their version numbers. For information about installing the toolbox, see the MATLAB Installation Guide for your platform. For the most up-to-date information about system requirements, see the system requirements page, available in the products area at The MathWorks Web site (www.mathworks.com). xiv About the Documentation About the Documentation This section: • Describes the structure of the Image Processing Toolbox documentation • Credits the sources of images used in the documentation • Explains the use of glossaries at the beginning of each major section of the documentation • Provides pointers to other sources of information Structure of the Documentation The documentation is organized into these major sections: • Chapter 1, “Getting Started” contains two step-by-step examples that will help you get started with using the Image Processing Toolbox. • Chapter 2, “Introduction” introduces the Image Processing Toolbox and its capabilities. • Chapter 3, “Displaying and Printing Images” describes how to display and print images in MATLAB. • Chapter 4, “Spatial Transformations” describes image cropping, resizing, rotating, and other geometric transformations you can perform with the Image Processing Toolbox. • Chapter 5, “Image Registration” describes how to align two images of the same scene using the Control Point Selection Tool. • Chapter 6, “Neighborhood and Block Operations” describes how to perform block operations on images. • Chapter 7, “Linear Filtering and Filter Design” describes how to create filters. • Chapter 8, “Transforms” discusses several important image transforms. • Chapter 9, “Morphological Operations” describes the functions in the toolbox that you can use to implement morphological image processing operations. • Chapter 10, “Analyzing and Enhancing Images” discusses working with image data and displaying images in MATLAB and the Image Processing Toolbox. xv Preface • Chapter 11, “Region-Based Processing” describes how to perform image processing on specific regions of an image. • Chapter 12, “Image Deblurring” describes the toolbox deblurring functions. • Chapter 13, “Color” describes how to handle color images. For detailed reference descriptions of each toolbox function, go to the MATLAB Help browser. Many reference descriptions also include examples, a description of the function’s algorithm, and references to additional reading material. Image Credits This table lists the copyright owners of the images used in the Image Processing Toolbox documentation. xvi Image Source cameraman Copyright Massachusetts Institute of Technology. Used with permission. cell Cancer cell from a rat’s prostate, courtesy of Alan W. Partin, M.D., Ph.D., Johns Hopkins University School of Medicine. circuit Micrograph of 16-bit A/D converter circuit, courtesy of Steve Decker and Shujaat Nadeem, MIT, 1993. concordaerial and westconcordaerial Visible color aerial photographs courtesy of mPower3/Emerge. concordorthophoto and westconcordorthophoto Orthoregistered photographs courtesy of Massachusetts Executive Office of Environmental Affairs, MassGIS. forest Photograph of Carmanah Ancient Forest, British Columbia, Canada, courtesy of Susan Cohen. About the Documentation Image Source LAN files Permission to use Landsat™ data sets provided by Space Imaging, LLC, Denver, Colorado. m83 M83 spiral galaxy astronomical image courtesy of Anglo-Australian Observatory, photography by David Malin. moon Copyright Michael Myers. Used with permission. trees Trees with a View, watercolor and ink on paper, copyright Susan Cohen. Used with permission. In addition, the following images are copyrighted: J. C. Russ, The Image Processing Handbook, Third Edition, 1998, CRC Press, Boca Raton, ISBN 0-8493-2532-3. Used with permission. afmsurf enamel rice alumgrns flowers saturn bacteria ic shot1 blood1 lily testpat1 bonemarr ngc4024l testpat2 circles ngc4024m text circlesm ngc4024s tire debye1 pearlite tissue1 Terminology At the beginning of each chapter (and sometimes at the beginning of a major section within a chapter) are tables that serve as glossaries of words you need to know to understand the information in the chapter. These tables clarify how xvii Preface we use terms that may be used in several different ways in image processing literature. For example: • Sometimes in the field of image processing, one word is used to describe more than one concept. For example the resolution of an image can describe the height and width of an image as a quantity of pixels in each direction, or it can describe the number of pixels per linear measure, such as 100 dots per inch. • Sometimes in the field of image processing, the same concepts are described by different terminology. For example, a grayscale image can also be called an intensity image. MATLAB Newsgroup If you read newsgroups on the Internet, you might be interested in the MATLAB newsgroup (comp.soft-sys.matlab). This newsgroup gives you access to an active MATLAB user community. It is an excellent way to seek advice and to share algorithms, sample code, and M-files with other MATLAB users. xviii Typographical Conventions Typographical Conventions This manual uses some or all of these conventions. Item Convention Example Example code Monospace font To assign the value 5 to A, enter A = 5 Function names, syntax, filenames, directory/folder names, user input, items in drop-down lists Monospace font Buttons and keys Boldface with book title caps Press the Enter key. Literal strings (in syntax descriptions in reference chapters) Monospace bold for literals f = freqspace(n,'whole') Italics for variables This vector represents the polynomial p = x2 + 2x + 3. Mathematical expressions MATLAB output The cos function finds the cosine of each array element. Syntax line example is MLGetVar ML_var_name Standard text font for functions, operators, and constants Monospace font MATLAB responds with A = 5 Menu and dialog box titles Boldface with book title caps Choose the File Options menu. New terms and for emphasis Italics An array is an ordered collection of information. Omitted input arguments (...) ellipsis denotes all of the input/output arguments from preceding syntaxes. [c,ia,ib] = union(...) String variables (from a finite list) Monospace italics sysc = d2c(sysd,'method') xix Preface Image Processing Demos The Image Processing Toolbox is supported by a full complement of demo applications. These are very useful as templates for your own end-user applications, or for seeing how to use and combine your toolbox functions for powerful image analysis and enhancement. The toolbox demos are located under the subdirectory, matlabroot\toolbox\images\imdemos where matlabroot represents your MATLAB installation directory. The table below lists the demos available. The easiest way to run an individual demo is to enter its name at the MATLAB command prompt. You can also launch MATLAB demos from the MATLAB Demo Window. To evoke this window select Demos from the Help menu of the main MATLAB window, or simply type demo at the command prompt. To see the list of available image processing demos, double-click on Toolboxes from the list on the left, then select Image Processing. Select the desired demo and press the Run button. If you want to know whether there is a demo that uses a particular function, check the function name in the index. If there is a demo that demonstrates this function, a subentry of “See also demoname” will appear, where demoname is the name of the demo. xx Demo Name Brief Description dctdemo DCT image compression: you choose the number of coefficients and it shows you a reconstructed image and an error image. edgedemo Edge detection: all supported types with optional manual control over threshold, direction, and sigma, as appropriate to the method used. firdemo 2-D Finite Impulse Response (FIR) filters: design your own filter by changing the cut-off frequency and filter order. Image Processing Demos imadjdemo Contrast Adjustment and Histogram Equalization: adjust intensity values using brightness, contrast, and gamma correction, or by using histogram equalization. ipexconformal Explore a Conformal Mapping: illustrates how to use spatial- and image-transformation functions to perform a conformal mapping. ipexdeconvblind Deblurring Images Using the Blind Deconvolution Algorithm: illustrates use of the deconvblind function. ipexdeconvlucy Deblurring Images Using the Lucy-Richardson Algorithm: illustrates use of the deconvlucy function. ipexdeconvreg Deblurring Images Using a Regularized Filter: illustrates use of the deconvreg function. ipexdeconvwnr Deblurring Images Using the Wiener Filter: illustrates use of the deconvwnr function. ipexgranulometry Finding the Granulometry of Stars in an Image: illustrates how to use morphology functions to perform granulometry. ipexmri Extracting Slices from a 3-Dimensional MRI Data Set: illustrates how to use the image transformation functions to interpolate and reslice a three-dimensional MRI data set, providing a convenient way to view a volume of data. ipexnormxcorr2 Registering an Image Using Normalized Cross-correlation: illustrates how to use translation to align two images. ipexregaerial Registering an Aerial Photo to an Orthophoto: illustrates how to use the Control Point Selection Tool to align two images. xxi Preface xxii ipexrotate Finding the Rotation and Scale of a Distorted Image: illustrates how to use the cp2tform function to get the rotation angle and scale factor of a distorted image. ipexsegcell Detecting a Cell Using Image Segmentation: illustrates how to use dilation and erosion to perform edge detection. ipexsegmicro Detecting Microstructures Using Image Segmentation: illustrates how to use morphological opening and closing to extract large objects from an image. ipexsegwatershed Detecting Touching Objects Using Watershed Segmentation: illustrates use of morphology functions to perform marker-control watershed segmentation. ipexshear Padding and Shearing an Image Simultaneously: illustrates how to use the padding options of the image transformation functions. ipextform Creating a Gallery of Transformed Images: illustrates how to use the imtransform function to perform many types of image transformations. ipss001 Connected Components Labelling: includes double thresholding, feature-based logic, and binary morphology. All operations are performed on one image. ipss002 Feature-based Logic: Shows object selection using AND operations on the ‘on’ pixels in two binary images and shows filtering and thresholding on a single image. ipss003 Correction of Nonuniform Illumination: creates a coarse approximation of the background, subtracts it from the image, and then adjusts the pixel intensity values to fill the entire range. Image Processing Demos nrfiltdemo Noise Reduction Using Linear and Non-linear Filters: allows you to add different types of noise with variable densities, and choose a filter neighborhood size. qtdemo Quadtree Decomposition: select a threshold and see a representation of the sparse matrix and a reconstruction of the original image. roidemo Region of Interest (ROI) Selection: select an ROI and apply operations such as unsharp and fill. Also displays the binary mask of the ROI. xxiii Preface xxiv 1 Getting Started This section contains two examples to get you started doing image processing using MATLAB and the Image Processing Toolbox. The examples contain cross-references to other sections in this manual that have in-depth discussions on the concepts presented in the examples. Example 1 — Some Basic Topics (p. 1-2) Guides you through an example of some of the basic image processing capabilities of the toolbox, including reading, writing, and displaying images Example 2 — Advanced Topics (p. 1-9) Guides you through some more advanced image processing topics including components labeling, object property measurement, image arithmetic, morphological image processing, and contrast enhancement Where to Go from Here (p. 1-23) Provides pointers to additional sources of information 1 Getting Started Example 1 — Some Basic Topics Before beginning with this exercise, start MATLAB. If you are new to MATLAB, you should first read the MATLAB Getting Started documentation. You should already have installed the Image Processing Toolbox, which runs seamlessly from MATLAB. For information about installing the toolbox, see the MATLAB Installation Guide for your platform. All of the images used in this example are supplied with the Image Processing Toolbox. Note that the images shown in this documentation differ slightly from what you see on your screen because the surrounding MATLAB figure window has been removed to save space. 1. Read and Display an Image Clear the MATLAB workspace of any variables and close open figure windows. clear, close all To read an image, use the imread command. Let’s read in a TIFF image named pout.tif (which is one of the sample images that is supplied with the Image Processing Toolbox), and store it in an array named I. I=imread('pout.tif'); Now call imshow to display I. imshow(I) 1-2 Example 1 — Some Basic Topics Here’s What Just Happened Step 1. The imread function recognized pout.tif as a valid TIFF file and stored it in the variable I. (For the list of graphics formats supported, see imread in the Image Processing Toolbox online “Function Reference.”) The functions imread and imshow read and display graphics images in MATLAB. In general, it is preferable to use imshow for displaying images because it handles the image-related MATLAB properties for you. (The MATLAB function image is for low-level programming tasks.) Note that if pout.tif were an indexed image, the appropriate syntax for imread would be, [X, map] = imread('pout.tif'); (For more information on the supported image types, see “Image Types in the Toolbox” on page 2-5.) 2. Check the Image in Memory Enter the whos command to see how I is stored in memory. whos MATLAB responds with Name I Size 291x240 Bytes Class 69840 uint8 array Grand total is 69840 elements using 69840 bytes 1-3 1 Getting Started Here’s What Just Happened Step 2. You called the whos command to see how pout.tif had been stored into the MATLAB workspace. As you saw, pout.tif is stored as a 291-by-240 array. Since pout.tif was an 8-bit image, it gets stored in memory as an uint8 array. MATLAB can store images in memory as uint8, uint16, or double arrays. (See “Reading a Graphics Image” on page 2-16 for an explanation of when the different storage classes are used.) 3. Perform Histogram Equalization As you can see, pout.tif is a somewhat low contrast image. To see the distribution of intensities in pout.tif in its current state, you can create a histogram by calling the imhist function. (Precede the call to imhist with the figure command so that the histogram does not overwrite the display of the image I in the current figure window.) figure, imhist(I) % Display a histogram of I in a new figure. 1600 1400 1200 1000 800 600 400 200 0 0 1-4 50 100 150 200 250 Example 1 — Some Basic Topics Notice how the intensity range is rather narrow. It does not cover the potential range of [0, 255], and is missing the high and low values that would result in good contrast. Now call histeq to spread the intensity values over the full range, thereby improving the contrast of I. Return the modified image in the variable I2. I2 = histeq(I); % Equalize I and output in new array I2. Display the new equalized image, I2, in a new figure window. figure, imshow(I2) % Display the new equalized image I2. Call imhist again, this time for I2. figure, imhist(I2) % Show the histogram for the new image I2. 1-5 1 Getting Started 1600 1400 1200 1000 800 600 400 200 0 0 50 100 150 200 250 See how the pixel values now extend across the full range of possible values. Here’s What Just Happened Step 3. You adjusted the contrast automatically by using the function histeq to evenly distribute the image’s pixel values over the full potential range for the storage class of the image. For an image X, with a storage class of uint8, the full range is 0 ≤ X ≤ 255 , for uint16 it is 0 ≤ X ≤ 65535 , and for double it is 0 ≤ X ≤ 1 . Note that the convention elsewhere in this user guide (and for all MATLAB documentation) is to denote the above ranges as [0,255], [0,65535], and [0,1], respectively. If you compare the two histograms, you can see that the histogram of I2 is more spread out and flat than the histogram of I1. The process that flattened and spread out this histogram is called histogram equalization. For more control over adjusting the contrast of an image (for example, if you want to choose the range over which the new pixel values should span), you can use the imadjust function, which is demonstrated under “5. Adjust the Image Contrast” on page 1-13 in Exercise 2. 1-6 Example 1 — Some Basic Topics 4. Write the Image Write the newly adjusted image I2 back to disk. Let’s say you’d like to save it as a PNG file. Use imwrite and specify a filename that includes the extension 'png'. imwrite (I2, 'pout2.png'); Here’s What Just Happened Step 4. MATLAB recognized the file extension of 'png' as valid and wrote the image to disk. It wrote it as an 8-bit image by default because it was stored as a uint8 intensity image in memory. If I2 had been an image array of type RGB and class uint8, it would have been written to disk as a 24-bit image. If you want to set the bit depth of your output image, use the BitDepth parameter with imwrite. This example writes a 4-bit PNG file. imwrite(I2, 'pout2.png', 'BitDepth', 4); Note that all output formats do not support the same set of output bit depths. See imwrite in the Reference for the list of valid bit depths for each format. See also “Writing a Graphics Image” on page 2-17 for a tutorial discussion on writing images using the Image Processing Toolbox. 5. Check the Contents of the Newly Written File Now, use the imfinfo function to see what was written to disk. Be sure not to end the line with a semicolon so that MATLAB displays the results. Also, be sure to use the same path (if any) as you did for the call to imwrite, above. imfinfo('pout2.png') MATLAB responds with ans = Filename:'pout2.png' FileModDate:'03-Jun-1999 15:50:25' FileSize:36938 Format:'png' FormatVersion:[] 1-7 1 Getting Started Width:240 Height:291 BitDepth:8 ColorType:'grayscale' . . . Note This example shows only a subset of all the fields returned by imfinfo. Also, the value in the FileModDate field for your file will be different from what is shown above. It will show the date and time that you used imwrite to create your image. Here’s What Just Happened Step 5. When you called imfinfo, MATLAB displayed all of the header fields for the PNG file format that are supported by the toolbox. You can modify many of these fields by using additional parameters in your call to imwrite. The additional parameters that are available for each file format are listed in tables in the reference entry for imwrite. (See “Querying a Graphics File” on page 2-18 for more information about using imfinfo.) 1-8 Example 2 — Advanced Topics Example 2 — Advanced Topics In this exercise you will work with another intensity image, rice.tif, and explore some more advanced operations. The goals of this exercise are to remove the nonuniform background from rice.tif, convert the resulting image to a binary image by using thresholding, use components labeling to return the number of objects (grains or partial grains) in the image, and compute object statistics. 1. Read and Display an Image Clear the MATLAB workspace of any variables and close open figure windows. Read and display the intensity image rice.tif. clear, close all I = imread('rice.tif'); imshow(I) 2. Use Morphological Opening to Estimate the Background Notice that the background illumination is brighter in the center of the image than at the bottom. Use the imopen function to estimate the background illumination. background = imopen(I,strel('disk',15)); To see the estimated background image, type imshow(background) 1-9 1 Getting Started Here’s What Just Happened Step 1. You used the toolbox functions imread and imshow to read and display an 8-bit intensity image. imread and imshow are discussed in Exercise 1, in “2. Check the Image in Memory” on page 1-3, under the “Here’s What Just Happened” discussion. Step 2. You performed a morphological opening operation by calling imopen with the input image, I, and a disk-shaped structuring element with a radius of 15. The structuring element was created by the strel function. The morphological opening has the effect of removing objects that cannot completely contain a disk of radius 15. For more information about morphological opening, see Chapter 9, “Dilation- and Erosion-Based Functions.” 3. Display the Background Approximation as a Surface Use the surf command to create a surface display of the background approximation, background. The surf function requires data of class double, however, so you first need to convert background using the double command. figure, surf(double(background(1:8:end,1:8:end))),zlim([0 255]); set(gca,'ydir','reverse'); The example uses MATLAB indexing syntax to view only 1 out of 8 pixels in each direction; otherwise the surface plot would be too dense. The example also sets the scale of the plot to better match the range of the uint8 data and reverses the y-axis of the display to provide a better view of the data (the pixels at the bottom of the image appear at the front of the surface plot). 1-10 Example 2 — Advanced Topics Here’s What Just Happened Step 3. You used the surf command to examine the background image. The surf command creates colored parametric surfaces that enable you to view mathematical functions over a rectangular region. In the surface display, [0, 0] represents the origin, or upper-left corner of the image. The highest part of the curve indicates that the highest pixel values of background (and consequently rice.tif) occur near the middle rows of the image. The lowest pixel values occur at the bottom of the image and are represented in the surface plot by the lowest part of the curve. The surface plot is a Handle Graphics® object, and you can therefore fine-tune its appearance by setting properties. For information on working with MATLAB graphics, see the MATLAB graphics documentation. 1-11 1 Getting Started 4. Subtract the Background Image from the Original Image Now subtract the background image, background, from the original image, I, to create a more uniform background. I2 = imsubtract(I,background); Now display the image with its more uniform background. figure, imshow(I2) Here’s What Just Happened Step 4. You subtracted a background approximation image from rice.tif. Because subtraction, like many of MATLAB mathematical operations, is only supported for data of class double, you must use the Image Processing Toolbox image arithmetic imsubtract function. The Image Processing Toolbox has a demo, ipss003, that approximates and removes the background from an image. For information on how to run this (and other demos), see “Image Processing Demos” in the Preface. 1-12 Example 2 — Advanced Topics 5. Adjust the Image Contrast The image is now a bit too dark. Use imadjust to adjust the contrast. I3 = imadjust(I2, stretchlim(I2), [0 1]); Display the newly adjusted image. figure, imshow(I3); Here’s What Just Happened Step 5. You used the imadjust command to increase the contrast in the image. The imadjust function takes an input image and can also take two vectors: [low high] and [bottom top]. The output image is created by mapping the value low in the input image to the value bottom in the output image, mapping the value high in the input image to the value top in the output image, and linearly scaling the values in between. See the reference pages for imadjust for more information. You called imadjust with stretchlim(I2) as the second argument. The stretchlim function automatically computes the right [low high] values to make imadjust increase (stretch) the contrast of the image. 1-13 1 Getting Started 6. Apply Thresholding to the Image Create a new binary thresholded image, bw, by using the functions graythresh and im2bw. level = graythresh(I3); bw = im2bw(I3,level); figure, imshow(bw) Now call the whos command to see what type of array the thresholded image bw is. whos MATLAB responds with Name I I2 I3 background bw level Size 256x256 256x256 256x256 256x256 256x256 1x1 Bytes Class 65536 65536 65536 65536 65536 8 uint8 array uint8 array uint8 array uint8 array logical array double array Grand total is 327681 elements using 327688 bytes 1-14 Example 2 — Advanced Topics Here’s What Just Happened Step 6. You called graythresh to automatically compute an appropriate threshold to use to convert the intensity image to binary. You then called im2bw to perform for thresholding, using the threshold, level, returned by graythresh. Notice that when you call the whos command, you see the expression logical listed after the class for bw. This indicates the presence of a logical flag. The flag indicates that bw is a logical matrix, and the Image Processing Toolbox treats logical matrices as binary images. Thresholding using MATLAB logical operators always results in a logical image. For more information about binary images and the logical flag, see “Binary Images” on page 2-8. 7. Determine the Number of Objects in the Image To determine the number of grains of rice in the image, use the bwlabel function. This function labels all of the connected components in the binary image bw and returns the number of objects it finds in the image in the output value, numobjects. [labeled,numObjects] = bwlabel(bw,4);% Label components. numObjects = 80 The accuracy of your results depends on a number of factors, including: • The size of the objects • The accuracy of your approximated background • Whether you set the connectivity parameter to 4 or 8 • Whether or not any objects are touching (in which case they may be labeled as one object) In the example, some grains of rice are touching, so bwlabel treats them as one object. 1-15 1 Getting Started Here’s What Just Happened Step 7. You called bwlabel to search for connected components and label them with unique numbers. bwlabel takes a binary input image and a value specifying the connectivity of objects. The parameter 4, passed to the bwlabel function, means that pixels must touch along an edge to be considered connected. For more information about the connectivity of objects, see “Pixel Connectivity” in Chapter 9. You can also determine the number of objects in a label matrix by asking for the maximum pixel value in the image. For example, max(labeled(:)) ans = 80 8. Examine the Label Matrix You may find it helpful to take a closer look at labeled to see what bwlabel has created. Use the imcrop command to select and display pixels in a region of labeled that includes an object and some background. To ensure that the output is displayed in the MATLAB window, do not end the line with a semicolon. In addition, choose a small rectangle for this exercise, so that the displayed pixel values don’t wrap in the MATLAB command window. The syntax shown below makes imcrop work interactively. Your mouse cursor becomes a cross-hair when placed over the image. Click at a position in labeled where you would like to select the upper left corner of a region. Drag the mouse to create the selection rectangle, and release the button when you are done. grain = imcrop(labeled) % Crop a portion of labeled. We chose the left edge of a grain and got the following results. grain = 1-16 Example 2 — Advanced Topics 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60 60 60 60 60 0 0 0 0 0 60 60 60 60 60 0 0 0 0 60 60 60 60 60 60 60 0 0 0 60 60 60 60 60 60 60 0 0 60 60 60 60 60 60 60 60 0 0 60 60 60 60 60 60 60 60 0 0 A good way to view a label matrix is to display it as a pseudo-color indexed image. In the pseudo-color image, the number that identifies each object in the label matrix maps to a different color in the associated colormap matrix. When you view a label matrix as an RGB image, the objects in the image are easier to distinguish. To view a label matrix in this way, use the label2rgb function. Using this function, you can specify the colormap, the background color, and how objects in the label matrix map to colors in the colormap. RGB_label = label2rgb(labeled, @spring, 'c', 'shuffle'); imshow(RGB_label); 1-17 1 Getting Started Here’s What Just Happened Step 8. You called imcrop and selected a portion of the image that contained both some background and part of an object. The pixel values were returned in the MATLAB window. If you examine the results above, you can see the corner of an object labeled with 60’s, which means that it was the 60th object labeled by bwlabel. The imcrop function can also take a vector specifying the coordinates for the crop rectangle. In this case, it does not operate interactively. For example, this call specifies a crop rectangle whose upper-left corner begins at (15, 25) and has a height and width of 10. rect = [15 25 10 10]; roi = imcrop(labeled, rect) You are not restricted to rectangular regions of interest. The toolbox also has a roipoly command that enables you to select polygonal regions of interest. Many image processing operations can be performed on regions of interest, including filtering and filling. See Chapter 11, “Region-Based Processing” for more information. The call to label2rgb illustrates a good way to visualize label matrices. The pixel values in the label matrix are used as indices into a colormap. Using label2rgb, you can specify your own colormap or use one of the MATLAB colormap-creating functions, including gray, pink, spring, and hsv. For information on these functions, see colormap in the MATLAB Function Reference. 9. Measure Object Properties in the Image The regionprops command measures object or region properties in an image and returns them in a structure array. When applied to an image with labeled components, it creates one structure element for each component. Use regionprops to create a structure array containing some basic properties for labeled. graindata = regionprops(labeled,'basic') MATLAB responds with 1-18 Example 2 — Advanced Topics graindata = 80x1 struct array with fields: Area Centroid BoundingBox To find the area of the component labeled with 51’s, use dot notation to access the Area field in the 51st element in the graindata structure array. Note that structure field names are case sensitive, so you need to capitalize the name as shown. graindata(51).Area returns the following results ans = 296 To find the smallest possible bounding box and the centroid (center of mass) for the same component, use this code: graindata(51).BoundingBox, graindata(51).Centroid ans = 142.5000 ans = 89.5000 155.3953 102.1791 24.0000 26.0000 To create a new vector, allgrains, which holds just the area measurement for each grain, use this code: allgrains = [graindata.Area]; whos allgrains Call the whos command to see how MATLAB allocated the allgrains variable. Name Size Bytes allgrains 1x80 640 Class double array 1-19 1 Getting Started Grand total is 80 elements using 640 bytes allgrains is a one-row array of 80 elements, where each element contains the area measurement of a grain. Check the area of the 51st element of allgrains. allgrains(51) returns ans = 296 which is the same result that you received when using dot notation to access the Area field of graindata(51). Here’s What Just Happened Step 9. You called regionprops to return a structure of basic property measurements for each thresholded grain of rice. The regionprops function supports many different property measurements, but setting the properties parameter to 'basic' is a convenient way to return three of the most commonly used measurements: the area, the centroid (or center of mass), and the bounding box. The bounding box represents the smallest rectangle that can contain a region, or in this case, a grain. The four-element vector returned by the BoundingBox field, [142.5000 89.5000 24.0000 26.0000] shows that the upper left corner of the bounding box is positioned at [142.5 89.5], and the box has a width of 24.0 and a height of 26.0. (The position is defined in spatial coordinates, hence the decimal values. For more information on the spatial coordinate system, see “Spatial Coordinates” on page 2-29.) For more information about working with MATLAB structure arrays, see “Structures” in the MATLAB programming and data types documentation. You used dot notation to access the Area field of all of the elements of graindata and stored this data to a new vector allgrains. This step simplifies analysis made on area measurements because you do not have to use field names to access the area. 1-20 Example 2 — Advanced Topics 10. Compute Statistical Properties of Objects in the Image Now use MATLAB functions to calculate some statistical properties of the thresholded objects. First use max to find the size of the largest grain. (If you have followed all of the steps in this exercise, the “largest grain” is actually two grains that are touching and have been labeled as one object). max(allgrains) returns ans = 695 Use the find command to return the component label of this large-sized grain. biggrain = find(allgrains==695) returns biggrain = 68 Find the mean grain size. mean(allgrains) returns ans = 249 Make a histogram containing 20 bins that show the distribution of rice grain sizes. hist(allgrains,20) 1-21 1 Getting Started Here’s What Just Happened Step 10. You used some of the MATLAB statistical functions, max, mean, and hist to return the statistical properties for the thresholded objects in rice.tif. The Image Processing Toolbox also has some statistical functions, such as mean2 and std2, which are well suited to image data because they return a single value for two-dimensional data. The functions mean and std were suitable here because the data in allgrains was one dimensional. The histogram shows that the most common sizes for rice grains in this image are in the range of 300 to 400 pixels. 1-22 Where to Go from Here Where to Go from Here For more information about the topics covered in these exercises, read the tutorial chapters that make up the remainder of this documentation. For reference information about any of the Image Processing Toolbox functions, see the online “Function Reference”, which complements the M-file help that is displayed in the MATLAB command window when you type help functionname For example, help imshow Online Help The Image Processing Toolbox User’s Guide is available online in both HTML and PDF formats. To access the HTML help, select Help from the menu bar of the MATLAB desktop. In the Help browser, expand the Image Processing Toolbox topic in the list. To access the PDF help, click on Image Processing Toolbox in the Contents tab of the Help browser, and go to the link under “Printable Documentation (PDF).” (Note that to view the PDF help, you must have Adobe's Acrobat Reader installed.) Toolbox Demos The Image Processing Toolbox includes many demo applications. The demos are useful for seeing the toolbox features put into action and for borrowing code for your own applications. See “Image Processing Demos” in the Preface for a complete list and summary of the demos, as well as instructions on how to run them. 1-23 1 Getting Started 1-24 2 Introduction This section introduces you to the fundamentals of image processing using MATLAB and the Image Processing Toolbox. Topics covered include Terminology (p. 2-2) Provides definitions of image processing terms used in this section Images in MATLAB and the Image Processing Toolbox (p. 2-4) Describes how images are represented in MATLAB and the Image Processing Toolbox Image Types in the Toolbox (p. 2-5) Describes the fundamental image types supported by the Image Processing Toolbox Working with Image Data (p. 2-15) Describes how to read, write, and perform other common image tasks Image Arithmetic (p. 2-21) Describes how to add, subtract, multiply, and divide images Coordinate Systems (p. 2-28) Explains image coordinate systems 2 Introduction Terminology An understanding of the following terms will help you to use this chapter. 2-2 Terms Definitions Binary image An image containing only black and white pixels. In MATLAB, a binary image is represented as a logical array of 0’s and 1’s (which usually represent black and white, respectively). This documentation often uses the variable name BW to represent a binary image in memory. Image type The defined relationship between array values and pixel colors. The toolbox supports binary, indexed, intensity, and RGB image types. Indexed image An image whose pixel values are direct indices into an RGB colormap. In MATLAB, an indexed image is represented by an array of class uint8, uint16, or double. The colormap is always an m-by-3 array of class double. We often use the variable name X to represent an indexed image in memory, and map to represent the colormap. Intensity image An image consisting of intensity (grayscale) values. In MATLAB, intensity images are represented by an array of class uint8, uint16, or double. While intensity images are not stored with colormaps, MATLAB uses a system colormap to display them. We often use the variable name I to represent an intensity image in memory. This term is synonymous with the term grayscale. Multiframe image An image file that contains more than one image, or frame. When in MATLAB memory, a multiframe image is a 4-D array where the fourth dimension specifies the frame number. This term is synonymous with the term multipage image. Terminology Terms Definitions RGB image An image in which each pixel is specified by three values—one each for the red, green, and blue components of the pixel’s color. In MATLAB, an RGB image is represented by an m-by-n-by-3 array of class uint8, uint16, or double. We often use the variable name RGB to represent an RGB image in memory. Storage class The numeric storage class used to store an image in MATLAB. The storage classes used in MATLAB are uint8, uint16, and double. Some function descriptions in the reference chapter of this User’s Guide have a section entitled “Class Support” that specifies which image classes the function can operate on. When this section is absent, the function can operate on all supported storage classes. 2-3 2 Introduction Images in MATLAB and the Image Processing Toolbox The basic data structure in MATLAB is the array, an ordered set of real or complex elements. This object is naturally suited to the representation of images, real-valued ordered sets of color or intensity data. MATLAB stores most images as two-dimensional arrays (i.e., matrices), in which each element of the matrix corresponds to a single pixel in the displayed image. (Pixel is derived from picture element and usually denotes a single dot on a computer display.) For example, an image composed of 200 rows and 300 columns of different colored dots would be stored in MATLAB as a 200-by-300 matrix. Some images, such as RGB, require a three-dimensional array, where the first plane in the third dimension represents the red pixel intensities, the second plane represents the green pixel intensities, and the third plane represents the blue pixel intensities. This convention makes working with images in MATLAB similar to working with any other type of matrix data, and makes the full power of MATLAB available for image processing applications. For example, you can select a single pixel from an image matrix using normal matrix subscripting. I(2,15) This command returns the value of the pixel at row 2, column 15 of the image I. Storage Classes in the Toolbox By default, MATLAB stores most data in arrays of class double. The data in these arrays is stored as double precision (64-bit) floating-point numbers. All MATLAB functions work with these arrays. For image processing, however, this data representation is not always ideal. The number of pixels in an image may be very large; for example, a 1000-by-1000 image has a million pixels. Since each pixel is represented by at least one array element, this image would require about 8 megabytes of memory. To reduce memory requirements, MATLAB supports storing image data in arrays as 8-bit or 16-bit unsigned integers, class uint8 and uint16. These arrays require one eighth or one fourth as much memory as double arrays. 2-4 Image Types in the Toolbox Image Types in the Toolbox The Image Processing Toolbox supports four basic types of images: • Index images • Intensity images • Binary images • RGB images This section discusses how MATLAB and the Image Processing Toolbox represent each of these image types. Indexed Images An indexed image consists of a data matrix, X, and a colormap matrix, map. The data matrix can be of class uint8, uint16, or double. The colormap matrix is an m-by-3 array of class double containing floating-point values in the range [0,1]. Each row of map specifies the red, green, and blue components of a single color. An indexed image uses direct mapping of pixel values to colormap values. The color of each image pixel is determined by using the corresponding value of X as an index into map. The value 1 points to the first row in map, the value 2 points to the second row, and so on. A colormap is often stored with an indexed image and is automatically loaded with the image when you use the imread function. However, you are not limited to using the default colormap—you can use any colormap that you choose. The figure below illustrates the structure of an indexed image. The pixels in the image are represented by integers, which are pointers (indices) to color values stored in the colormap. 2-5 2 Introduction 75 10 12 21 40 53 53 75 14 17 21 21 53 53 75 8 5 8 10 30 15 51 15 18 31 31 18 16 56 31 18 31 31 31 31 0 0.0627 0.2902 0 0.2902 0.3882 0.4510 0.2588 0 0.0627 0.0314 0 0.0627 0.0314 0.0627 0.1608 . . . 0 0.0314 0 1.0000 0.0627 0.0941 0 0.0627 Figure 2-1: Pixel Values Are Indices to a Colormap in Indexed Images Class and Colormap Offsets The relationship between the values in the image matrix and the colormap depends on the class of the image matrix. If the image matrix is of class double, the value 1 points to the first row in the colormap, the value 2 points to the second row, and so on. If the image matrix is of class uint8 or uint16, there is an offset—the value 0 points to the first row in the colormap, the value 1 points to the second row, and so on. The offset is also used in graphics file formats to maximize the number of colors that can be supported. In the image above, the image matrix is of class double. Because there is no offset, the value 5 points to the fifth row of the colormap. Limitations to uint16 Support Note that the toolbox provides limited support for indexed images of class uint16. You can read these images into MATLAB and display them, but before you can process a uint16 indexed image you must first convert it to either a double or a uint8. To convert to a double, call im2double; to reduce the image 2-6 Image Types in the Toolbox to 256 colors or fewer (uint8) call imapprox. For more information, see the reference pages for im2double and imapprox. Intensity Images An intensity image is a data matrix, I, whose values represent intensities within some range. MATLAB stores an intensity image as a single matrix, with each element of the matrix corresponding to one image pixel. The matrix can be of class double, uint8, or uint16.While intensity images are rarely saved with a colormap, MATLAB uses a colormap to display them. The elements in the intensity matrix represent various intensities, or gray levels, where the intensity 0 usually represents black and the intensity 1, 255, or 65535 usually represents full intensity, or white. The figure below depicts an intensity image of class double. .2251 0.5342 0.2051 0.5342 0.1789 0.4308 0.2483 3344 0.2563 0.2157 0.1307 0.2624 0.2624 0.2826 0.2826 0.1789 0.3344 0.3344 0.2826 0.3822 0.2051 0.3344 0.3344 0.4 0.4391 0.4391 0.3256 0.2483 0.2624 0.2549 0.33 Figure 2-2: Pixel Values in an Intensity Image Define Gray Levels 2-7 2 Introduction Binary Images In a binary image, each pixel assumes one of only two discrete values. Essentially, these two values correspond to on and off. A binary image is stored as a logical array of 0’s (off pixels) and 1’s (on pixels). The figure below depicts a binary image. 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Figure 2-3: Pixels in a Binary Image Have Two Possible Values: 0 or 1 RGB Images An RGB image, sometimes referred to as a truecolor image, is stored in MATLAB as an m-by-n-by-3 data array that defines red, green, and blue color components for each individual pixel. RGB images do not use a palette. The color of each pixel is determined by the combination of the red, green, and blue intensities stored in each color plane at the pixel’s location. Graphics file formats store RGB images as 24-bit images, where the red, green, and blue components are 8 bits each. This yields a potential of 16 million colors. The precision with which a real-life image can be replicated has led to the commonly used term truecolor image. An RGB MATLAB array can be of class double, uint8, or uint16. In an RGB array of class double, each color component is a value between 0 and 1. A pixel whose color components are (0,0,0) displays as black, and a pixel whose color components are (1,1,1) displays as white. The three color components for each 2-8 Image Types in the Toolbox pixel are stored along the third dimension of the data array. For example, the red, green, and blue color components of the pixel (10,5) are stored in RGB(10,5,1), RGB(10,5,2), and RGB(10,5,3), respectively. Figure 2-4 depicts an RGB image of class double. 0.5804 0.2235 0.1294 0.2902 0.4196 0.4824 0.4824 Blue 0.5804 0.2902 0.0627 0.2902 0.2902 0.4824 0.4824 0.5804 0.0627 0.0627 0.0627 0.2235 0.2588 0.2588 Green 0.5176 0.2588 0.0627 0.0941 0.0941 0.0627 0.0627 0.5176 0.1922 0.0627 0.1294 0.1922 0.2588 0.2588 0.4510 0.0941 0.0627 0.0941 0.0941 0.0941 0.0941 0.5176 0.1294 0.1608 0.1294 0.1294 0.2588 0.2588 0.5176 0.1608 0.0627 0.1608 0.1922 0.2588 0.2588 0.41960.2235 0.2588 0.3529 0.4196 0.4196 0.3529 0.2902 Red 0.5490 0.5490 0.5804 0.7412 0.7765 0.7765 0.45100.3882 0.4196 0.3529 0.4196 0.4196 0.4196 0.4196 0.5490 0.5176 0.5804 0.5804 0.7765 0.7765 0.5490 0.2588 0.2902 0.2588 0.2235 0.4824 0.2235 0.4196 0.2235 0.1608 0.2588 0.2588 0.1608 0.2588 0.4510 0.2588 0.1608 0.2588 0.2588 0.2588 0.2588 Figure 2-4: The Color Planes of an RGB Image To determine the color of the pixel at (2,3), you would look at the RGB triplet stored in (2,3,1:3). Suppose (2,3,1) contains the value 0.5176, (2,3,2) contains 0.1608, and (2,3,3) contains 0.0627. The color for the pixel at (2,3) is 0.5176 0.1608 0.0627 To further illustrate the concept of the three separate color planes used in an RGB image, the code sample below creates a simple RGB image containing uninterrupted areas of red, green, and blue, and then creates one image for each of its separate color planes (red, green, and blue). It displays each color plane image separately, and also displays the original image. 2-9 2 Introduction RGB=reshape(ones(64,1)*reshape(jet(64),1,192),[64,64,3]); R=RGB(:,:,1); G=RGB(:,:,2); B=RGB(:,:,3); imshow(R) figure, imshow(G) figure, imshow(B) figure, imshow(RGB) Figure 2-5: The Separated Color Planes of an RGB Image 2-10 Image Types in the Toolbox Notice that each separated color plane in the figure contains an area of white. The white corresponds to the highest values (purest shades) of each separate color. For example, in the Red Plane image, the white represents the highest concentration of pure red values. As red becomes mixed with green or blue, gray pixels appear. The black region in the image shows pixel values that contain no red values, i.e., R == 0. Multiframe Image Arrays For some applications, you may need to work with collections of images related by time or view, such as magnetic resonance imaging (MRI) slices or movie frames. The Image Processing Toolbox provides support for storing multiple images in the same array. Each separate image is called a frame. If an array holds multiple frames, they are concatenated along the fourth dimension. For example, an array with five 400-by-300 RGB images would be 400-by-300-by-3-by-5. A similar multiframe intensity or indexed image would be 400-by-300-by-1-by-5. Use the cat command to store separate images into one multiframe file. For example, if you have a group of images A1, A2, A3, A4, and A5, you can store them in a single array using A = cat(4,A1,A2,A3,A4,A5) You can also extract frames from a multiframe image. For example, if you have a multiframe image MULTI, this command extracts the third frame. FRM3 = MULTI(:,:,:,3) Note that in a multiframe image array, each image must be the same size and have the same number of planes. In a multiframe indexed image, each image must also use the same colormap. Multiframe Support Limitations Many of the functions in the toolbox operate only on the first two or first three dimensions. You can still use four-dimensional arrays with these functions, but you must process each frame individually. For example, this call displays the seventh frame in the array MULTI. imshow(MULTI(:,:,:,7)) 2-11 2 Introduction If you pass an array to a function and the array has more dimensions than the function is designed to operate on, your results may be unpredictable. In some cases, the function simply processes the first frame of the array, but in other cases the operation does not produce meaningful results. See the reference pages for information about how individual functions work with the dimensions of an image array. For more information about displaying multiframe images, see Chapter 3, “Displaying and Printing Images.” Summary of Image Types and Numeric Classes This table summarizes the way MATLAB interprets data matrix elements as pixel colors, depending on the image type and storage class. Image Type Storage Class Interpretation Binary logical An array of zeros (0) and ones (1) Indexed1 double An array of integers in the range [1, p] uint8 or uint16 An array of integers in the range [0, p-1] double An array of floating-point values. The typical range of values is [0, 1]. uint8 or uint16 An array of integers. The typical range of values is [0, 255] or [0, 65535]. double An m-by-n-by-3 array of floating-point values in the range [0, 1]. uint8 or uint16 An m-by-n-by-3 array of integers in the range [0, 255] or [0, 65535]. Intensity1 RGB (Truecolor) 1. The associated colormap is a p-by-3 array of floating-point values in the range [0, 1]. For intensity images the colormap is typically grayscale. Converting Image Types For certain operations, it is helpful to convert an image to a different image type. For example, if you want to filter a color image that is stored as an 2-12 Image Types in the Toolbox indexed image, you should first convert it to RGB format. When you apply the filter to the RGB image, MATLAB filters the intensity values in the image, as is appropriate. If you attempt to filter the indexed image, MATLAB simply applies the filter to the indices in the indexed image matrix, and the results may not be meaningful. Note When you convert an image from one format to another, the resulting image may look different from the original. For example, if you convert a color indexed image to an intensity image, the resulting image is grayscale, not color. For more information about how the image conversion functions listed below, see their reference pages. The following table lists all the image conversion functions in the Image Processing Toolbox. Function Description dither Create a binary image from a grayscale intensity image by dithering; create an indexed image from an RGB image by dithering gray2ind Create an indexed image from a grayscale intensity image grayslice Create an indexed image from a grayscale intensity image by thresholding im2bw Create a binary image from an intensity image, indexed image, or RGB image, based on a luminance threshold ind2gray Create a grayscale intensity image from an indexed image ind2rgb Create an RGB image from an indexed image mat2gray Create a grayscale intensity image from data in a matrix, by scaling the data 2-13 2 Introduction Function Description rgb2gray Create a grayscale intensity image from an RGB image rgb2ind Create an indexed image from an RGB image You can also perform certain conversions just using MATLAB syntax. For example, you can convert an intensity image to RGB format by concatenating three copies of the original matrix along the third dimension. RGB = cat(3,I,I,I); The resulting RGB image has identical matrices for the red, green, and blue planes, so the image displays as shades of gray. In addition to these standard conversion tools, there are some functions that return a different image type as part of the operation they perform. For example, the region-of-interest routines each return a binary image that you can use to mask an indexed or intensity image for filtering or for other operations. Color Space Conversions The Image Processing Toolbox represents colors as RGB values, either directly (in an RGB image) or indirectly (in an indexed image). However, there are other methods for representing colors. For example, a color can be represented by its hue, saturation, and value components (HSV). Different methods for representing colors are called color spaces. The toolbox provides a set of routines for converting between RGB and other color spaces. The image processing functions themselves assume all color data is RGB, but you can process an image that uses a different color space by first converting it to RGB, and then converting the processed image back to the original color space. For more information about color space conversion routines, see Chapter 13, “Color.” 2-14 Working with Image Data Working with Image Data Images are most commonly stored in MATLAB using the logical, uint8, uint16 and double data types. This section describes how to work with image data in MATLAB. Topics include: • Reading images into the MATLAB workspace • Writing images to files, in many standard graphics file formats • Querying graphics image files for information stored in header fields • Converting images between image storage classes • Converting images between graphics file formats For information about displaying and printing images, see “Displaying and Printing Images”. You can also perform many standard MATLAB array manipulations on uint8 and uint16 image data, including: • Indexing, including logical indexing • Reshaping, reordering, and concatenating • Reading from and writing to MAT-files, • Using relational operators Certain MATLAB functions, including the find, all, any, conv2, convn, fft2, fftn, and sum functions accept uint8 or uint16 data but return data in double precision format. The basic MATLAB arithmetic operators, however, do not accept uint8 or uint16 data. For example, if you attempt to add two uint8 images, A and B, you get an error, such as, C = A + B ??? Function '+' not defined for variables of class 'uint8'. Because these arithmetic operations are an important part of many image-processing operations, the Image Processing Toolbox includes functions that support these operations on uint8 and uint16 data, as well as the other numeric data types. See “Image Arithmetic” for more information. 2-15 2 Introduction Reading a Graphics Image The function imread reads an image from any supported graphics image file format, in any of the supported bit depths. Most image file formats use 8 bits to store pixel values. When these are read into memory, MATLAB stores them as class uint8. For file formats that support 16-bit data, PNG and TIFF, MATLAB stores the images as class uint16. As with MATLAB-generated images, once a image is displayed, it becomes a Handle Graphics Image object. Note For indexed images, imread always reads the colormap into an array of class double, even though the image array itself may be of class uint8 or uint16. For example, this code reads the image ngc6543a.jpg into the MATLAB workspace as the variable RGB. RGB = imread('ngc6543a.jpg'); In this example, imread infers the file format to use from the contents of the file. You can also specify the file format as an argument to imread. MATLAB supports the following graphics file formats: • BMP (Microsoft Windows Bitmap) • CUR (Microsoft Windows Cursor resource) • GIF (Graphics Interchange Format) • HDF (Hierarchical Data Format) • ICO (Windows Icon resource • JPEG (Joint Photographic Experts Group) • PBM (Portable Bitmap) • PCX (Windows Paintbrush) • PGM (Portable Graymap) • PNG (Portable Network Graphics) • PPM (Portable Pixmap) • RAS (Sun Raster image) • TIFF (Tagged Image File Format) • XWD (X Window Dump) 2-16 Working with Image Data For the latest information concerning the bit depths and/or image formats supported, see the reference pages for imread. Writing a Graphics Image The function imwrite writes an image to a graphics file in one of the supported formats. The most basic syntax for imwrite takes the image variable name and a filename. If you include an extension in the filename, MATLAB infers the desired file format from it. This example loads an image of a clown from a MAT-file, and then creates a BMP file containing the clown image. load clown whos Name Size Bytes X caption map 200x320 2x1 81x3 512000 4 1944 Class double array char array double array Grand total is 64245 elements using 513948 bytes imwrite(X,map,'clown.bmp') For some graphics formats, you can specify additional parameters. One of the additional parameters for PNG files sets the bit depth. This example writes an intensity image I to a 4-bit PNG file. imwrite(I,'clown.png','BitDepth',4); The bit depths and image types supported for each format are shown in the reference pages for imwrite. This example writes an image A to a JPEG file with a compression quality setting of 100 (the default is 75). imwrite(A, 'myfile.jpg', 'Quality', 100); 2-17 2 Introduction Output File Storage Classes imwrite uses the following rules to determine the storage class used in the output image. Storage Class of Image Storage Class of Output Image File logical If the output image file format specified supports 1-bit images, imwrite creates a 1-bit image file. If the output image file format specified does not support 1-bit images, such as JPEG, imwrite converts the image to a class uint8 intensity image. uint8 If the output image file format specified supports 8-bit images, imwrite creates an 8-bit image file. uint16 If the output image file format specified supports 16-bit images (PNG or TIFF), imwrite creates a 16-bit image file. If the output image file format specified does not support 16-bit images, imwrite scales the image data to class uint8 and creates an 8-bit image file. double MATLAB scales the image data to uint8 and creates an 8-bit image file because most image file formats use 8-bits. See the reference entry for imwrite for more information. Querying a Graphics File The imfinfo function enables you to obtain information about graphics files that are in any of the formats supported by the toolbox. The information you obtain depends on the type of file, but it always includes at least the following: • Name of the file • File format • Version number of the file format 2-18 Working with Image Data • File modification date • File size in bytes • Image width in pixels • Image height in pixels • Number of bits per pixel • Image type: RGB (truecolor), intensity (grayscale), or indexed See the reference entry for imfinfo for more information. Converting Image Storage Classes You can convert uint8 and uint16 data to double precision using the MATLAB function, double. However, converting between storage classes changes the way MATLAB and the toolbox interpret the image data. If you want the resulting array to be interpreted properly as image data, you need to rescale or offset the data when you convert it. For easier conversion of storage classes, use one of these toolbox functions: im2double, im2uint8, and im2uint16. These functions automatically handle the rescaling and offsetting of the original data. For example, this command converts a double-precision RGB image with data in the range [0,1] to a uint8 RGB image with data in the range [0,255]. RGB2 = im2uint8(RGB1); Losing Information in Conversions When you convert to a class that uses fewer bits to represent numbers, you generally lose some of the information in your image. For example, a uint16 intensity image is capable of storing up to 65,536 distinct shades of gray, but a uint8 intensity image can store only 256 distinct shades of gray. When you convert a uint16 intensity image to a uint8 intensity image, im2uint8 quantizes the gray shades in the original image. In other words, all values from 0 to 127 in the original image become 0 in the uint8 image, values from 128 to 385 all become 1, and so on. This loss of information is often not a problem, however, since 256 still exceeds the number of shades of gray that your eye is likely to discern. 2-19 2 Introduction Converting Indexed Images It is not always possible to convert an indexed image from one storage class to another. In an indexed image, the image matrix contains only indices into a colormap, rather than the color data itself, so no quantization of the color data is possible during the conversion. For example, a uint16 or double indexed image with 300 colors cannot be converted to uint8, because uint8 arrays have only 256 distinct values. If you want to perform this conversion, you must first reduce the number of the colors in the image using the imapprox function. This function performs the quantization on the colors in the colormap, to reduce the number of distinct colors in the image. See “Reducing Colors in an Indexed Image” on page 13-12 for more information. Converting Graphics File Formats To change the graphics format of an image, use imread to read in the image and then save the image with imwrite, specifying the appropriate format. For example, to convert an image from a BMP to a PNG, read the BMP image using imread, convert the storage class if necessary, and then write the image using imwrite, with 'PNG' specified as your target format. bitmap = imread('mybitmap.bmp','bmp'); imwrite(bitmap,'mybitmap.png','png'); For the specifics of which bit depths are supported for the different graphics formats, and for how to specify the format type when writing an image to file, see the reference entries for imread and imwrite. 2-20 Image Arithmetic Image Arithmetic Image arithmetic is the implementation of standard arithmetic operations, such as addition, subtraction, multiplication, and division, on images. Image arithmetic has many uses in image processing both as a preliminary step in more complex operations and by itself. For example, image subtraction can be used to detect differences between two or more images of the same scene or object. You can do image arithmetic using the MATLAB arithmetic operators; however, you must convert the images to class double to use these operators. To make working with images more convenient, the Image Processing Toolbox includes a set of functions that implement arithmetic operations for all numeric, nonsparse data types. The advantages to using these functions include: • No conversion to the double data type is necessary. The functions accept any numeric data type, including uint8, uint16, and double, and return the result image in the same format. Note that the functions perform the operations in double precision, on an element-by-element basis, but do not convert images to double precision values in the MATLAB workspace. • Overflow is handled automatically. The functions truncate return values to fit the data type. For details about this truncation, see “Image Arithmetic Truncation Rules” on page 2-22. See “Summary of Image Arithmetic Functions” on page 2-22 for a complete list. For more information about using these functions to perform arithmetic operations, see these sections: • “Adding Images” on page 2-23 • “Subtracting Images” on page 2-24 • “Multiplying Images” on page 2-25 • “Dividing Images” on page 2-27 • “Nesting Calls to Image Arithmetic Functions” on page 2-27 2-21 2 Introduction Summary of Image Arithmetic Functions The following table lists the image arithmetic functions. For more complete descriptions, see their reference pages. Function Description imabsdiff Absolute difference of two images imadd Add two images imcomplement Complement an image imdivide Divide two images imlincomb Compute linear combination of two images immultiply Multiply two images imsubtract Subtract two images Image Arithmetic Truncation Rules The results of integer arithmetic can easily overflow the data type allotted for storage. For example, the maximum value you can store in uint8 data is 255. Arithmetic operations can also result in fractional values, which cannot be represented using integer arrays. The image arithmetic functions use these rules for integer arithmetic: • Values that exceed the range of the integer type are truncated to that range • Fractional values are rounded For example, if the data type is uint8, results greater than 255 (including Inf) are set to 255. The following table lists some additional examples. 2-22 Result Class Truncated Value 300 uint8 255 -45 uint8 0 10.5 uint8 11 Image Arithmetic Adding Images To add two images or add a constant value to an image, use the imadd function. imadd adds the value of each pixel in one of the input images with the corresponding pixel in the other input image and returns the sum in the corresponding pixel of the output image. Image addition has many uses in image processing. For example, the following code fragment uses addition to superimpose one image on top of another. The images must be the same size and class. I = imread('rice.tif'); J = imread('cameraman.tif'); K = imadd(I,J); imshow(K) You can also use addition to brighten an image by adding a constant value to each pixel. For example, the following code brightens an RGB image. RGB = imread('flowers.tif'); RGB2 = imadd(RGB,50); subplot(1,2,1); imshow(RGB); subplot(1,2,2); imshow(RGB2); 2-23 2 Introduction Original Image Image After Addition Handling Overflow When you add the pixel values of two images, the result can easily overflow the maximum value supported by the data type, especially for uint8 data. When overflow occurs, imadd truncates the value to the maximum value supported by the data type. This is an effect known as saturation. For example, imadd truncates uint8 data at 255. To avoid saturation, convert the image to a larger data type, such as uint16, before performing the addition. Subtracting Images To subtract one image from another, or subtract a constant value from an image, use the imsubtract function. imsubtract subtracts each pixel value in one of the input images from the corresponding pixel in the other input image and returns the result in the corresponding pixel in an output image. Image subtraction can be used as a preliminary step in more complex image processing or by itself. For example, you can use image subtraction to detect changes in a series of images of the same scene. This code fragment subtracts the background from an image of rice grains. The images must be the same size and class. rice= imread('rice.tif'); background = imopen(rice, strel('disk',15)); 2-24 Image Arithmetic rice2 = imsubtract(rice,background); imshow(rice),figure,imshow(rice2); Original image Difference Image To subtract a constant from each pixel in I, replace Y with a constant, as in the following example. Z = imsubtract(I,50); Handling Negative Values Subtraction can result in a negative values for certain pixels. When this occurs with unsigned data types, such as uint8 or uint16, the imsubtract function truncates the negative value to zero (0), which displays as black. To avoid negative values, but preserve the value differentiation of these pixels, use the imabsdiff function. The imabsdiff function calculates the absolute difference between each corresponding pixel in the two images so the result is always nonnegative. Multiplying Images To multiply two images, use the immultiply function. immultiply does an element-by-element multiplication (.*) of each corresponding pixel in a pair of input images and returns the product of these multiplications in the corresponding pixel in an output image. Image multiplication by a constant, referred to as scaling, is a common image processing operation. When used with a scaling factor greater than one, scaling brightens an image; a factor less than one darkens an image. Scaling generally 2-25 2 Introduction produces a much more natural brightening/darkening effect than simply adding an offset to the pixels, since it preserves the relative contrast of the image better. For example, this code scales an image by a constant factor. I = imread('moon.tif'); J = immultiply(I,1.2); imshow(I); figure, imshow(J) Original image Image after multiplication Handling Overflow Multiplication of uint8 images very often results in overflow. The immultiply function truncates values that overflow the data type to the maximum value. To avoid truncation, convert uint8 images to a larger data type, such as uint16, before performing multiplication. 2-26 Image Arithmetic Dividing Images To divide two images, use the imdivide function. The imdivide function does an element-by-element division (./) of each corresponding pixel in a pair of input images. The immultiply function returns the result in the corresponding pixel in an output image. Image division, like image subtraction, can be used to detect changes in two images. However, instead of giving the absolute change for each pixel, division gives the fractional change or ratio between corresponding pixel values. Image division is also called ratioing. For example, the following code divides the rice grain image by a morphologically opened version of the itself. (For information about morphological image processing, see Chapter 9, “Morphological Operations.”) The images must be the same size and class. I = imread('rice.tif'); background = imopen(I, strel('disk',15)); Ip = imdivide(I,background); imshow(Ip,[]) Nesting Calls to Image Arithmetic Functions You can use the image arithmetic functions in combination to perform a series of operations. For example, to calculate the average of two images, A+B C = -------------2 You could enter I = imread('rice.tif'); I2 = imread('cameraman.tif'); K = imdivide(imadd(I,I2), 2); % not recommended However, when used with uint8 or uint16 data, each arithmetic function truncates its result before passing it on to the next operation. This truncation can significantly reduce the amount of information in the output image. A better way to perform this series of calculations is to use the imlincomb function. imlincomb performs all the arithmetic operations in the linear combination in double precision and only truncates the final result. K = imlincomb(.5,I,.5,I2); % recommended 2-27 2 Introduction Coordinate Systems Locations in an image can be expressed in various coordinate systems, depending on context. This section discusses the two main coordinate systems used in the Image Processing Toolbox, and the relationship between them. These two coordinate systems are described in: • “Pixel Coordinates” • “Spatial Coordinates” on page 2-29 Pixel Coordinates Generally, the most convenient method for expressing locations in an image is to use pixel coordinates. In this coordinate system, the image is treated as a grid of discrete elements, ordered from top to bottom and left to right, as illustrated by the following figure. 1 2 3 c 1 2 3 r Figure 2-6: The Pixel Coordinate System For pixel coordinates, the first component r (the row) increases downward, while the second component c (the column) increases to the right. Pixel coordinates are integer values and range between 1 and the length of the row or column. There is a one-to-one correspondence between pixel coordinates and the coordinates MATLAB uses for matrix subscripting. This correspondence makes the relationship between an image’s data matrix and the way the image 2-28 Coordinate Systems displays easy to understand. For example, the data for the pixel in the fifth row, second column is stored in the matrix element (5,2). Spatial Coordinates In the pixel coordinate system, a pixel is treated as a discrete unit, uniquely identified by a single coordinate pair, such as (5,2). From this perspective, a location such as (5.3,2.2) is not meaningful. At times, however, it is useful to think of a pixel as a square patch. From this perspective, a location such as (5.3,2.2) is meaningful, and is distinct from (5,2). In this spatial coordinate system, locations in an image are positions on a plane, and they are described in terms of x and y (not r and c as in the pixel coordinate system). The following figure illustrates the spatial coordinate system used for images. Notice that y increases downward. 0.5 0.5 1 1.5 2 2.5 3 3.5 x 1 1.5 2 2.5 3 3.5 y Figure 2-7: The Spatial Coordinate System This spatial coordinate system corresponds closely to the pixel coordinate system in many ways. For example, the spatial coordinates of the center point of any pixel are identical to the pixel coordinates for that pixel. There are some important differences, however. In pixel coordinates, the upper-left corner of an image is (1,1), while in spatial coordinates, this location by default is (0.5,0.5). This difference is due to the pixel coordinate system being discrete, while the spatial coordinate system is continuous. Also, the upper-left corner is always (1,1) in pixel coordinates, but you can specify a 2-29 2 Introduction nondefault origin for the spatial coordinate system. See “Using a Nondefault Spatial Coordinate System” on page 2-30 for more information. Another potentially confusing difference is largely a matter of convention: the order of the horizontal and vertical components is reversed in the notation for these two systems. As mentioned earlier, pixel coordinates are expressed as (r,c), while spatial coordinates are expressed as (x,y). In the reference pages, when the syntax for a function uses r and c, it refers to the pixel coordinate system. When the syntax uses x and y, it refers to the spatial coordinate system. Using a Nondefault Spatial Coordinate System By default, the spatial coordinates of an image correspond with the pixel coordinates. For example, the center point of the pixel in row 5, column 3 has spatial coordinates x=3, y=5. (Remember, the order of the coordinates is reversed.) This correspondence simplifies many of the toolbox functions considerably. Several functions primarily work with spatial coordinates rather than pixel coordinates, but as long as you are using the default spatial coordinate system, you can specify locations in pixel coordinates. In some situations, however, you may want to use a nondefault spatial coordinate system. For example, you could specify that the upper-left corner of an image is the point (19.0,7.5), rather than (0.5,0.5). If you call a function that returns coordinates for this image, the coordinates returned will be values in this nondefault spatial coordinate system. To establish a nondefault spatial coordinate system, you can specify the XData and YData image properties when you display the image. These properties are two-element vectors that control the range of coordinates spanned by the image. By default, for an image A, XData is [1 size(A,2)], and YData is [1 size(A,1)]. For example, if A is a 100 row by 200 column image, the default XData is [1 200], and the default YData is [1 100]. The values in these vectors are actually the coordinates for the center points of the first and last pixels (not the pixel edges), so the actual coordinate range spanned is slightly larger; for instance, if XData is [1 200], the x-axis range spanned by the image is [0.5 200.5]. These commands display an image using nondefault XData and YData. A = magic(5); 2-30 Coordinate Systems x = [19.5 23.5]; y = [8.0 12.0]; image(A,'XData',x,'YData',y), axis image, colormap(jet(25)) For information about the syntax variations that specify nondefault spatial coordinates, see the reference page for imshow. 2-31 2 Introduction 2-32 3 Displaying and Printing Images This section introduces the image display techniques supported by the Image Processing Toolbox for each image type supported by the toolbox (e.g., RGB, intensity). Topics covered include Terminology (p. 3-2) Provides definitions of image processing terms used in this section Displaying Images (p. 3-3) Describes how to use the imshow function to display all types of images Special Display Techniques (p. 3-12) Describes how to display multiframe images or include a colorbar with an image Setting Toolbox Display Preferences (p. 3-23) Describes how to set preferences that impact toolbox display functions Zooming in on a Region of an Image (p. 3-26) Describes how to zoom in and out on an image Texture Mapping (p. 3-28) Describes how to map an image onto a parametric surface, such as a sphere, or below a surface plot Printing Images (p. 3-29) Describes how to print an image from within MATLAB Troubleshooting (p. 3-30) Provides tips about handling common image display problems 3 Displaying and Printing Images Terminology An understanding of the following terms will help you to use this chapter. Terms Definitions Color approximation There are two ways in which this term is used in MATLAB: • The method by which MATLAB chooses the best colors for an image whose number of colors you are decreasing • The automatic choice of screen colors MATLAB makes when displaying on a system with limited color display capability 3-2 Screen bit depth The number of bits per screen pixel Screen color resolution The number of distinct colors that can be produced by the screen Displaying Images Displaying Images In MATLAB, the primary way to display images is by using the image function. This function creates a Handle Graphics® image object, and it includes syntax for setting the various properties of the object. MATLAB also includes the imagesc function, which is similar to image but which automatically scales the input data. The Image Processing Toolbox includes an additional display routine called imshow. Like image and imagesc, this function creates a Handle Graphics image object. However, imshow also automatically sets various Handle Graphics properties and attributes of the image to optimize the display. This section discusses displaying images using imshow. In general, using imshow for image processing applications is preferable to using image and imagesc. It is easier to use and in most cases, displays an image using one image pixel per screen pixel. (For more information about image and imagesc, see their pages in the MATLAB Function Reference or see the MATLAB graphics documentation.) Note One of the most common toolbox usage errors is using the wrong syntax of imshow for your image type. This chapter shows which syntax is appropriate for each type of image. If you need help determining what type of image you are working with, see “Image Types in the Toolbox” on page 2-5. Displaying Indexed Images To display an indexed image with imshow, specify both the image matrix and the colormap. imshow(X,map) For each pixel in X, imshow displays the color stored in the corresponding row of map. The relationship between the values in the image matrix and the colormap depends on whether the image matrix is of class double, uint16, or uint8. If the image matrix is of class double, the value 1 points to the first row in the colormap, the value 2 points to the second row, and so on. If the image matrix is of class uint8 or uint16, there is an offset; the value 0 points to the first row in the colormap, the value 1 points to the second row, and so on. (The 3-3 3 Displaying and Printing Images offset is handled automatically by the image object, and is not controlled through a Handle Graphics property.) imshow directly maps each pixel in an indexed image to its corresponding colormap entry. If the colormap contains a greater number of colors than the image, imshow ignores the extra colors in the colormap. If the colormap contains fewer colors than the image requires, imshow sets all image pixels over the limits of the colormap’s capacity to the last color in the colormap. For example, if an image of class uint8 contains 256 colors, and you display it with a colormap that contains only 16 colors, all pixels with a value of 15 or higher are displayed with the last color in the colormap. The Image and Axes Properties of an Indexed Image When you display an indexed image, imshow sets the Handle Graphics properties that control how colors display, as follows: • The image CData property is set to the data in X. • The image CDataMapping property is set to direct. • The axes CLim property does not apply, because CDataMapping is set to direct. • The figure Colormap property is set to the data in map. Displaying Intensity Images To display a intensity (grayscale) image, the most basic syntax is imshow(I) imshow displays the image by scaling the intensity values to serve as indices into a grayscale colormap. If I is double, a pixel value of 0.0 is displayed as black, a pixel value of 1.0 is displayed as white, and pixel values in between are displayed as shades of gray. If I is uint8, then a pixel value of 255 is displayed as white. If I is uint16, then a pixel value of 65535 is displayed as white. Intensity images are similar to indexed images in that each uses an m-by-3 RGB colormap, but normally, you will not specify a colormap for an intensity image. MATLAB displays intensity images by using a grayscale system colormap (where R=G=B). By default, the number of levels of gray in the colormap is 256 on systems with 24-bit color, and 64 or 32 on other systems. (See “Working with Different Screen Bit Depths” on page 13-3 for a detailed explanation.) 3-4 Displaying Images Another syntax form of imshow for intensity images enables you to explicitly specify the number of gray levels to use. To display an image I with 32 gray levels, specify a value for n. imshow(I,32) Because MATLAB scales intensity images to fill the colormap range, a colormap of any size can be used. Larger colormaps enable you to see more detail, but they also use up more color slots. The availability of color slots is discussed further in “Displaying Multiple Images” on page 3-17, and also in “Working with Different Screen Bit Depths” on page 13-3. Displaying Intensity Images That Have Unconventional Ranges In some cases, you may have data you want to display as an intensity image, even though the data is outside the conventional toolbox range (i.e., [0,1] for double arrays, [0,255] for uint8 arrays, or [0,65535] for uint16 arrays). For example, if you filter an intensity image, some of the output data may fall outside the range of the original data. To display unconventional range data as an image, you can specify the data range directly, using imshow(I,[low high]) If you use an empty matrix ([]) for the data range, imshow scales the data automatically, setting low and high to the minimum and maximum values in the array. The next example filters an intensity image, creating unconventional range data. imshow is then called using an empty matrix. I = imread('testpat1.tif'); J = filter2([1 2;-1 -2],I); imshow(J,[]); 3-5 3 Displaying and Printing Images When you use this syntax, imshow sets the axes CLim property to [min(J(:)) max(J(:))]. CDataMapping is always scaled for intensity images, so that the value min(J(:)) is displayed using the first colormap color, and the value max(J(:)) is displayed using the last colormap color. The Image and Axes Properties of an Intensity Image When you display an intensity image, imshow sets the Handle Graphics properties that control how colors display, as follows: • The image CData property is set to the data in I. • The image CDataMapping property is set to scaled. • The axes CLim property is set to [0 1] if the image matrix is of class double, [0 255] if the matrix is of class uint8, or [0 65535] if it is of class uint16. • The figure Colormap property is set to a grayscale colormap whose values range from black to white. Displaying Binary Images To display a binary image, the syntax is BW = imread('circles.tif'); imshow(BW) 3-6 Displaying Images In MATLAB, a binary image is of class logical. Binary images contain only 0’s and 1’s. Pixels with the value 0 display as black; pixels with the value 1 display as white. Note For the toolbox to interpret the image as binary, it must be of class logical. Intensity images that happen to contain only 0’s and 1’s are not binary images. Changing the Display Colors of a Binary Image You may prefer to invert binary images when you display them, so that 0 values display as white and 1 values display as black. To do this, use the NOT (~) operator in MATLAB. (In this example, a box is drawn around the image to show the image boundary.) For example, imshow(~BW) 3-7 3 Displaying and Printing Images You can also display a binary image using a colormap. For example, the following command displays 0’s as red and 1’s as blue. imshow(BW,[1 0 0; 0 0 1]) Reading and Writing Binary Images In certain file formats, a binary image can be stored in a 1-bit format. When you read in a binary image in 1-bit format, MATLAB represents it in the workspace as a logical array. By default, MATLAB writes binary images as 1-bit images, if the file format supports it. imwrite(BW,'test.tif'); % MATLAB supports writing 1-bit TIFFs. 3-8 Displaying Images To verify the bit depth of test.tif, call imfinfo. As you will see, the BitDepth field indicates that it has been saved as a 1-bit image, with the beginning of your output looking something like this. imfinfo('test.tif') ans = Filename: 'd:\mystuff\grid.tif' FileModDate: '25-Nov-1998 11:36:17' FileSize: 340 Format: 'tif' FormatVersion: [] Width: 20 Height: 20 BitDepth: 1 ColorType: 'grayscale' FormatSignature: [73 73 42 0] ByteOrder: 'little-endian' NewSubfileType: 0 BitsPerSample: 1 Compression: 'CCITT 1D' ... Note You may have noticed that the ColorType field of the binary image queried above has a value of 'grayscale'. MATLAB sets this field to one of three values: 'grayscale', 'indexed', and 'truecolor'. When reading an image, MATLAB evaluates the image type by checking both the BitDepth and the ColorType fields. The Image and Axes Properties of a Binary Image imshow sets the Handle Graphics properties that control how colors display, as follows: • The image CData is set to the data in BW. • The image CDataMapping property is set to direct. • The axes CLim property is set to [0 1]. • The figure Colormap property is set to a grayscale colormap whose values range from black to white. 3-9 3 Displaying and Printing Images Displaying RGB Images RGB images, also called truecolor images, represent color values directly, rather than through a colormap. To display an RGB image, the most basic syntax is imshow(RGB) RGB is m-by-n-by-3 array. For each pixel (r,c) in RGB, imshow displays the color represented by the triplet (r,c,1:3). Systems that use 24 bits per screen pixel can display truecolor images directly, because they allocate 8 bits (256 levels) each to the red, green, and blue color planes. On systems with fewer colors, MATLAB displays the image using a combination of color approximation and dithering. See “Working with Different Screen Bit Depths” on page 13-3 for more information. The Image and Axes Properties of an RGB Image When you display an RGB image, imshow sets the Handle Graphics properties that control how colors display, as follows: • The image CData property is set to the data in RGB. The data will be three-dimensional. When CData is three-dimensional, MATLAB interprets the array as truecolor data, and ignores the values of the CDataMapping, CLim, and Colormap properties. • The image CDataMapping property is ignored. • The axes CLim property is ignored. • The figure Colormap property is ignored. 3-10 Displaying Images Displaying Images Directly from Disk Generally, when you want to display an image, you will first use imread to load it and the data is stored as one or more variables in the MATLAB workspace. However, if you do not want to load an image directly before displaying it, you can display a file directly using this syntax. imshow filename The file must be in the current directory or on the MATLAB path. For example, to display a file named flowers.tif, imshow flowers.tif If the image has multiple frames, imshow will only display the first frame. For information on the display options for multiframe images, see “Displaying Multiframe Images” on page 3-13. This syntax is very useful for scanning through images. Note, however, that when you use this syntax, the image data is not stored in the MATLAB workspace. If you want to bring the image into the workspace, use the getimage function, which gets the image data from the current Handle Graphics image object. For example, rgb = getimage; will assign flowers.tif to rgb if the figure window in which it is displayed is currently active. 3-11 3 Displaying and Printing Images Special Display Techniques In addition to imshow, the toolbox includes functions that perform specialized display operations, or exercise more direct control over the display format. These functions, together with the MATLAB graphics functions, provide a range of image display options. This section includes the following topics: • “Adding a Colorbar” on page 3-12 • “Displaying Multiframe Images” on page 3-13 • “Displaying Multiple Images” on page 3-17 • “Zooming in on a Region of an Image” on page 3-26 • “Texture Mapping” on page 3-28 Adding a Colorbar Use the colorbar function to add a colorbar to an axes object. If you add a colorbar to an axes object that contains an image object, the colorbar indicates the data values that the different colors in the image correspond to. Seeing the correspondence between data values and the colors displayed by using a colorbar is especially useful if you are displaying unconventional range data as an image, as described under “Displaying Intensity Images That Have Unconventional Ranges” on page 3-5. In the example below, a grayscale image of class uint8 is filtered, resulting in data that is no longer in the range [0,255]. I = imread('saturn.tif'); h = [1 2 1; 0 0 0; -1 -2 -1]; I2 = filter2(h,I); imshow(I2,[]), colorbar 3-12 Special Display Techniques 400 300 200 100 0 −100 −200 −300 Displaying Multiframe Images A multiframe image is an image file that contains more than one image. The MATLAB-supported formats that enable the reading and writing of multiframe images are HDF and TIFF. See “Multiframe Image Arrays” on page 2-11 for more information about reading and writing multiframe images. Once read into MATLAB, the image frames of a multiframe image are always handled in the fourth dimension. Multiframe images can be loaded from disk using a special syntax of imread, or created using MATLAB. Multiframe images can be displayed in several different ways; to display a multiframe image, you can • Display the frames individually, using the imshow function. See “Displaying the Frames of a Multiframe Image Individually” on page 3-14 below. • Display all of the frames at once, using the montage function. See “Displaying All Frames of a Multiframe Image at Once” on page 3-15. • Convert the frames to a movie, using the immovie function. See “Converting a Multiframe Image to a Movie” on page 3-16. 3-13 3 Displaying and Printing Images Displaying the Frames of a Multiframe Image Individually In MATLAB, the frames of a multiframe image are handled in the fourth dimension. To view an individual frame, call imshow and specify the frame using standard MATLAB indexing notation. For example, to view the seventh frame in the intensity array I, imshow(I(:,:,:,7)) The following example loads mri.tif and displays the third frame. % Initialize an array to hold the 27 frames of mri.tif mri = uint8(zeros(128,128,1,27)); for frame=1:27 % Read each frame into the appropriate frame in memory [mri(:,:,:,frame),map] = imread('mri.tif',frame); end imshow(mri(:,:,:,3),map); Intensity, indexed, and binary multiframe images have a dimension of m-by-n-by-1-by-k, where k represents the total number of frames, and 1 signifies that the image data has just one color plane. Therefore, the following call, imshow(mri(:,:,:,3),map); is equivalent to, imshow(mri(:,:,1,3),map); RGB multiframe images have a dimension of m-by-n-by-3-by-k, where k represents the total number of frames, and 3 signifies the existence of the three color planes used in RGB images. This example, imshow(RGB(:,:,:,7)); 3-14 Special Display Techniques shows all three color planes of the seventh frame, and is not equivalent to imshow(RGB(:,:,3,7)); which shows only the third color plane (blue) of the seventh frame. These two calls will only yield the same results if the image is RGB grayscale (R=G=B). Displaying All Frames of a Multiframe Image at Once To view all of the frames in a multiframe array at one time, use the montage function. montage divides a figure into multiple display regions and displays each image in a separate region. The syntax for montage is similar to the imshow syntax. To display a multiframe intensity image, the syntax is montage(I) To display a multiframe indexed image, the syntax is montage(X,map) Note All of the frames in a multiframe indexed array must use the same colormap. This example loads and displays all frames of a multiframe indexed image. % Initialize an array to hold the 27 frames of mri.tif. mri = uint8(zeros(128,128,1,27)); for frame=1:27 % Read each frame into the appropriate frame in memory. [mri(:,:,:,frame),map] = imread('mri.tif',frame); end montage(mri,map); 3-15 3 Displaying and Printing Images Figure 3-1: All Frames of Multiframe Image Displayed in One Figure Notice that montage displays images in a row-wise manner. The first frame appears in the first position of the first row, the next frame in the second position of the first row, and so on. montage arranges the frames so that they roughly form a square. Converting a Multiframe Image to a Movie To create a MATLAB movie from a multiframe image array, use the immovie function. This call creates a movie from a multiframe indexed image X mov = immovie(X,map); 3-16 Special Display Techniques where X is a four-dimensional array of images that you want to use for the movie. You can play the movie in MATLAB using the movie function. movie(mov); This example loads the multiframe image mri.tif and makes a movie out of it. It won’t do any good to show the results here, so try it out; it’s fun to watch. % Initialize and array to hold the 27 frames of mri.tif. mri = uint8(zeros(128,128,1,27)); for frame=1:27 % Read each frame into the appropriate frame in memory. [mri(:,:,:,frame),map] = imread('mri.tif',frame); end mov = immovie(mri,map); movie(mov); Note that immovie displays the movie as it is being created, so you will actually see the movie twice. The movie runs much faster the second time (using movie). Note MATLAB movies require MATLAB in order to be run. To make a movie that can be run outside of MATLAB, you can use the MATLAB avifile and addframe functions to create an AVI file. AVI files can be created using indexed and RGB images of classes uint8 and double, and don’t require a multiframe image. For instructions on creating an AVI file, see the Development Environment section in the MATLAB documentation. Displaying Multiple Images MATLAB does not place any restrictions on the number of images you can display simultaneously. However, there are usually system limitations that are dependent on the computer hardware you are using. The sections below describe how to display multiple figures separately, or within the same figure. The main limitation is the number of colors your system can display. This number depends primarily on the number of bits that are used to store the color information for each pixel. Most systems use either 8, 16, or 24 bits per pixel. 3-17 3 Displaying and Printing Images If you are using a system with 16 or 24 bits per pixel, you are unlikely to run into any problems, regardless of the number of images you display. However, if your system uses 8 bits per pixel, it can only display a maximum of 256 different colors, and you can therefore quickly run out of color slots if you display multiple images. (Actually, the total number of colors you can display is slightly fewer than 256, because some color slots are reserved for Handle Graphics objects. The operating system usually reserves a few colors as well.) To determine the number of bits per pixel on your system, enter this command. get(0,'ScreenDepth') See “Working with Different Screen Bit Depths” on page 13-3 for more information. This section discusses: • Displaying each image in a separate figure • Displaying multiple images in the same figure It also includes information about working around system limitations. Displaying Each Image in a Separate Figure The simplest way to display multiple images is to display them in different figure windows. imshow always displays an image in the current figure, so if you display two images in succession, the second image replaces the first image. To avoid replacing the image in the current figure, use the figure command to explicitly create a new empty figure before calling imshow for the next image. For example, imshow(I) figure, imshow(I2) figure, imshow(I3) When you use this approach, the figures you create are empty initially. If you have an 8-bit display, you must make sure that the total number of colormap entries does not exceed 256. For example, if you try to display three images, each having a different colormap with 128 entries, at least one of the images will display with the wrong colors. (If all three images have identical colormaps, there will not be a problem, because only 128 color slots are used.) Remember that intensity images are also displayed using colormaps, so the color slots used by these images count toward the 256-color total. 3-18 Special Display Techniques In the next example, two indexed images are displayed on an 8-bit display. Since these images do not have similar colormaps and due to the limitation of the screen color resolution, the first image is forced to use the colormap of the second image, resulting in an inaccurate display. [X1,map1]=imread('forest.tif'); [X2,map2]=imread('trees.tif'); imshow(X1,map1),figure,imshow(X2,map2); Figure 3-2: Displaying Two Indexed Images on an 8-bit Screen As X2 is displayed, X1 is forced to use X2’s colormap (and now you can’t see the forest for the trees). Note that the actual display results of this example will 3-19 3 Displaying and Printing Images vary depending on what other application windows are open and using up system color slots. One way to avoid these display problems is to manipulate the colormaps to use fewer colors. There are various ways to do this, such as using the imapprox function. See “Reducing the Number of Colors in an Image” on page 13-6 for information. Another solution is to convert images to RGB (truecolor) format for display, because MATLAB automatically uses dithering and color approximation to display these images. Use the ind2rgb function to convert indexed images to RGB. imshow(ind2rgb(X,map)) Or, simply use the cat command to display an intensity image as an RGB image. imshow(cat(3,I,I,I)) Displaying Multiple Images in the Same Figure You can display multiple images in a single figure window with some limitations. This discussion shows you how to do this in one of two ways: 1 By using imshow in conjunction with subplot 2 By using subimage in conjunction with subplot subplot divides a figure into multiple display regions. The syntax of subplot is subplot(m,n,p) This syntax divides the figure into an m-by-n matrix of display regions and makes the pth display region active. For example, if you want to display two images side by side, use [X1,map1]=imread('forest.tif'); [X2,map2]=imread('trees.tif'); subplot(1,2,1), imshow(X1,map2) subplot(1,2,2), imshow(X2,map2) 3-20 Special Display Techniques Figure 3-3: Two Images in Same Figure Using the Same Colormap If sharing a colormap (using the subplot function) produces unacceptable display results as Figure 3-3 shows, use the subimage function (shown below). Or, as another alternative, you can map all images to the same colormap as you load them. See “Colormap Mapping” on page 13-11 for more information. subimage converts images to RGB before displaying and therefore circumvents the colormap sharing problem. This example displays the same two images shown in Figure 3-3 with better results. [X1,map1]=imread('forest.tif'); [X2,map2]=imread('trees.tif'); subplot(1,2,1), subimage(X1,map1) subplot(1,2,2), subimage(X2,map2) 3-21 3 Displaying and Printing Images Figure 3-4: Two Images in Same Figure Using Separate Colormaps 3-22 Setting Toolbox Display Preferences Setting Toolbox Display Preferences The behavior of imshow is influenced in part by the current settings of the toolbox preferences. Depending on the arguments you specify and the current settings of the toolbox preferences, imshow may: • Suppress the display of axes and tick marks. • Include or omit a “border” around the image. • Call the truesize function to display the image without interpolation. • Set other figure and axes properties to tailor the display. All of these settings can be changed by using the iptsetpref function.The truesize preference can also be changed by setting the display_option parameter of imshow. This section describes how to set the toolbox preferences and how to use the display_option parameter. When you display an image using the imshow function, MATLAB also sets the Handle Graphics figure, axes, and image properties, which control the way image data is interpreted. These settings are optimized for each image type. The specific properties set are described under the following sections: • “The Image and Axes Properties of an Indexed Image” on page 3-4 • “The Image and Axes Properties of an Intensity Image” on page 3-6 • “The Image and Axes Properties of a Binary Image” on page 3-9 • “The Image and Axes Properties of an RGB Image” on page 3-10 Toolbox Preferences The toolbox preferences affect the behavior of imshow for the duration of the current MATLAB session. You can change these settings at any time by using the iptsetpref function. To preserve your preference settings from one session 3-23 3 Displaying and Printing Images to the next, make your settings in your startup.m file. These are the preferences that you may set. Toolbox Preference Description ImshowBorder Controls whether imshow displays the figure window as larger than the image (leaving a border between the image axes and the edges of the figure), or the same size as the image (leaving no border). ImshowAxesVisible Controls whether imshow displays images with the axes box and tick labels. ImshowTruesize Controls whether imshow calls the truesize function. This preference can be overridden for a single call to imshow; see “Using the truesize Function” below for more details TrueSizeWarning Controls whether you will receive a warning message if an image is too large for the screen. This example call to iptsetpref resizes the figure window so that it fits tightly around displayed images. iptsetpref('ImshowBorder', 'tight'); To determine the current value of a preference, use the iptgetpref function. For more information about toolbox preferences and the values they accept, see the reference entries for iptgetpref and iptsetpref. Using the truesize Function The truesize function assigns a single screen pixel to each image pixel, e.g., a 200-by-300 image will be 200 screen pixels in height and 300 screen pixels in width. This is generally the preferred way to display an image. In most situations, when the toolbox is operating under default behavior, imshow calls the truesize command automatically before displaying an image. In some cases, you may not want imshow to automatically call truesize (for example, if you are working with a small image). If you display an image 3-24 Setting Toolbox Display Preferences without calling truesize, the image displays at the default axis size. In such cases, MATLAB must use interpolation to determine the values for screen pixels that do not directly correspond to elements in the image matrix. (See “Interpolation” on page 4-3 for more information.) You can affect whether MATLAB automatically calls truesize by using either of these methods: • Set the preference for the current MATLAB session. This example sets the ImshowTruesize preference to 'manual', meaning that truesize will not be automatically called by imshow. iptsetpref('ImshowTruesize','manual') • Set the preference for a single imshow command by setting the display_option parameter. This example sets the display_option parameter to truesize, so that truesize is called for the image displayed, regardless of the current preference setting. imshow(X, map,'truesize') For more information see the reference descriptions for imshow and truesize. 3-25 3 Displaying and Printing Images Zooming in on a Region of an Image The simplest way to zoom in on a region of an image is to use the zoom buttons provided on the figure window. To enable zooming from the command line, use the zoom command.When you zoom in, the figure window remains the same size, but only a portion of the image is displayed, at a higher magnification. (zoom works by changing the axis limits; it does not change the image data in the figure.) Once zooming in is enabled, there are two ways to zoom in on an image: 1 Single mouse click: click on a spot in the image by placing the cursor on the spot and the pressing the left mouse button. The image is magnified and the center of the new view is the spot where you clicked. 2 Click and drag the mouse: select a region by clicking on the image, holding down the left mouse button, and dragging the mouse. This creates a dotted rectangle. When you release the mouse button, the region enclosed by the rectangle is displayed with magnification. Zooming In or Out with the Zoom Buttons The zoom buttons in the MATLAB figure enable you to zoom in or out on an image using your mouse. To zoom in, click the “magnifying glass” button with the plus sign in it. There are two ways to zoom in on an image after selecting the zoom in button. See “Zooming in on a Region of an Image” above. To zoom out, click the “magnifying glass” button with the minus sign in it. Click your left mouse button over the spot in the image you would like to zoom out from. Zooming In or Out from the Command Line The zoom command enables you to zoom in or out on an image using your mouse. To enable zooming (in or out), type zoom on 3-26 Zooming in on a Region of an Image There are two ways to zoom in on an image. See “Zooming in on a Region of an Image” on page 3-26. To zoom out, click on the image with the right mouse button. (If you have a single-button mouse, hold down the Shift key and click.) To zoom out completely and restore the original view, enter zoom out To disable zooming, enter zoom off 3-27 3 Displaying and Printing Images Texture Mapping When you use the imshow command, MATLAB displays the image in a two-dimensional view. However, it is also possible to map an image onto a parametric surface, such as a sphere, or below a surface plot. The warp function creates these displays by texture mapping the image. Texture mapping is a process that maps an image onto a surface grid using interpolation. This example texture maps an image of a test pattern onto a cylinder. [x,y,z] = cylinder; I = imread('testpat1.tif'); warp(x,y,z,I); 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −1 1 −0.5 0 0.5 −0.4 1 −1 −0.8 −0.2 0 0.2 0.4 0.6 0.8 −0.6 Figure 3-5: An Image Texture Mapped onto a Cylinder The image may not map onto the surface in the way that you had expected. One way to modify the way the texture map appears is to change the settings of the Xdir, Ydir, and Zdir properties. For more information, see Changing Axis Direction in the MATLAB Graphics documentation. For more information about texture mapping, see the reference entry for the warp function. 3-28 Printing Images Printing Images If you want to output a MATLAB image to use in another application (such as a word-processing program or graphics editor), use imwrite to create a file in the appropriate format. See “Writing a Graphics Image” on page 2-17 for details. If you want to print the contents of a MATLAB figure (including nonimage elements such as labels), use the MATLAB print command, or choose the Print option from the File menu of the figure window. Note that if you produce output in either of these ways, the results reflect the settings of various Handle Graphics properties. In some cases, you may need to change the settings of certain properties to get the results you want. Here are some tips that may be helpful when you print images: • Image colors print as shown on the screen. This means that images are not affected by the InvertHardcopy figure property. • To ensure that printed images have the proper size and aspect ratio, you should set the figure’s PaperPositionMode property to auto. When PaperPositionMode is set to auto, the width and height of the printed figure are determined by the figure’s dimensions on the screen. By default, the value of PaperPositionMode is manual. If you want the default value of PaperPositionMode to be auto, you can add this line to your startup.m file. set(0,'DefaultFigurePaperPositionMode','auto') For detailed information about printing with File/Print or the print command (and for information about Handle Graphics), see “Printing and Exporting Figures with MATLAB” in the MATLAB Graphics documentation. For a complete list of options for the print command, enter help print at the MATLAB command line prompt or see print in the MATLAB Function Reference. 3-29 3 Displaying and Printing Images Troubleshooting This section contains three common scenarios (in bold text) which can occur unexpectedly, and what you can do to derive the expected results. Color Image Displays as Grayscale If a color image displays as a grayscale image, it can indicate that the image is an indexed image and you have not specified the associated colormap. To display an indexed image using imshow, you must specify the colormap, using this syntax: imshow(X,map); You should also make sure that you used the correct syntax for loading an indexed image, which is, [X, map]=imread('filename.ext'); For more information about displaying indexed images, see “Displaying Indexed Images” on page 3-3. Displaying Multiframe Images If you load a multiframe image and display it, MATLAB only displays one frame. To view the other frames in a multiframe image, you must load each frame separately. This can be done using a for loop. It may be helpful to first use imfinfo to find out how many frames the image file contains, and what their dimensions are. To see an example, go to “Displaying the Frames of a Multiframe Image Individually” on page 3-14. 3-30 4 Spatial Transformations This section describes the spatial transformation functions in the Image Processing Toolbox. Spatial transformations map pixel locations in an input image to new locations in an output image. Topics covered include Terminology (p. 4-2) Provides definitions of image processing terms used in this section Interpolation (p. 4-3) Defines interpolation, the process used to estimate the value of a pixel in an output image when that pixel does not appear in the input image Image Resizing (p. 4-5) Describes how to use the imresize function to change the size of an image Image Rotation (p. 4-8) Describes how to use the imrotate function to rotate an image Image Cropping (p. 4-10) Describes how to use the imcrop function to extract a rectangular portion of an image Performing General Spatial Transformations (p. 4-12) Describes the general spatial transformation capabilities of the toolbox 4 Spatial Transformations Terminology An understanding of the following terms will help you to use this chapter. 4-2 Terms Definitions Aliasing Artifacts in an image that can appear as a result of reducing an image’s size. When the size of an image is reduced, original pixels are downsampled to create fewer pixels. Aliasing that occurs as a result of size reduction normally appears as “stair-step” patterns (especially in high contrast images), or as “Moire” (ripple-effect) patterns. Anti-aliasing Any method for correcting aliasing (see above). The method discussed in this chapter is low-pass filtering (see below). Bicubic interpolation Output pixel values are calculated from a weighted average of pixels in the nearest 4-by-4 neighborhood. Bilinear interpolation Output pixel values are calculated from a weighted average of pixels in the nearest 2-by-2 neighborhood. Geometric operation An operation that modifies the spatial relations between pixels in an image. Examples include resizing (growing or shrinking), rotating, and shearing. Interpolation The process by which we estimate an image value at a location in between image pixels. Nearest neighbor interpolation Output pixel values are assigned the value of the pixel that the point falls within. No other pixels are considered. Interpolation Interpolation Interpolation is the process by which we estimate an image value at a location in between image pixels. For example, if you resize an image so it contains more pixels than it did originally, the software obtains values for the additional pixels through interpolation. The imresize and imrotate geometric functions use two-dimensional interpolation as part of the operations they perform. (The improfile image analysis function also uses interpolation. See “Intensity Profile” on page 10-4 for information about this function.) The Image Processing Toolbox provides three interpolation methods: • Nearest neighbor interpolation • Bilinear interpolation • Bicubic interpolation The interpolation methods all work in a fundamentally similar way. In each case, to determine the value for an interpolated pixel, you find the point in the input image that the output pixel corresponds to. You then assign a value to the output pixel by computing a weighted average of some set of pixels in the vicinity of the point. The weightings are based on the distance each pixel is from the point. The methods differ in the set of pixels that are considered: • For nearest neighbor interpolation, the output pixel is assigned the value of the pixel that the point falls within. No other pixels are considered. • For bilinear interpolation, the output pixel value is a weighted average of pixels in the nearest 2-by-2 neighborhood. • For bicubic interpolation, the output pixel value is a weighted average of pixels in the nearest 4-by-4 neighborhood. The number of pixels considered affects the complexity of the computation. Therefore the bilinear method takes longer than nearest neighbor interpolation, and the bicubic method takes longer than bilinear. However, the greater the number of pixels considered, the more accurate the effect is, so there is a trade-off between processing time and quality. 4-3 4 Spatial Transformations Image Types The functions that use interpolation take an argument that specifies the interpolation method. For most of these functions, the default method is nearest neighbor interpolation. This method produces acceptable results for all image types, and is the only method that is appropriate for indexed images. For intensity and RGB images, however, you should generally specify bilinear or bicubic interpolation, because these methods produce better results than nearest neighbor interpolation. For RGB images, interpolation is performed on the red, green, and blue image planes individually. For binary images, interpolation has effects that you should be aware of. If you use bilinear or bicubic interpolation, the computed values for the pixels in the output image will not all be 0 or 1. The effect on the resulting output image depends on the class of the input image: • If the class of the input image is double, the output image is a grayscale image of class double. The output image is not binary, because it includes values other than 0 and 1. • If the class of the input image is uint8, the output image is a binary image of class uint8. The interpolated pixel values are rounded off to 0 and 1 so the output image can be of class uint8. Note For bicubic interpolation, you may need to damp doubles to within the [0 1] range. If you use nearest neighbor interpolation, the result is always binary, because the values of the interpolated pixels are taken directly from pixels in the input image. 4-4 Image Resizing Image Resizing To change the size of an image, use the imresize function. Using imresize, you can: • Specify the size of the output image. • Specify the interpolation method used • Specify the filter to use to prevent aliasing Specifying the Size of the Output Image Using imresize, you can specify the size of the output image in two ways • By specifying the magnification factor to be used on the image • By specifying the dimensions of the output image Using the Magnification Factor To enlarge an image, specify a magnification factor greater than 1. To reduce an image, specify a magnification factor between 0 and 1. For example, the command below increases the size of the image, I, by 1.25 times. I = imread('ic.tif'); J = imresize(I,1.25); imshow(I) figure, imshow(J) 4-5 4 Spatial Transformations Specifying the Size of the Output Image You can specify the size of the output image by passing a vector that contains the number of row and columns in the output image. The command below creates an output image, Y, with 100 rows and 150 columns. Y = imresize(X,[100 150]) Note If the specified size does not produce the same aspect ratio as the input image, the output image will be distorted. Specifying the Interpolation Method By default, imresize uses nearest neighbor interpolation to determine the values of pixels in the output image but you can specify other interpolation methods. This table lists the supported interpolation methods. See “Interpolation” for more information about these methods. Argument Value Interpolation Method 'nearest' Nearest neighbor (the default) 'bilinear' Bilinear interpolation 'bicubic' Bicubic interpolation In this example, imresize uses the bilinear interpolation method. Y = imresize(X,[100 150],'bilinear') Using Filters to Prevent Aliasing Reducing the size of an image can introduce artifacts, such as aliasing, in the output image because information is always lost when you reduce the size of an image. Aliasing appears as ripple patterns (called Moire patterns) in the output image. When you reduce the size of the image using either bilinear or bicubic interpolation, imresize automatically applies a low-pass filter to the image 4-6 Image Resizing before interpolation, to limit the impact of aliasing on the output image. You can specify the size of this filter or specify a different filter. Note Even with low-pass filtering, resizing can introduce artifacts, because information is always lost when you reduce the size of an image. The imresize function does not apply a low-pass filter if nearest neighbor interpolation is used. Nearest neighbor interpolation is primarily used for indexed images, and low-pass filtering is not appropriate for these images. You can also specify a filter of your own creation. For more information about specifying a filter, see the reference page for imresize. 4-7 4 Spatial Transformations Image Rotation To rotate an image, use the imrotate function. imrotate accepts two primary arguments: • The image to be rotated • The rotation angle You specify the rotation angle in degrees. If you specify a positive value, imrotate rotates the image counterclockwise; if you specify a negative value, imrotate rotates the image clockwise. This example rotates the image, I, 35 degrees in the counterclockwise direction. J = imrotate(I,35); As optional arguments to imrotate, you can also specify: • The interpolation method • The size of the output image Specifying the Interpolation Method By default, imrotate uses nearest neighbor interpolation to determine the value of pixels in the output image but you can specify other interpolation methods. This table lists the supported interpolation methods. See “Interpolation” for more information about these methods. Argument Value Interpolation Method 'nearest' Nearest neighbor (the default) 'bilinear' Bilinear interpolation 'bicubic' Bicubic interpolation For example, these commands rotate an image 35° counterclockwise and use bilinear interpolation. I = imread('ic.tif'); J = imrotate(I,35,'bilinear'); imshow(I) figure, imshow(J) 4-8 Image Rotation Specifying the Size of the Output Image By default, imrotate creates an output image large enough to include the entire original image. Pixels that fall outside the boundaries of the original image are set to 0 and appear as a black background in the output image. If you specify the text string crop' as an argument, imrotate crops the output image to be the same size as the input image. (See the reference page for imrotate for an example of cropping.) 4-9 4 Spatial Transformations Image Cropping To extract a rectangular portion of an image, use the imcrop function. imcrop accepts two primary arguments: • The image to be cropped • The coordinates of a rectangle that defines the crop area If you call imcrop without specifying the crop rectangle, the cursor changes to a cross hair when it is over the image. Click on one corner of the region you want to select, and while holding down the mouse button, drag across the image. imcrop draws a rectangle around the area you are selecting. When you release the mouse button, imcrop creates a new image from the selected region. In this example, you display an image and call imcrop. The rectangle you select is shown in red. imshow ic.tif I = imcrop; The imcrop function waits for you to draw the cropping rectangle on the image. 4-10 Image Cropping Now call imshow to display the cropped image. If you call imcrop without specifying any output arguments, imcrop displays the image in a new figure. imshow(I) 4-11 4 Spatial Transformations Performing General Spatial Transformations To perform general 2-D spatial transformations, use the imtransform function. (For information about performing more advanced transformations, see “Advanced Spatial Transformation Techniques” on page 4-15.) The imtransform function accepts two primary arguments: • The image to be transformed • A spatial transformation structure, called a TFORM, that specifies the type of transformation you want to perform Specifying the Transformation Type You specify the type of transformation you want to perform in a TFORM structure. There are two ways to create a TFORM struct: • Using the maketform function • Using the cp2tform function Using maketform When you use the maketform function, you can specify the type of transformation you want to perform. The following table lists the types of transformations maketform supports. 4-12 Transformation Description 'affine' A transformation that may include translation, rotation, scaling, stretching and shearing. Straight lines remain straight, and parallel lines remain parallel, but rectangles may become parallelograms. 'projective' A transformation in which straight lines remain straight, but parallel lines converge toward “vanishing points.” (The vanishing points may fall inside or outside the image—even at infinity.) 'box' A special case of an affine transformation where each dimension is shifted and scaled independently. Performing General Spatial Transformations Transformation Description 'custom' A user-defined transformation, providing the forward and/or inverse functions that are called by imtransform. 'composite' A composition of two or more transformations. The 'custom' and 'composite' capabilities of maketform allow a virtually limitless variety of spatial transformations to be used with imtransform and/or tformarray. Using cp2tform You use cp2tform to create the TFORM when you want to perform a transformation that requires fitting of data points, such as a polynomial transformation. Chapter 5, “Image Registration” explains how to use the cp2tform function to fit a 2-D transformation to a set of control points selected in a pair of images. Note When used with imtransform, TFORM structures must define a 2-D spatial transformation. If an image contains more than two dimensions, such as an RGB image, the same 2-D transformation is automatically applied to all 2-D planes along the higher dimensions. For arbitrary-dimensional array transformations, use the tformarray function. 4-13 4 Spatial Transformations Performing the Transformation Once you define the transformation in a TFORM struct, you can perform the transformation by calling imtransform. For example, this code uses imtransform to perform a projective transformation of a checkerboard image. I = checkerboard(20,1,1); figure; imshow(I) T = maketform('projective',[1 1; 41 1; 41 41; 1 41],... [5 5; 40 5; 35 30; -10 30]); R = makeresampler('cubic','circular'); K = imtransform(I,T,R,'Size',[100 100],'XYScale',1); figure, imshow(K) Original image Transformed image The imtransform function options let you control many aspects of the transformation. For example, note how the transformed image appears to contain multiple copies of the original image. This is accomplished by using the 'Size' option, to make the output image larger than the input image, and then specifying a padding method that extends the input image by repeating the pixels in a circular pattern. The Image Processing Toolbox Image Transformation demos provide more examples of using the imtransform function, and related functions, to perform different types of spatial transformations. 4-14 Performing General Spatial Transformations Advanced Spatial Transformation Techniques The following functions, when used in combination, provide a vast array of options for defining and working with 2-D, N-D, and mixed-D spatial transformations: • maketform • fliptform • tformfwd • tforminv • findbounds • makeresampler • tformarray • imtransform The imtransform, findbounds, and tformarray functions use the tformfwd and tforminv functions internally to encapsulate the forward transformations needed to determine the extent of an output image or array and/or to map the output pixels/array locations back to input locations. You can use tformfwd and tforminv to explore the geometric effects of a transformation by applying them to points and lines and plotting the results. They support a consistent handling of both image and pointwise data. The previous example, “Performing the Transformation” on page 4-14, used the makeresampler function with a standard interpolation method. You can also use it to obtain special effects or custom processing. For example, you could specify your own separable filtering/interpolation kernel, build a custom resampler around the MATLAB interp2 or interp3 functions, or even implement an advanced anti-aliasing technique. And, as noted, you can use tformarray to work with arbitrary-dimensional array transformations. The arrays do not even need to have the same dimensions. The output can either have a lower or higher number of dimensions than the input. For example, if you are sampling 3-D data on a 2-D slice or manifold, the input array might have a lower dimensionality. The output dimensionality might be higher, for example, if you combine multiple 2-D transformations into a single 2-D to 3-D operation. 4-15 4 Spatial Transformations 4-16 5 Image Registration This section describes the image registration capabilities of the Image Processing Toolbox. Image registration is the process of aligning two or more images of the same scene. Image registration is often used as a preliminary step in other image processing applications. Topics covered include Terminology (p. 5-2) Provides definitions of image processing terms used in this section Registering an Image (p. 5-4) Steps you through an example of the image registration process Types of Supported Transformations (p. 5-13) Lists the types of supported transformations Selecting Control Points (p. 5-15) Describes how to use the Control Point Selection Tool (cpselect) to select control points in pairs of images Using Correlation to Improve Control Points (p. 5-33) Describes how to use the cpcorr function to fine-tune your control point selections 5 Image Registration Terminology An understanding of the following terms will help you to use this chapter. 5-2 Terms MATLAB Definition Aligned image The output image after registration has been performed. The output image is derived by applying a transformation to the input image (see below) that brings it into alignment with the base image (see below). Base image This is the image against which you compare the image to be registered. It is also often called the reference image. Control point pairs Matching locations, also referred to as landmarks, in the input image and the base image. Distortion The differences in one image as compared to another of the same subject. These may have occurred as a result of terrain relief and other changes in perspective when imaging the same scene from different viewpoints, lens and other internal sensor distortions, or differences between sensors and sensor types. Global transformation A transformation in which a single mathematical expression applies to an entire image. Input image This refers to the image that you wish to register. It is often called the observed image. Terminology Terms MATLAB Definition Local transformation A transformation in which different mathematical expressions (usually differing in parameters rather than form) apply to different regions within an image. Spatial transformation The mapping of locations of points in one image to new locations in another image. 5-3 5 Image Registration Registering an Image Image registration is the process of aligning two or more images of the same scene. Typically, one image, called the base image, is considered the reference to which the other images, called input images, are compared. The object of image registration is to bring the input image into alignment with the base image by applying a spatial transformation to the input image. A spatial transformation maps locations in one image to new locations in another image. (For more details, see Chapter 4, “Spatial Transformations.”) Determining the parameters of the spatial transformation needed to bring the images into alignment is key to the image registration process. Image registration is often used as a preliminary step in other image processing applications. For example, you can use image registration to align satellite images of the earth’s surface or images created by different medical diagnostic modalities (MRI and SPECT). After registration, you can compare features in the images to see how a river has migrated, how an area is flooded, or to see if a tumor is visible in an MRI or SPECT image. Point Mapping The Image Processing Toolbox provides tools to support point mapping to determine the parameters of the transformation required to bring an image into alignment with another image. In point mapping, you pick points in a pair of images that identify the same feature or landmark in the images. Then, a spatial mapping is inferred from the positions of these control points. Image registration using point mapping involves these steps: 1 Read the images into the MATLAB workspace. 2 Specify control point pairs in the images. 3 Save the control point pairs. 4 Fine tune the control points using cross-correlation. (This is an optional step.) 5 Specify the type of transformation to be used and infer its parameters from the control point pairs. 5-4 Registering an Image 6 Transform the unregistered image to bring it into alignment. This process is best understood by looking at an example. See “Example: Registering to a Digital Orthophoto” on page 5-6 for an extended example. Note You may need to perform several iterations of this process, experimenting with different types of transformations, before you achieve a satisfactory result. In some cases, you may perform successive registrations, removing gross global distortions first, and then removing smaller local distortions in subsequent passes. The following figure, Overview of Image Registration Process, provides a graphic illustration of this process. 5-5 5 Image Registration Image to be registered Input Image Base Image Image you are comparing it to Specify control points in input and base images using cpselect Fine tune control points with cpcorr (optional) Determine parameters of spatial transformation using cp2tform tform structure imtransform Aligned image Figure 5-1: Overview of Image Registration Process Example: Registering to a Digital Orthophoto This example registers a digital aerial photograph to a digital orthophoto covering the same area. Both images are centered on the business district of West Concord, Massachusetts. The aerial image is geometrically uncorrected: it includes camera perspective, terrain and building relief, and internal (lens) distortions, and it does not have any particular alignment or registration with respect to the earth. 5-6 Registering an Image The orthophoto, supplied by the Massachusetts Geographic Information System (MassGIS), has been orthorectified to remove camera, perspective, and relief distortions (via a specialized image transformation process). It is also georegistered (and geocoded)—the columns and rows of the digital orthophoto image are aligned to the axes of the Massachusetts State Plane coordinate system, each pixel center corresponds to a definite geographic location, and every pixel is 1 meter square in map units. Step 1: Read the Images into MATLAB In this example, the base image is westconcordorthophoto.png, the MassGIS georegistered orthophoto. It is a panchromatic (grayscale) image. The image to be registered is westconcordaerial.png, a digital aerial photograph supplied by mPower3/Emerge, and is a visible-color RGB image. orthophoto = imread('westconcordorthophoto.png'); figure, imshow(orthophoto) unregistered = imread('westconcordaerial.png'); figure, imshow(unregistered) You do not have to read the images into the MATLAB workspace. The cpselect function accepts file specifications for grayscale images. However, if you want to use cross-correlation to tune your control point positioning, the images must be in the workspace. 5-7 5 Image Registration Aerial Photo Image Orthophoto Image Step 2: Choose Control Points in the Images The toolbox provides an interactive tool, called the Control Point Selection Tool, that you can use to pick pairs of corresponding control points in both images. Control points are landmarks that you can find in both images, like a road intersection, or a natural feature. To start this tool, enter cpselect at the MATLAB prompt, specifying as arguments the input and base images. Note The unregistered image is an RGB image. Because the Control Point Selection Tool only accepts grayscale images, the example passes only one plane of the color image to cpselect. cpselect(unregistered(:,:,1),orthophoto) The cpselect function displays two views of both the input image and the base image in which you can pick control points by pointing and clicking. For more 5-8 Registering an Image information, see “Selecting Control Points” on page 5-15. This figure shows the Control Point Selection Tool with four pairs of control points selected. The number of control point pairs you pick is at least partially determined by the type of transformation you want to perform (specified in Step 5). See “Types of Supported Transformations” on page 5-13 for information about the minimum number of points required by each transformation. Step 3: Save the Control Point Pairs to the MATLAB Workspace In the Control Point Selection Tool, click on the File menu and choose the Save Points to Workspace option. See “Saving Control Points” on page 5-30 for more information. 5-9 5 Image Registration For example, the Control Point Selection Tool returns the following set of control points in the input image. These values represent spatial coordinates; the left column are x coordinates, the right column are y coordinates. input_points = 120.7086 93.9772 319.2222 78.9202 127.9838 291.6312 352.0729 281.1445 Step 4: Fine-tune the Control Point Pair Placement This is an optional step that uses cross-correlation to adjust the position of the control points you selected with cpselect. See “Using Correlation to Improve Control Points” on page 5-33 for more information. Note cpcorr can only adjust points for images that are the same scale and have the same orientation. The images cannot be rotated relative to each other. Because the Concord image is rotated in relation to the base image, cpcorr cannot tune the control points. When it cannot tune the points, cpcorr returns the input points, unmodified. input_points_corr = cpcorr(input_points,base_points,... unregistered(:,:,1),orthophoto) input_points_corr = 120.7086 93.9772 319.2222 78.9202 127.1046 289.8935 352.0729 281.1445 Step 5: Specify the Type of Transformation and Infer its Parameters In this step, you pass the control points to the cp2tform function that determines the parameters of the transformation needed to bring the image into alignment. cp2tform is a data-fitting function that determines the transformation based on the geometric relationship of the control points. cp2tform returns the parameters in a geometric transformation structure, called a TFORM structure. 5-10 Registering an Image When you use cp2tform, you must specify the type of transformation you want to perform. The cp2tform function can infer the parameters for five types of transformations. You must choose which transformation will correct the type of distortion present in the input image. See “Types of Supported Transformations” on page 5-13 for more information. Images can contain more than one type of distortion. The predominant distortion in the aerial image of West Concord (the input image) results from the camera perspective. Ignoring terrain relief, which is minor in this area, image registration can correct for this using a projective transformation. The projective transformation also rotates the image into alignment with the map coordinate system underlying the base image (the digital orthophoto). Other distortions could be corrected simultaneously by first creating a composite transformation with maketform, given sufficient information about the terrain and camera. (See “Performing General Spatial Transformations” on page 4-12 for more information.) mytform = cp2tform(input_points,base_points,'projective'); Step 6: Transform the Unregistered Image As the final step in image registration, you transform the input image to bring it into alignment with the base image. You use imtransform to perform the transformation, passing it the input image and the TFORM structure, which defines the transformation. imtransform returns the transformed image. For more information about using imtransform, see Chapter 4, “Spatial Transformations.” registered = imtransform(unregistered,mytform) Note The transformation defined in mytform, which is based on control points picked in only one plane of the RGB image, is applied to all three planes of the input image. 5-11 5 Image Registration Compare the transformed image to the base image to see how the registration came out. Registered Image 5-12 Orthophoto Image Types of Supported Transformations Types of Supported Transformations The cp2tform function can infer the parameters for six types of transformations. This table lists the transformations in order of complexity, with examples of each type of distortion. The first four transformations, 'linear conformal', 'affine', 'projective' and 'polynomial' are global transformations. In these transformations, a single mathematical expression applies to an entire image. The last two transformations, 'piecewise linear' and 'lwm' (local weighted mean), are local transformations. In these transformations, different mathematical expressions apply to different regions within an image. When exploring how different transformations affect the images you are working with, try the global transformations first. If these transformation are not satisfactory, try the local transformations; the piecewise linear transformation first and then the local weighted mean transformation. Transformation Type Description Minimum Control Points 'linear conformal' Use this transformation when shapes in the input image are unchanged, but the image is distorted by some combination of translation, rotation, and scaling. Straight lines remain straight, and parallel lines are still parallel. 2 pairs 'affine' Use this transformation when shapes in the input image exhibit shearing. Straight lines remain straight, and parallel lines remain parallel, but rectangles become parallelograms. 3 pairs Example 5-13 5 Image Registration 5-14 'projective' Use this transformation when the scene appears “tilted.” Straight lines remain straight, but parallel lines converge toward “vanishing points” (which may or may not fall within the image. 4 pairs 'polynomial' Use this transformation when objects in the image are curved. The higher the order of the polynomial, the better the fit, but the result can contain more curves than the base image. 6 pairs (order 2) 'piecewise linear' Use this transformation when parts of the image appear distorted differently. 4 pairs 'lwm' Use this transformation (local weighted mean), when the distortion varies locally and piecewise linear is not sufficient. 6 pairs (12 pairs recommended) 10 pairs (order 3) 16 pairs (order 4) Selecting Control Points Selecting Control Points The toolbox includes an interactive tool that enables you to specify control points in the images you want to register. The tool displays the images side-by-side. When you are satisfied with the number and placement of the control points, you can save the control points. Using the Control Point Selection Tool To specify control points in a pair of images you want to register, use the Control Point Selection Tool, cpselect. The tool displays the image you want to register, called the input image, next to the image you want to compare it to, called the base image or reference image. Specifying control points is a four-step process: 1 Start the tool, specifying the input image and the base image. 2 View the images, looking for visual elements that you can identify in both images. cpselect provides many ways to navigate around the image, panning and zooming to view areas of the image in more detail. 3 Specify matching control point pairs in the input image and the base image. 4 Save the control points in the MATLAB workspace. Figure 5-2, Control Point Selection Tool shows the default appearance of the tool when you first start it. To get more information about any part of the interface, click on it in this figure. 5-15 5 Image Registration Use menu options to save points, delete points, and customize the tool Select points Use point prediction Zoom in and out Move the detail image Lock relative magnification of images Default cursor Specify magnification Detail views Overview windows Detail rectangle Figure 5-2: Control Point Selection Tool Starting the Control Point Selection Tool To use the Control Point Selection Tool, enter the cpselect command at the MATLAB prompt. As arguments, specify the image you want to register (the input image), and the image you want to compare it to (the base image). To illustrate, this code fragment reads an image into a variable, moon_base, in the MATLAB workspace. It then creates another version of the image with a 5-16 Selecting Control Points deliberate size distortion, called moon_input. This is the image that needs registration to remove the size distortion. The code then starts the cpselect tool, specifying the two images. moon_base = imread('moon.tif ); moon_input = imresize(moon_base, 1.2); cpselect(moon_input, moon_base); The cpselect command has other optional arguments. For example, you can restart a control point selection session by including a cpstruct structure as the third argument. For more information about restarting sessions, see “Saving Control Points” on page 5-30. For complete details, see the cpselect reference page. Default Views of the Images When the Control Point Selection Tool starts, it contains four image display windows. The top two windows are called the Detail windows. These windows show a closeup view of a portion of the images you are working with. The input image is on the left and the base image is on the right. The two windows at the bottom of the interface are called the Overview windows. These windows show the images in their entirety, at the largest scale that fits the window. The input overview image is on the left and the base overview image is on the right. Superimposed on the image in the Overview windows is a rectangle, called the Detail Rectangle. This rectangle defines the part of the image that is visible in the Detail window. By default, at startup, the Detail Rectangle covers one quarter of the entire image and is positioned over the center of the image. 5-17 5 Image Registration Input Detail windows Base Overview windows Detail rectangles Viewing the Images By default, cpselect displays the entire base and input images in the Overview windows and displays a close up view of a portion of these images in the Detail windows. However, to find visual elements that are common to both images, you may want to change the section of the image displayed in the detail view or zoom in on a part of the image to view it in more detail. The following sections describe the different ways to change your view of the images. • “Using Scroll Bars to View Other Parts of an Image” on page 5-19 • “Using the Detail Rectangle to Change the View” on page 5-19 • “Moving the Image Displayed in the Detail Window” on page 5-19 • “Zooming In and Out on an Image” on page 5-20 • “Specifying the Magnification of the Images” on page 5-21 5-18 Selecting Control Points • “Locking the Relative Magnification of the Input and Base Images” on page 5-22 Using Scroll Bars to View Other Parts of an Image To view parts of an image that are not visible in the Detail or Overview windows, use the scroll bars provided in each window. As you scroll the image in the Detail window, note how the Detail Rectangle moves over the image in the Overview window. The position of the Detail Rectangle always shows the portion of the image in the Detail window. Using the Detail Rectangle to Change the View To get a closer view of any part of the image, move the Detail Rectangle in the Overview window over that section of the image. cpselect displays that section of the image in the Detail window at a higher magnification than the overview window. To move the Detail Rectangle: 1 Click on the default cursor button in the button bar. 2 Move the pointer into the Detail Rectangle. The cursor changes to the fleur shape, ,indicating the directions in which it can be moved. 3 Press and hold the mouse button to drag the Detail Rectangle anywhere on the image. Note As you move the Detail Rectangle over the image in the Overview window, the view of the image displayed in the Detail window changes. Moving the Image Displayed in the Detail Window To change the section of the image displayed in the Detail window, use the Hand tool to move the image in the window. To use the Hand tool: 1 Click on the Hand tool button in the button bar. 5-19 5 Image Registration 2 Move the pointer over the image in the Detail window. The cursor changes shape, , indicating the directions in which it can be moved. 3 Press and hold the mouse button and drag the image in the Detail window. Note As you move the image in the Detail window, the Detail Rectangle in the Overview window moves. Zooming In and Out on an Image To enlarge an image to get a closer look or shrink an image to see the whole image in context, use the Zoom buttons on the button bar. (You can also zoom in or out on an image by changing the magnification. See “Specifying the Magnification of the Images” on page 5-21 for more information.) To zoom in or zoom out on the base or input images: 1 Click the appropriate “magnifying glass” button. Zoom in Zoom out 2 Move the pointer over the image you want to zoom in or out on. The cursor changes to the cross-hair cursor . You can zoom in or out on either the input or the base images, in either the Detail or Overview windows. To keep the relative magnifications of the base and input images synchronized, click the Lock ratio check box. See “Locking the Relative Magnification of the Input and Base Images” on page 5-22 for more information. 5-20 Selecting Control Points Note If you zoom in close on the image displayed in the Overview window, the Detail Rectangle may no longer be visible. You can use the zoom tool in two ways: - Position the cursor over a location in the image and click the mouse. With each click, cpselect changes the magnification of the image by a preset amount. (See “Specifying the Magnification of the Images” on page 5-21 for a list of some of these magnifications.) cpselect centers the new view of the image on the spot where you clicked. - Alternately, you can position the cursor over a location in the image and, while pressing and holding the mouse button, draw a rectangle defining the area you want to zoom in or out on. cpselect magnifies the image so that the chosen section fills the Detail window. cpselect resizes the Detail Rectangle in the Overview window as well. Note When you zoom-in or -out on an image, notice how the magnification value changes. Specifying the Magnification of the Images To enlarge an image to get a closer look or to shrink an image to see the whole image in context, use the magnification edit box. (You can also use the Zoom buttons to enlarge or shrink an image. See “Zooming In and Out on an Image” on page 5-20 for more information.) To change the magnification of an image: 1 Move the cursor into the magnification edit box of the window you want to change. The cursor changes to the text entry cursor. Note Each Detail window and Overview window has its own magnification edit box. 5-21 5 Image Registration 2 Type a new value in the magnification edit box and press Enter, or click on the menu associated with the edit box and choose from a list of preset magnifications. cpselect changes the magnification of the image and displays the new view in the appropriate window. Magnification edit box Magnification menu Locking the Relative Magnification of the Input and Base Images To keep the relative magnification of the input and base images automatically synchronized in the Detail or Overview windows, click on the Lock Ratio checkbox. The two Detail windows and the two Overview windows each have their own Lock ratio checkboxes. When the Lock Ratio check box is selected, cpselect changes the magnification of both the input and base images when you zoom-in or -out on either one of the images or specify a magnification value for either of the images. Lock magnification ratio checkbox Specifying Matching Control Point Pairs The primary function of the Control Point Selection Tool is to enable you to pick control points in the image to be registered, the input image, and the image to 5-22 Selecting Control Points which you are comparing it, the base image. When you start cpselect, the point selection tool is enabled, by default. You specify control points by pointing and clicking in the input and base images, in either the Detail or the Overview windows. Each point you specify in the input image must have a match in the base image. The following sections describe the ways you can use the Control Point Selection Tool to choose control point pairs: • “Picking Control Point Pairs Manually” • “Using Control Point Prediction” on page 5-25 This section also describes how to move control points after you’ve created them and how to delete control points. Picking Control Point Pairs Manually To specify a pair of control points in your images: 1 Click on the Control point selection button . Point selection mode is active by default. 2 Position the cursor over a feature you have visually selected in any of the images displayed. The cursor changes to a pointing finger . You can pick control points in either of the Detail windows, input or base, or in either of the Overview windows, input or base. You also can work in either direction: input-to-base image, or base-to-input image. 3 Click the mouse button. cpselect places a control point symbol at the position you specified, in both the Detail window and the Overview window. (The appearance of the control point symbol indicates its current state. Initially, control points are in an active, unmatched state. See “Control Point States” for more information. Note Depending on where in the image you pick control points, the symbol for the point may be visible in the Overview window, but not in the Detail window. 5-23 5 Image Registration 4 To create the match for this control point, move the cursor into the corresponding Detail or Overview window. (For example, if you started in an input window, move the cursor to a base window.) 5 Click the mouse button. cpselect places a control point symbol at the position you specified, in both the Detail and Overview windows. Because this control point completes a pair, the appearance of this symbol indicates an active, matched state. Note that the appearance of the first control point you selected (in step 3) also changes to an active, matched state. You pick pairs of control points by moving from a view of the input image to a view of the base image, or vice versa. You can pick several control points in one view of the image, and then move to the corresponding window to locate their matches. To match an unmatched control point, select it to make it active, and then pick a point in the corresponding view window. When you select a match for a control point, the symbols for both points change to indicate their matched state. You can move or delete control points after you create them. The following figure illustrates control points in several states. 5-24 Selecting Control Points Active unmatched Unmatched Matched Using Control Point Prediction Instead of picking matching control points by moving the cursor between corresponding Detail or Overview windows, you can let the Control Point Selection Tool estimate the match for the control points you specify, automatically. The Control Point Selection Tool determines the position of the matching control point based on the geometric relationship of the previously selected control points. 5-25 5 Image Registration Note By default, the Control Point Selection Tool does not include predicted points in the set of valid control points returned in input_points or base_points. To include predicted points, you must accept them by selecting the points and fine-tuning their position with the cursor. When you move a predicted point, the Control Point Selection Tool changes the symbol to indicate that it has changed to a standard control point. For more information, see “Moving Control Points” on page 5-28. To illustrate point prediction, this figure shows four control points selected in the input image, where the points form the four corners of a square. (The control points selections in the figure do not attempt to identify any landmarks in the image.) The figure shows the picking of a fourth point, in the left window, and the corresponding predicted point in the right window. Note how the Control Point Selection Tool places the predicted point at the same location relative to the other control points, forming the bottom right corner of the square. Control point selected manually Predicted control point Note Because the Control Point Selection Tool predicts control point locations based on the locations of the previous control points, you cannot use point prediction until you have a minimum of two pairs of matched points. Until this minimum is met, the point prediction button is disabled. 5-26 Selecting Control Points To use control point prediction: 1 Click on the Control point prediction button . 2 Position the cursor anywhere in any of the images displayed. The cursor changes to a pointing finger . You can pick control points in either of the Detail windows, input or base, or in either of the Overview windows, input or base. You also can work in either direction: input-to-base image or base-to-input image. 3 Click either mouse button. The Control Point Selection Tool places a control point symbol at the position you specified and places another control point symbol for a matching point in all the other windows. The symbol for the predicted point contains the letter “P”, indicating that it’s a predicted control point. This figure illustrates predicted points in active unmatched, matched, and predicted states. For a complete description of all point states, see “Control Point States” on page 5-27. Predicted control point Active predicted control point Control Point States The appearance of control point symbols indicate their current state. When you first pick a control point, its state is active and unmatched. When you pick the 5-27 5 Image Registration match for a control point, the appearance of both symbols changes to indicate their matched status. This table lists all the possible control point states with their symbols. cpselect displays this list in a separate window called a Legend. The Legend is visible by default, but you can control it’s visibility using the Legend option from the View menu. Table 5-1: Control Point States Symbol State Description Active unmatched The point is currently selected but does not have a matching point. This is the initial state of most points. Active matched The point is currently selected and has a matching point. Active predicted The point is a predicted point. If you move its position, the point changes to active matched state. Unmatched The point is not selected and it is unmatched. You must select it before you can create its matching point. Matched The point has a matching point. Predicted This point was added by cpselect during point prediction. Moving Control Points To move a control point 1 Click on the Control point selection button or the default cursor button . 2 Position the cursor over the control point you want to move. 3 Press and hold the mouse button and drag the control point. The state of the control point changes to active when you click on it. 5-28 Selecting Control Points If you move a predicted control point, the state of the control point changes to a regular (nonpredicted) control point. Deleting Control Points To delete a control point, and optionally its matching point: 1 Click on the Control point selection button or the default cursor button . 2 Click on the control point you want to delete. Its state changes to active. If the control point has a match, both points become active. 3 Delete the point (or points) using one of these methods: - Pressing the Backspace key - Pressing the Delete key - Choosing one of the delete options from the Edit menu. Using this menu you can delete individual points or pairs of matched points, in the input or base images. Delete options Undoing and Redoing Control Point Selections You can undo a deletion or series of deletions using the Undo Delete option on the cpselect Edit menu. Undo options After undoing a deletion, you can delete the points again using the Redo option, also on the Edit menu. 5-29 5 Image Registration Saving Control Points After you specify control point pairs, you must save them in the MATLAB workspace to make them available for the next step in image registration, processing by cp2tform. To save control points to the MATLAB workspace: 1 Select File on the Control Point Selection Tool menu bar. 2 Choose the Save Points to Workspace option. The Control Point Selection Tool displays this dialog box: By default, the Control Point Selection Tool saves the x and y coordinates that specify the locations of the control points you selected in two arrays named input_points and base_points, although you can specify other names. These are n-by-2 arrays, where n is the number of valid control point pairs you selected. For example, this is an example of the input_points array if you picked four pairs of control points. The values in the left column represent the x coordinates; the values in the right column represent the y coordinates. input_points = 215.6667 225.7778 156.5556 270.8889 262.3333 311.3333 340.1111 368.8889 Whenever you exit the Control Point Selection Tool, it asks if you want to save your control points. 5-30 Selecting Control Points Saving Your Control Point Selection Session To save the current state of the Control Point Selection Tool, select the Structure with all points check box in the Save Points to Workspace dialog box. 5-31 5 Image Registration This option saves the position of all the control points you specified and their current state in a cpstruct structure. cpstruct = inputPoints: basePoints: inputBasePairs: ids: inputIdPairs: baseIdPairs: isInputPredicted: isBasePredicted: [4x2 [4x2 [4x2 [4x1 [4x2 [4x2 [4x1 [4x1 double] double] double] double] double] double] double] double] You can use the cpstruct to restart a control point selection session at the point where you left off. This option is useful if you are picking many points over a long time and want to preserve unmatched and predicted points when you resume work. The Control Point Selection Tool does not include unmatched and predicted points in the input_points and base_points arrays. To extract the arrays of valid control point coordinates from a cpstruct, use the cpstruct2pairs function. 5-32 Using Correlation to Improve Control Points Using Correlation to Improve Control Points You may want to fine-tune the control points you selected using cpselect. Points selected by eye using the interactive tool can sometimes be improved using cross-correlation. To use cross-correlation, pass sets of control points in the input and base images, along with the images themselves, to the cpcorr function. input_pts_adj= cpcorr(input_points, base_points, input, base); The cpcorr function defines 11-by-11 regions around each control point in the input image and around the matching control point in the base image, and then calculates the correlation between the values at each pixel in the region. Next, the cpcorr function looks for the position with the highest correlation value and uses this as the optimal position of the control point. The cpcorr function only moves control points up to 4 pixels based on the results of the cross correlation. Note Features in the two images must be at the same scale and have the same orientation. They cannot be rotated relative to each other. If cpcorr cannot correlate some of the control points, it returns their values in input_points, unmodified. 5-33 5 Image Registration 5-34 6 Neighborhood and Block Operations This section discusses these generic block processing functions. Topics covered include Terminology (p. 6-2) Provides definitions of image processing terms used in this section Block Processing Operations (p. 6-3) Provides an overview of the types of block processing operations supported by the toolbox Sliding Neighborhood Operations (p. 6-5) Defines sliding neighborhood operations and describes how you can use them to implement many types of filtering operations Distinct Block Operations (p. 6-9) Describes block operations Column Processing (p. 6-12) Describes how to process sliding neighborhoods or distinct blocks as columns 6 Neighborhood and Block Operations Terminology An understanding of the following terms will help you to use this chapter. 6-2 Terms Definitions Block operation An operation in which an image is processed in blocks rather than all at once. The blocks have the same size across the image. Some operation is applied to one block at a time. The blocks are reassembled to form an output image. Border padding Additional rows and columns temporarily added to the border(s) of an image when some of the blocks extend outside the image. The additional rows and columns normally contain zeros. Center pixel The pixel at the center of a neighborhood. Column processing An operation in which neighborhoods are reshaped into columns before processing in order to speed up computation time. Distinct block operation A block operation in which the blocks do not overlap. Neighborhood operation An operation in which each output pixel is computed from a set of neighboring input pixels. Convolution, dilation, and median filtering are examples of neighborhood operations. A neighborhood operation can also be called a sliding neighborhood operation. Overlap Extra rows and columns of pixels outside a block whose values are taken into account when processing the block. These extra pixels cause distinct blocks to overlap one another. The blkproc function enables you to specify an overlap. Block Processing Operations Block Processing Operations Certain image processing operations involve processing an image in sections called blocks, rather than processing the entire image at once. The Image Processing Toolbox provides several functions for specific operations that work with blocks, for example, the imdilate function for image dilation. In addition, the toolbox provides more generic functions for processing an image in blocks. This chapter discusses these generic block processing functions. To use one of the functions, you supply information about the size of the blocks, and specify a separate function to use to process the blocks. The block processing function does the work of breaking the input image into blocks, calling the specified function for each block, and reassembling the results into an output image. Types of Block Processing Operations Using these functions, you can perform various block processing operations, including sliding neighborhood operations and distinct block operations. • In a sliding neighborhood operation, the input image is processed in a pixelwise fashion. That is, for each pixel in the input image, some operation is performed to determine the value of the corresponding pixel in the output image. The operation is based on the values of a block of neighboring pixels. • In a distinct block operation, the input image is processed a block at a time. That is, the image is divided into rectangular blocks, and some operation is performed on each block individually to determine the values of the pixels in the corresponding block of the output image. In addition, the toolbox provides functions for column processing operations. These operations are not actually distinct from block operations; instead, they are a way of speeding up block operations by rearranging blocks into matrix columns. Note that even if you do not use these block processing functions, the information here may be useful to you, as it includes concepts fundamental to many areas of image processing. In particular, the discussion of sliding neighborhood operations is applicable to linear filtering and morphological 6-3 6 Neighborhood and Block Operations operations. See Chapter 7, “Linear Filtering and Filter Design” and Chapter 9, “Morphological Operations” for information about these applications. 6-4 Sliding Neighborhood Operations Sliding Neighborhood Operations A sliding neighborhood operation is an operation that is performed a pixel at a time, with the value of any given pixel in the output image being determined by applying some algorithm to the values of the corresponding input pixel’s neighborhood. A pixel’s neighborhood is some set of pixels, defined by their locations relative to that pixel, which is called the center pixel. The neighborhood is a rectangular block, and as you move from one element to the next in an image matrix, the neighborhood block slides in the same direction. The following figure shows the neighborhood blocks for some of the elements in a 6-by-5 matrix with 2-by-3 sliding blocks. The center pixel for each neighborhood is marked with a dot. Figure 6-1: Neighborhood Blocks in a 6-by-5 Matrix The center pixel is the actual pixel in the input image being processed by the operation. If the neighborhood has an odd number of rows and columns, the center pixel is actually in the center of the neighborhood. If one of the dimensions has even length, the center pixel is just to the left of center or just above center. For example, in a 2-by-2 neighborhood, the center pixel is the upper left one. For any m-by-n neighborhood, the center pixel is floor(([m n]+1)/2) In the 2-by-3 block shown in Figure 6-1, the center pixel is (1,2), or, the pixel in the second column of the top row of the neighborhood. 6-5 6 Neighborhood and Block Operations To perform a sliding neighborhood operation: 1 Select a single pixel. 2 Determine the pixel’s neighborhood. 3 Apply a function to the values of the pixels in the neighborhood. This function must return a scalar. 4 Find the pixel in the output image whose position corresponds to that of the center pixel in the input image. Set this output pixel to the value returned by the function. 5 Repeat steps 1 through 4 for each pixel in the input image. For example, suppose Figure 6-1 represents an averaging operation. The function might sum the values of the six neighborhood pixels and then divide by 6. The result is the value of the output pixel. Padding Borders As Figure 6-1 shows, some of the pixels in a neighborhood may be missing, especially if the center pixel is on the border of the image. Notice that in the figure, the upper left and bottom right neighborhoods include “pixels” that are not part of the image. To process these neighborhoods, sliding neighborhood operations pad the borders of the image, usually with 0’s. In other words, these functions process the border pixels by assuming that the image is surrounded by additional rows and columns of 0’s. These rows and columns do not become part of the output image and are used only as parts of the neighborhoods of the actual pixels in the image. Linear and Nonlinear Filtering You can use sliding neighborhood operations to implement many kinds of filtering operations. One example of a sliding neighbor operation is convolution, which is used to implement linear filtering. MATLAB provides the conv and filter2 functions for performing convolution, and the toolbox provides the imfilter function. See Chapter 7, “Linear Filtering and Filter Design” for more information about these functions. 6-6 Sliding Neighborhood Operations In addition to convolution, there are many other filtering operations you can implement through sliding neighborhoods. Many of these operations are nonlinear in nature. For example, you can implement a sliding neighborhood operation where the value of an output pixel is equal to the standard deviation of the values of the pixels in the input pixel’s neighborhood. You can use the nlfilter function to implement a variety of sliding neighborhood operations. nlfilter takes as input arguments an image, a neighborhood size, and a function that returns a scalar, and returns an image of the same size as the input image. The value of each pixel in the output image is computed by passing the corresponding input pixel’s neighborhood to the function. For example, this call computes each output pixel by taking the standard deviation of the values of the input pixel’s 3-by-3 neighborhood (that is, the pixel itself and its eight contiguous neighbors). I2 = nlfilter(I,[3 3],'std2'); You can write an M-file to implement a specific function, and then use this function with nlfilter. For example, this command processes the matrix I in 2-by-3 neighborhoods with a function called myfun.m. nlfilter(I,[2 3],@myfun); @myfun is an example of a function handle. You can also use an inline function. For example, f = inline('sqrt(min(x(:)))'); I2 = nlfilter(I,[2 2],f); The example below uses nlfilter to set each pixel to the maximum value in its 3-by-3 neighborhood. I = imread('tire.tif'); f = inline('max(x(:))'); I2 = nlfilter(I,[3 3],f); imshow(I); figure, imshow(I2); 6-7 6 Neighborhood and Block Operations Figure 6-2: Each Output Pixel Set to Maximum Input Neighborhood Value Many operations that nlfilter can implement run much faster if the computations are performed on matrix columns rather than rectangular neighborhoods. For information about this approach, see the reference page for colfilt. Note nlfilter is an example of a “function function.” For more information on how to use this kind of function, see Function Functions in the MATLAB documentation. For more information on inline functions, see inline in the MATLAB Function Reference. For more information on function handles, see function_handle in the MATLAB Function Reference. 6-8 Distinct Block Operations Distinct Block Operations Distinct blocks are rectangular partitions that divide a matrix into m-by-n sections. Distinct blocks overlay the image matrix starting in the upper-left corner, with no overlap. If the blocks don’t fit exactly over the image, the toolbox adds zero padding so that they do. Figure 6-3 shows a 15-by-30 matrix divided into 4-by-8 blocks. Figure 6-3: An Image Divided into Distinct Blocks The zero padding process adds 0’s to the bottom and right of the image matrix, as needed. After zero padding, the matrix is size 16-by-32. The function blkproc performs distinct block operations. blkproc extracts each distinct block from an image and passes it to a function you specify. blkproc assembles the returned blocks to create an output image. For example, the command below processes the matrix I in 4-by-6 blocks with the function myfun. I2 = blkproc(I,[4 6],@myfun); You can specify the function as an inline function. For example, f = inline('mean2(x)*ones(size(x))'); I2 = blkproc(I,[4 6],f); 6-9 6 Neighborhood and Block Operations The example below uses blkproc to set every pixel in each 8-by-8 block of an image matrix to the average of the elements in that block. I = imread('tire.tif'); f = inline('uint8(round(mean2(x)*ones(size(x))))'); I2 = blkproc(I,[8 8],f); imshow(I) figure, imshow(I2); Notice that inline computes the mean of the block and then multiplies the result by a matrix of ones, so that the output block is the same size as the input block. As a result, the output image is the same size as the input image. blkproc does not require that the images be the same size; however, if this is the result you want, you must make sure that the function you specify returns blocks of the appropriate size. Note blkproc is an example of a “function function.” For more information on how to use this kind of function, see the Function Functions section in the MATLAB documentation. Overlap When you call blkproc to define distinct blocks, you can specify that the blocks overlap each other, that is, you can specify extra rows and columns of pixels outside the block whose values are taken into account when processing the block. When there is an overlap, blkproc passes the expanded block (including the overlap) to the specified function. 6-10 Distinct Block Operations Figure 6-4 shows the overlap areas for some of the blocks in a 15-by-30 matrix with 1-by-2 overlaps. Each 4-by-8 block has a one-row overlap above and below, and a two-column overlap on each side. In the figure, shading indicates the overlap. The 4-by-8 blocks overlay the image matrix starting in the upper-left corner. Figure 6-4: An Image Divided into Distinct Blocks With Specified Overlaps To specify the overlap, you provide an additional input argument to blkproc. To process the blocks in the figure above with the function myfun, the call is B = blkproc(A,[4 8],[1 2],@myfun) Overlap often increases the amount of zero padding needed. For example, in Figure 6-3, the original 15-by-30 matrix became a 16-by-32 matrix with zero padding. When the 15-by-30 matrix includes a 1-by-2 overlap, the padded matrix becomes an 18-by-36 matrix. The outermost rectangle in the figure delineates the new boundaries of the image after padding has been added to accommodate the overlap plus block processing. Notice that in the figure above, padding has been added to the left and top of the original image, not just to the right and bottom. 6-11 6 Neighborhood and Block Operations Column Processing The toolbox provides functions that you can use to process sliding neighborhoods or distinct blocks as columns. This approach is useful for operations that MATLAB performs columnwise; in many cases, column processing can reduce the execution time required to process an image. For example, suppose the operation you are performing involves computing the mean of each block. This computation is much faster if you first rearrange the blocks into columns, because you can compute the mean of every column with a single call to the mean function, rather than calling mean for each block individually. You can use the colfilt function to implement column processing. This function: 1 Reshapes each sliding or distinct block of an image matrix into a column in a temporary matrix 2 Passes the temporary matrix to a function you specify 3 Rearranges the resulting matrix back into the original shape Sliding Neighborhoods For a sliding neighborhood operation, colfilt creates a temporary matrix that has a separate column for each pixel in the original image. The column corresponding to a given pixel contains the values of that pixel’s neighborhood from the original image. Figure 6-5 illustrates this process. In this figure, a 6-by-5 image matrix is processed in 2-by-3 neighborhoods. colfilt creates one column for each pixel in the image, so there are a total of 30 columns in the temporary matrix. Each pixel’s column contains the value of the pixels in its neighborhood, so there are six rows. colfilt zero pads the input image as necessary. For example, the neighborhood of the upper left pixel in the figure has two zero-valued neighbors, due to zero padding. 6-12 Column Processing Figure 6-5: colfilt Creates a Temporary Matrix for Sliding Neighborhood The temporary matrix is passed to a function, which must return a single value for each column. (Many MATLAB functions work this way, for example, mean, median, std, sum, etc.) The resulting values are then assigned to the appropriate pixels in the output image. colfilt can produce the same results as nlfilter with faster execution time; however, it may use more memory. The example below sets each output pixel to the maximum value in the input pixel’s neighborhood, producing the same result as the nlfilter example shown in Figure 6-2. I2 = colfilt(I,[3 3],'sliding',@max); Distinct Blocks For a distinct block operation, colfilt creates a temporary matrix by rearranging each block in the image into a column. colfilt pads the original image with 0’s, if necessary, before creating the temporary matrix. Figure 6-6 illustrates this process. In this figure, a 6-by-16 image matrix is processed in 4-by-6 blocks. colfilt first zero pads the image to make the size 8-by-18 (six 4-by-6 blocks), and then rearranges the blocks into 6 columns of 24 elements each. 6-13 6 Neighborhood and Block Operations Figure 6-6: colfilt Creates a Temporary Matrix for Distinct Block Operation After rearranging the image into a temporary matrix, colfilt passes this matrix to the function. The function must return a matrix of the same size as the temporary matrix. If the block size is m-by-n, and the image is mm-by-nn, the size of the temporary matrix is (m*n)-by-(ceil(mm/m)*ceil(nn/n)). After the function processes the temporary matrix, the output is rearranged back into the shape of the original image matrix. This example sets all the pixels in each 8-by-8 block of an image to the mean pixel value for the block, producing the same result as the blkproc example in “Distinct Block Operations” on page 6-9. 6-14 Column Processing I = im2double(imread('tire.tif')); f = inline('ones(64,1)*mean(x)'); I2 = colfilt(I,[8 8],'distinct',f); Notice that the inline function computes the mean of the block and then multiplies the result by a vector of ones, so that the output block is the same size as the input block. As a result, the output image is the same size as the input image. Restrictions You can use colfilt to implement many of the same distinct block operations that blkproc performs. However, colfilt has certain restrictions that blkproc does not: • The output image must be the same size as the input image. • The blocks cannot overlap. For situations that do not satisfy these constraints, use blkproc. 6-15 6 Neighborhood and Block Operations 6-16 7 Linear Filtering and Filter Design The Image Processing Toolbox provides a number of functions for designing and implementing two-dimensional linear filters for image data. This section describes these functions and how to use them effectively. Topics covered include Terminology (p. 7-2) Provides definitions of image processing terms used in this section Linear Filtering (p. 7-4) Provides an explanation of linear filtering and how it is implemented in the toolbox. This topic describes filtering in terms of the spatial domain, and is accessible to anyone doing image processing. Filter Design (p. 7-16) Discusses designing two-dimensional finite impulse response (FIR) filters. This section assumes you are familiar with working in the frequency domain. 7 Linear Filtering and Filter Design Terminology An understanding of the following terms will help you to use this chapter. Note that this table includes brief definitions of terms related to filter design; a detailed discussion of these terms and the theory behind filter design is outside the scope of this User’s Guide. 7-2 Terms Definitions Convolution A neighborhood operation in which each output pixel is a weighted sum of neighboring input pixels. The weights are defined by the convolution kernel. Image processing operations implemented with convolution include smoothing, sharpening, and edge enhancement. Convolution kernel A matrix of weights used to perform convolution. A convolution kernel is a correlation kernel that has been rotated 180 degrees. Correlation A neighborhood operation in which each output pixel is a weighted sum of neighboring input pixels. The weights are defined by the correlation kernel. Correlation is closely related mathematically to convolution. Correlation kernel A matrix of weights used to perform correlation. The filter design functions in the Image Processing Toolbox return correlation kernels. A correlation kernel is a convolution kernel that has been rotated 180 degrees. FIR filter A filter whose response to a single point, or impulse, has finite extent. FIR stands for finite impulse response. An FIR filter can be implemented using convolution. All filter design functions in the Image Processing Toolbox return FIR filters. Frequency response A mathematical function describing the gain of a filter in response to different input frequencies. Neighborhood operation An operation in which each output pixel is computed from a set of neighboring input pixels. Convolution, dilation, and median filtering are examples of neighborhood operations. Terminology Terms Definitions Ripples Oscillations around a constant value. The frequency response of a practical filter often has ripples where the frequency response of an ideal filter is flat. Window method A filter design method that multiples the ideal impulse response by a window function, which tapers the ideal impulse response. The resulting filter’s frequency response approximates a desired frequency response. 7-3 7 Linear Filtering and Filter Design Linear Filtering Filtering is a technique for modifying or enhancing an image. For example, you can filter an image to emphasize certain features or remove other features. Filtering is a neighborhood operation, in which the value of any given pixel in the output image is determined by applying some algorithm to the values of the pixels in the neighborhood of the corresponding input pixel. A pixel’s neighborhood is some set of pixels, defined by their locations relative to that pixel. (See Chapter 6, “Neighborhood and Block Operations”, for a general discussion of neighborhood operations.) Linear filtering is filtering in which the value of an output pixel is a linear combination of the values of the pixels in the input pixel’s neighborhood. This section discusses linear filtering in MATLAB and the Image Processing Toolbox. It includes: • A description of filtering, using convolution and correlation • A description of how to use the imfilter function to perform filtering • A discussion about using predefined filter types See “Filter Design” on page 7-16 for information about how to design filters. Convolution Linear filtering of an image is accomplished through an operation called convolution. In convolution, the value of an output pixel is computed as a weighted sum of neighboring pixels. The matrix of weights is called the convolution kernel, also known as the filter. For example, suppose the image is A = [17 23 4 10 11 24 5 6 12 18 1 7 13 19 25 8 14 20 21 2 15 16 22 3 9] and the convolution kernel is 7-4 Linear Filtering h = [8 3 4 1 5 9 6 7 2] Then Figure 7-1 shows how to compute the (2,4) output pixel using these steps: 1 Rotate the convolution kernel 180 degrees about its center element. 2 Slide the center element of the convolution kernel so that lies on top of the (2,4) element of A. 3 Multiply each weight in the rotated convolution kernel by the pixel of A underneath. 4 Sum up the individual products from step 3. Hence the (2,4) output pixel is 1 ⋅ 2 + 8 ⋅ 9 + 15 ⋅ 4 + 7 ⋅ 7 + 14 ⋅ 5 + 16 ⋅ 3 + 13 ⋅ 6 + 20 ⋅ 1 + 22 ⋅ 8 = 575 Values of rotated convolution kernel Image pixel values 17 24 1 23 5 7 2 7 6 8 9 5 14 1 15 4 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9 3 Center of kernel 8 Figure 7-1: Computing the (2,4) Output of Convolution 7-5 7 Linear Filtering and Filter Design Correlation The operation called correlation is closely related to convolution. In correlation, the value of an output pixel is also computed as a weighted sum of neighboring pixels. The difference is that the matrix of weights, in this case called the correlation kernel, is not rotated during the computation. Figure 7-2 shows how to compute the (2,4) output pixel of the correlation of A, assuming h is correlation kernel instead of a convolution kernel, using these steps: 1 Slide the center element of the correlation kernel so that lies on top of the (2,4) element of A. 2 Multiply each weight in the correlation kernel by the pixel of A underneath. 3 Sum up the individual products from step 3. The (2,4) output pixel from the correlation is 1 ⋅ 8 + 8 ⋅ 1 + 15 ⋅ 6 + 7 ⋅ 3 + 14 ⋅ 5 + 16 ⋅ 7 + 13 ⋅ 4 + 20 ⋅ 9 + 22 ⋅ 2 = 585 Values of correlation kernel Image pixel values 17 24 1 23 5 7 8 3 4 1 8 14 5 9 15 16 4 6 13 10 12 19 21 3 11 18 25 2 9 20 6 22 7 Center of kernel 2 Figure 7-2: Computing the (2,4) Output of Correlation 7-6 Linear Filtering Filtering Using imfilter Filtering of images, either by correlation or convolution, can be performed using the toolbox function imfilter. This example filters the image in the file blood1.tif with a 5-by-5 filter containing equal weights. Such a filter is often called an averaging filter. I = imread('blood1.tif'); h = ones(5,5) / 25; I2 = imfilter(I,h); imshow(I), title('Original image') figure, imshow(I2), title('Filtered image') Original Image Filtered Image Data Types The imfilter function handles data types similar to the way the image arithmetic functions do, as described in “Image Arithmetic Truncation Rules” on page 2-22. The output image has the same data type, or numeric class, as the input image. The imfilter function computes the value of each output pixel using double-precision, floating-point arithmetic. If the result exceeds the range of the data type, the imfilter function truncates the result to that data type's allowed range. If it is an integer data type, imfilter rounds fractional values. Because of the truncation behavior, you may sometimes want to consider converting your image to a different data type before calling imfilter. In this example, the output of imfilter has negative values when the input is of class double. 7-7 7 Linear Filtering and Filter Design A = magic(5) A = 17 23 4 10 11 24 5 6 12 18 1 7 13 19 25 8 14 20 21 2 15 16 22 3 9 14 9 9 -16 -16 -8 -14 -20 -21 -2 h = [-1 0 1] h = -1 0 1 imfilter(A,h) ans = 24 5 6 12 18 -16 -16 9 9 14 -16 9 14 9 -16 Notice that the result has negative values. Now suppose A was of class uint8, instead of double. A = uint8(magic(5)); imfilter(A,h) ans = 24 5 6 12 18 0 0 9 9 14 0 9 14 9 0 14 9 9 0 0 0 0 0 0 0 Since the input to imfilter is of class uint8, the output also is of class uint8, and so the negative values are truncated to 0. In such cases, it may be appropriate to convert the image to another type, such as a signed integer type, single, or double, before calling imfilter. 7-8 Linear Filtering Correlation and Convolution Options The imfilter function can perform filtering using either correlation or convolution. It uses correlation by default, because the filter design functions, described in “Filter Design” on page 7-16, and the fspecial function, described in “Using Predefined Filter Types” on page 7-14, produce correlation kernels. However, if you want to perform filtering using convolution instead, you can pass the string 'conv' as optional input argument to imfilter. For example, A = magic(5); h = [-1 0 1] imfilter(A,h) ans = 24 5 6 12 18 -16 -16 9 9 14 % filter using correlation -16 9 14 9 -16 imfilter(A,h,'conv') 14 9 9 -16 -16 -8 -14 -20 -21 -2 % filter using convolution ans = -24 -5 -6 -12 -18 16 16 -9 -9 -14 16 -9 -14 -9 16 -14 -9 -9 16 16 8 14 20 21 2 7-9 7 Linear Filtering and Filter Design Boundary Padding Options When computing an output pixel at the boundary of an image, a portion of the convolution or correlation kernel is usually off the edge of the image, as illustrated in the figure below. What value should these outside pixels have? ? 8 3 1 ? 5 ? 17 24 1 23 5 7 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9 4 8 14 9 15 6 7 16 Center of kernel 2 Figure 7-3: When the Values of the Kernel Fall Outside the Image The imfilter function normally fills in these “off-the-edge” image pixels by assuming that they are 0. This is called zero-padding and is illustrated in the figure below. 7-10 Linear Filtering Outside pixels are assumed to be 0. 0 17 24 1 8 3 4 0 8 1 5 9 6 0 7 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9 Center of kernel 2 Figure 7-4: Zero-Padding of Outside Pixels When filtering an image, zero-padding can result in a dark band around the edge of the image, as shown in this example. I = imread('blood1.tif'); h = ones(5,5)/25; I2 = imfilter(I,h); imshow(I), title('Original image') figure, imshow(I2), title('Filtered image') 7-11 7 Linear Filtering and Filter Design Original Image Image with Black Border To eliminate the zero-padding artifacts around the edge of the image, imfilter offers an alternative boundary padding method called border replication. In border replication, the value of any pixel outside the image is determined by replicating the value from the nearest border pixel. This is illustrated in the figure below. These pixel values are replicated from boundary pixels. 1 17 24 1 8 3 4 8 8 1 5 9 15 15 23 5 7 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9 14 16 6 7 Center of kernel 2 Figure 7-5: Replicated Boundary Pixels 7-12 Linear Filtering To filter using border replication, pass the additional optional argument 'replicate' to imfilter. I3 = imfilter(I,h,'replicate'); figure, imshow(I3), title('Filtered with border replication') Image Border with Replication The imfilter function supports other boundary padding options, such as 'circular' and 'symmetric'. See the reference page for imfilter for details. Multidimensional Filtering The imfilter function can handle both multidimensional images and multidimensional filters. A convenient property of filtering is that filtering a three-dimensional image with a two-dimensional filter is equivalent to filtering each plane of the three-dimensional image individually with the same two-dimensional filter. This property makes it easy, for example, to filter each color plane of a truecolor image with the same filter. rgb = imread('flowers.tif'); h = ones(5,5) / 25; rgb2 = imfilter(rgb,h); imshow(rgb), title('Original image') figure, imshow(rgb2), title('Filtered image') 7-13 7 Linear Filtering and Filter Design Original Image Filtered Image Relationship to Other Filtering Functions MATLAB has several two-dimensional and multidimensional filtering functions. The function filter2 performs two-dimensional correlation; conv2 performs two-dimensional convolution; and convn performs multidimensional convolution. Each of these other filtering functions always converts the input to double, and the output is always double. Also, each of these other filtering functions always assumes the input is zero-padded, and they do not support other padding options. In contrast, the imfilter function does not convert input images to double. The imfilter function also offers a flexible set of boundary padding options, as described in “Boundary Padding Options” on page 7-10. Using Predefined Filter Types The fspecial function produces several kinds of predefined filters, in the form of correlation kernels. After creating a filter with fspecial, you can apply it directly to your image data using imfilter. This example illustrates applying an unsharp masking filter to an intensity image. The unsharp masking filter has the effect of making edges and fine detail in the image more crisp. 7-14 Linear Filtering I = imread('moon.tif'); h = fspecial('unsharp'); I2 = imfilter(I,h); imshow(I), title('Original image') figure, imshow(I2), title('Filtered image') Original Image Filtered Image 7-15 7 Linear Filtering and Filter Design Filter Design This section describes working in the frequency domain to design filters. Topics discussed include: • Finite impulse response (FIR) filters, the class of linear filter that the toolbox supports • The frequency transformation method, which transforms a one-dimensional FIR filter into a two-dimensional FIR filter • The frequency sampling method, which creates a filter based on a desired frequency response • The windowing method, which multiplies the ideal impulse response with a window function to generate the filter • Creating the desired frequency response matrix • Computing the frequency response of a filter This section assumes you are familiar with working in the frequency domain. This topic is discussed in many signal processing and image processing textbooks. Note Most of the design methods described in this section work by creating a two-dimensional filter from a one-dimensional filter or window created using functions from the Signal Processing Toolbox. Although this toolbox is not required, you may find it difficult to design filters in the Image Processing Toolbox if you do not have the Signal Processing Toolbox as well. FIR Filters The Image Processing Toolbox supports one class of linear filter, the two-dimensional finite impulse response (FIR) filter. FIR filters have several characteristics that make them ideal for image processing in the MATLAB environment: • FIR filters are easy to represent as matrices of coefficients. • Two-dimensional FIR filters are natural extensions of one-dimensional FIR filters. 7-16 Filter Design • There are several well-known, reliable methods for FIR filter design. • FIR filters are easy to implement. • FIR filters can be designed to have linear phase, which helps prevent distortion. Another class of filter, the infinite impulse response (IIR) filter, is not as suitable for image processing applications. It lacks the inherent stability and ease of design and implementation of the FIR filter. Therefore, this toolbox does not provide IIR filter support. Frequency Transformation Method The frequency transformation method transforms a one-dimensional FIR filter into a two-dimensional FIR filter. The frequency transformation method preserves most of the characteristics of the one-dimensional filter, particularly the transition bandwidth and ripple characteristics. This method uses a transformation matrix, a set of elements that defines the frequency transformation. The toolbox function ftrans2 implements the frequency transformation method. This function’s default transformation matrix produces filters with nearly circular symmetry. By defining your own transformation matrix, you can obtain different symmetries. (See Jae S. Lim, Two-Dimensional Signal and Image Processing, 1990, for details.) The frequency transformation method generally produces very good results, as it is easier to design a one-dimensional filter with particular characteristics than a corresponding two-dimensional filter. For instance, the next example designs an optimal equiripple one-dimensional FIR filter and uses it to create a two-dimensional filter with similar characteristics. The shape of the one-dimensional frequency response is clearly evident in the two-dimensional response. b = remez(10,[0 0.4 0.6 1],[1 1 0 0]); h = ftrans2(b); [H,w] = freqz(b,1,64,'whole'); colormap(jet(64)) plot(w/pi-1,fftshift(abs(H))) figure, freqz2(h,[32 32]) 7-17 7 Linear Filtering and Filter Design 1.4 1.2 1.4 1.2 1 Magnitude 1 0.8 0.6 0.8 0.6 0.4 0.2 0.4 0 1 1 0.5 0.2 0.5 0 0 −0.5 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Frequency −0.5 −1 −1 Frequency Figure 7-6: A One-Dimensional Frequency Response (left) and the Corresponding Two-Dimensional Frequency Response (right) Frequency Sampling Method The frequency sampling method creates a filter based on a desired frequency response. Given a matrix of points that defines the shape of the frequency response, this method creates a filter whose frequency response passes through those points. Frequency sampling places no constraints on the behavior of the frequency response between the given points; usually, the response ripples in these areas. The toolbox function fsamp2 implements frequency sampling design for two-dimensional FIR filters. fsamp2 returns a filter h with a frequency response that passes through the points in the input matrix Hd. The example below creates an 11-by-11 filter using fsamp2, and plots the frequency response of the resulting filter. (The freqz2 function in this example calculates the two-dimensional frequency response of a filter. See “Computing the Frequency Response of a Filter” on page 7-21 for more information.) Hd = zeros(11,11); Hd(4:8,4:8) = 1; [f1,f2] = freqspace(11,'meshgrid'); mesh(f1,f2,Hd), axis([-1 1 -1 1 0 1.2]), colormap(jet(64)) h = fsamp2(Hd); figure, freqz2(h,[32 32]), axis([-1 1 -1 1 0 1.2]) 7-18 Filter Design 1 0.8 Magnitude 1 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 1 0 1 1 0.5 0.5 0 1 0.5 0.5 0 0 −0.5 0 −0.5 −0.5 −1 −1 Frequency −0.5 −1 −1 Frequency Figure 7-7: Desired Two-Dimensional Frequency Response (left) and Actual Two-Dimensional Frequency Response (right) Notice the ripples in the actual frequency response, compared to the desired frequency response. These ripples are a fundamental problem with the frequency sampling design method. They occur wherever there are sharp transitions in the desired response. You can reduce the spatial extent of the ripples by using a larger filter. However, a larger filter does not reduce the height of the ripples, and requires more computation time for filtering. To achieve a smoother approximation to the desired frequency response, consider using the frequency transformation method or the windowing method. Windowing Method The windowing method involves multiplying the ideal impulse response with a window function to generate a corresponding filter. Like the frequency sampling method, the windowing method produces a filter whose frequency response approximates a desired frequency response. The windowing method, however, tends to produce better results than the frequency sampling method. The toolbox provides two functions for window-based filter design, fwind1 and fwind2. fwind1 designs a two-dimensional filter by using a two-dimensional window that it creates from one or two one-dimensional windows that you specify. fwind2 designs a two-dimensional filter by using a specified two-dimensional window directly. 7-19 7 Linear Filtering and Filter Design fwind1 supports two different methods for making the two-dimensional windows it uses: • Transforming a single one-dimensional window to create a two-dimensional window that is nearly circularly symmetric, by using a process similar to rotation • Creating a rectangular, separable window from two one-dimensional windows, by computing their outer product The example below uses fwind1 to create an 11-by-11 filter from the desired frequency response Hd. Here, the hamming function from the Signal Processing Toolbox is used to create a one-dimensional window, which fwind1 then extends to a two-dimensional window. Hd = zeros(11,11); Hd(4:8,4:8) = 1; [f1,f2] = freqspace(11,'meshgrid'); mesh(f1,f2,Hd), axis([-1 1 -1 1 0 1.2]), colormap(jet(64)) h = fwind1(Hd,hamming(11)); figure, freqz2(h,[32 32]), axis([-1 1 -1 1 0 1.2]) 1 0.8 Magnitude 1 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 1 0 1 0.5 1 0.5 0 0.5 1 0.5 0 0 −0.5 0 −0.5 −0.5 −1 −1 Frequency −0.5 −1 −1 Frequency Figure 7-8: Desired Two-Dimensional Frequency Response (left) and Actual Two-Dimensional Frequency Response (right) Creating the Desired Frequency Response Matrix The filter design functions fsamp2, fwind2, and fwind2 all create filters based on a desired frequency response magnitude matrix. You can create an appropriate desired frequency response matrix using the freqspace function. 7-20 Filter Design freqspace returns correct, evenly spaced frequency values for any size response. If you create a desired frequency response matrix using frequency points other than those returned by freqspace, you may get unexpected results, such as nonlinear phase. For example, to create a circular ideal lowpass frequency response with cutoff at 0.5 use [f1,f2] = freqspace(25,'meshgrid'); Hd = zeros(25,25); d = sqrt(f1.^2 + f2.^2) < 0.5; Hd(d) = 1; mesh(f1,f2,Hd) 1 0.8 0.6 0.4 0.2 0 1 0.5 1 0.5 0 0 −0.5 −0.5 −1 −1 Figure 7-9: Ideal Circular Lowpass Frequency Response Note that for this frequency response, the filters produced by fsamp2, fwind1, and fwind2 are real. This result is desirable for most image processing applications. To achieve this in general, the desired frequency response should be symmetric about the frequency origin (f1 = 0, f2 = 0). Computing the Frequency Response of a Filter The freqz2 function computes the frequency response for a two-dimensional filter. With no output arguments, freqz2 creates a mesh plot of the frequency response. For example, consider this FIR filter, h =[0.1667 0.6667 0.1667 0.6667 -3.3333 0.6667 0.1667 0.6667 0.1667]; 7-21 7 Linear Filtering and Filter Design This command computes and displays the 64-by-64 point frequency response of h. freqz2(h) 6 5 Magnitude 4 3 2 1 0 1 0.5 1 0.5 0 0 −0.5 Frequency −0.5 −1 −1 Frequency Figure 7-10: The Frequency Response of a Two-Dimensional Filter To obtain the frequency response matrix H and the frequency point vectors f1 and f2, use output arguments [H,f1,f2] = freqz2(h); freqz2 normalizes the frequencies f1 and f2 so that the value 1.0 corresponds to half the sampling frequency, or π radians. For a simple m-by-n response, as shown above, freqz2 uses the two-dimensional fast Fourier transform function fft2. You can also specify vectors of arbitrary frequency points, but in this case freqz2 uses a slower algorithm. See “Fourier Transform” on page 8-3 for more information about the fast Fourier transform and its application to linear filtering and filter design. 7-22 8 Transforms The usual mathematical representation of an image is a function of two spatial variables: f ( x, y ) . The value of the function at a particular location ( x, y ) represents the intensity of the image at that point. The term transform refers to an alternative mathematical representation of an image. This section defines several important transforms and shows examples of their application to image processing. Topics covered include Terminology (p. 8-2) Provides definitions of image processing terms used in this section Fourier Transform (p. 8-3) Defines the Fourier transform and some of its applications in image processing Discrete Cosine Transform (p. 8-16) Describes the Discrete Cosine Transform (DCT) of an image and its application, particularly in image compression Radon Transform (p. 8-20) Describes how the Image Processing Toolbox radon function computes projections of an image matrix along specified directions. 8 Transforms Terminology An understanding of the following terms will help you to use this chapter. Note that this table includes brief definitions of terms related to transforms; a detailed discussion of these terms and the theory behind transforms is outside the scope of this User’s Guide. 8-2 Terms Definitions Discrete transform A transform whose input and output values are discrete samples, making it convenient for computer manipulation. Discrete transforms implemented by MATLAB and the Image Processing Toolbox include the discrete Fourier transform (DFT) and the discrete cosine transform (DCT). Frequency domain The domain in which an image is represented by a sum of periodic signals with varying frequency. Inverse transform An operation that when performed on a transformed image, produces the original image. Spatial domain The domain in which an image is represented by intensities at given points in space. This is the most common representation for image data. Transform An alternative mathematical representation of an image. For example, the Fourier transform is a representation of an image as a sum of complex exponentials of varying magnitudes, frequencies, and phases. Transforms are useful for a wide range of purposes, including convolution, enhancement, feature detection, and compression. Fourier Transform Fourier Transform The Fourier transform is a representation of an image as a sum of complex exponentials of varying magnitudes, frequencies, and phases. The Fourier transform plays a critical role in a broad range of image processing applications, including enhancement, analysis, restoration, and compression. This section includes the following subsections: • “Definition of Fourier Transform” • “Discrete Fourier Transform” on page 8-8, including a discussion of fast Fourier transform • “Applications” on page 8-11 (sample applications using Fourier transforms) Definition of Fourier Transform If f ( m, n ) is a function of two discrete spatial variables m and n, then we define the two-dimensional Fourier transform of f ( m, n ) by the relationship F ( ω 1, ω 2 ) = ∞ ∞ ∑ ∑ f ( m, n )e – jω 1 m e – jω 2 n m = –∞ n = –∞ The variables ω1 and ω2 are frequency variables; their units are radians per sample. F ( ω 1, ω 2 ) is often called the frequency-domain representation of f ( m, n ) . F ( ω 1, ω 2 ) is a complex-valued function that is periodic both in ω 1 and ω 2 , with period 2π . Because of the periodicity, usually only the range – π ≤ ω 1, ω 2 ≤ π is displayed. Note that F ( 0, 0 ) is the sum of all the values of f ( m, n ) . For this reason, F ( 0, 0 ) is often called the constant component or DC component of the Fourier transform. (DC stands for direct current; it is an electrical engineering term that refers to a constant-voltage power source, as opposed to a power source whose voltage varies sinusoidally.) The inverse two-dimensional Fourier transform is given by 1 f ( m, n ) = --------2 4π π ∫ω π 1 = –π ∫ω 2 = –π F ( ω 1, ω 2 )e jω 1 m jω 2 n e dω 1 dω 2 Roughly speaking, this equation means that f ( m, n ) can be represented as a sum of an infinite number of complex exponentials (sinusoids) with different 8-3 8 Transforms frequencies. The magnitude and phase of the contribution at the frequencies ( ω 1, ω 2 ) are given by F ( ω 1, ω 2 ) . Example Consider a function f ( m, n ) that equals 1 within a rectangular region and 0 everywhere else. n f(m,n) m Figure 8-1: A Rectangular Function To simplify the diagram, f ( m, n ) is shown as a continuous function, even though the variables m and n are discrete. 8-4 Fourier Transform Figure 8-2 shows the magnitude of the Fourier transform, F ( ω 1, ω 2 ) , of Figure 8-1 as a mesh plot. The mesh plot of the magnitude is a common way to visualize the Fourier transform. ω 2 ( ve r t i ca l fr e que n cy ) ω1 on (horiz ta l f r e quency ) Figure 8-2: Magnitude Image of a Rectangular Function The peak at the center of the plot is F ( 0, 0 ) , which is the sum of all the values in f ( m, n ) . The plot also shows that F ( ω 1, ω 2 ) has more energy at high horizontal frequencies than at high vertical frequencies. This reflects the fact that horizontal cross sections of f ( m, n ) are narrow pulses, while vertical cross sections are broad pulses. Narrow pulses have more high-frequency content than broad pulses. 8-5 8 Transforms Another common way to visualize the Fourier transform is to display log F ( ω 1, ω 2 ) as an image, as in 5 4 3 ω2 2 1 0 1 ω1 Figure 8-3: The Log of the Fourier Transform of a Rectangular Function Using the logarithm helps to bring out details of the Fourier transform in regions where F ( ω 1, ω 2 ) is very close to 0. 8-6 Fourier Transform Examples of the Fourier transform for other simple shapes are shown below. Figure 8-4: Fourier Transforms of Some Simple Shapes 8-7 8 Transforms Discrete Fourier Transform Working with the Fourier transform on a computer usually involves a form of the transform known as the discrete Fourier transform (DFT). There are two principal reasons for using this form: • The input and output of the DFT are both discrete, which makes it convenient for computer manipulations. • There is a fast algorithm for computing the DFT known as the fast Fourier transform (FFT). The DFT is usually defined for a discrete function f ( m, n ) that is nonzero only over the finite region 0 ≤ m ≤ M – 1 and 0 ≤ n ≤ N – 1 . The two-dimensional M-by-N DFT and inverse M-by-N DFT relationships are given by F ( p, q ) = M–1 N–1 ∑ ∑ f ( m, n )e – j ( 2π ⁄ M )pm – j ( 2π ⁄ N )qn e p = 0, 1, …, M – 1 q = 0, 1, …, N – 1 m=0 n=0 M–1 N–1 1 f ( m, n ) = ---------MN ∑ ∑ F ( p, q )e j ( 2π ⁄ M )pm j ( 2π ⁄ N )qn e m = 0, 1, …, M – 1 n = 0, 1, …, N – 1 p=0 q=0 The values F ( p, q ) are the DFT coefficients of f ( m, n ) . The zero-frequency coefficient, F ( 0, 0 ) is often called the “DC component.” DC is an electrical engineering term that stands for direct current. (Note that matrix indices in MATLAB always start at 1 rather than 0; therefore, the matrix elements f(1,1) and F(1,1) correspond to the mathematical quantities f ( 0, 0 ) and F ( 0, 0 ) , respectively.) The MATLAB functions fft, fft2, and fftn implement the fast Fourier transform algorithm for computing the one-dimensional DFT, two-dimensional DFT, and N-dimensional DFT, respectively. The functions ifft, ifft2, and ifftn compute the inverse DFT. 8-8 Fourier Transform Relationship to the Fourier Transform The DFT coefficients F ( p, q ) are samples of the Fourier transform F ( ω 1, ω 2 ) . F ( p, q ) = F ( ω 1, ω 2 ) ω 1 = 2πp ⁄ M ω 2 = 2πq ⁄ N p = 0, 1, …, M – 1 q = 0, 1, …, N – 1 Example Let’s construct a matrix f that is similar to the function f(m,n) in the example in “Definition of Fourier Transform” on page 8-4. Remember that f(m,n) is equal to 1 within the rectangular region and 0 elsewhere. We use a binary image to represent f(m,n). f = zeros(30,30); f(5:24,13:17) = 1; imshow(f,'notruesize') Compute and visualize the 30-by-30 DFT of f with these commands. F = fft2(f); F2 = log(abs(F)); imshow(F2,[-1 5],'notruesize'); colormap(jet); colorbar 8-9 8 Transforms 5 4 3 2 1 0 1 Figure 8-5: A Discrete Fourier Transform Computed Without Padding This plot differs from the Fourier transform displayed on Figure 8-3. First, the sampling of the Fourier transform is much coarser. Second, the zero-frequency coefficient is displayed in the upper-left corner instead of the traditional location in the center. We can obtain a finer sampling of the Fourier transform by zero-padding f when computing its DFT. The zero-padding and DFT computation can be performed in a single step with this command. F = fft2(f,256,256); This command zero-pads f to be 256-by-256 before computing the DFT. imshow(log(abs(F)),[-1 5]); colormap(jet); colorbar 8-10 Fourier Transform 5 4 3 2 1 0 1 Figure 8-6: A Discrete Fourier Transform Computed with Padding The zero-frequency coefficient, however, is still displayed in the upper-left corner rather than the center. You can fix this problem by using the function fftshift, which swaps the quadrants of F so that the zero-frequency coefficient is in the center. F = fft2(f,256,256); F2 = fftshift(F); imshow(log(abs(F2)),[-1 5]); colormap(jet); colorbar The resulting plot is identical to the one on Figure 8-3. Applications This section presents a few of the many image processing-related applications of the Fourier transform. Frequency Response of Linear Filters The Fourier transform of the impulse response of a linear filter gives the frequency response of the filter. The function freqz2 computes and displays a filter’s frequency response. The frequency response of the Gaussian 8-11 8 Transforms convolution kernel shows that this filter passes low frequencies and attenuates high frequencies. h = fspecial('gaussian'); freqz2(h) 1 0.9 Magnitude 0.8 0.7 0.6 0.5 0.4 0.3 1 0.5 1 0.5 0 0 0.5 Frequency 0.5 1 1 Frequency Figure 8-7: The Frequency Response of a Gaussian Filter See “Linear Filtering and Filter Design” on page 7-1 for more information about linear filtering, filter design, and frequency responses. Fast Convolution A key property of the Fourier transform is that the multiplication of two Fourier transforms corresponds to the convolution of the associated spatial functions. This property, together with the fast Fourier transform, forms the basis for a fast convolution algorithm. Suppose that A is an M-by-N matrix and B is a P-by-Q matrix. The convolution of A and B can be computed using the following steps: 1 Zero-pad A and B so that they are at least (M+P–1)-by-(N+Q–1). (Often A and B are zero-padded to a size that is a power of 2 because fft2 is fastest for these sizes.) 8-12 Fourier Transform 2 Compute the two-dimensional DFT of A and B using fft2. 3 Multiply the two DFTs together. 4 Using ifft2, compute the inverse two-dimensional DFT of the result from step 3. For example, A = magic(3); B = ones(3); A(8,8) = 0; B(8,8) = 0; C = ifft2(fft2(A).*fft2(B)); C = C(1:5,1:5); C = real(C) % Remove imaginary % Zero pad A to be 8 by 8 % Zero pad B to be 8 by 8 % Extract the nonzero portion part caused by roundoff error C = 8.0000 11.0000 15.0000 7.0000 4.0000 9.0000 17.0000 30.0000 21.0000 13.0000 15.0000 30.0000 45.0000 30.0000 15.0000 7.0000 19.0000 30.0000 23.0000 11.0000 6.0000 13.0000 15.0000 9.0000 2.0000 The FFT-based convolution method is most often used for large inputs. For small inputs it is generally faster to use imfilter. Locating Image Features The Fourier transform can also be used to perform correlation, which is closely related to convolution. Correlation can be used to locate features within an image; in this context correlation is often called template matching. For instance, suppose you want to locate occurrences of the letter “a” in an image containing text. This example reads in text.tif and creates a template image by extracting a letter “a” from it. bw = imread('text.tif'); a=bw(59:71,81:91); %Extract one of the letters imshow(bw); figure, imshow(a); a from the image. 8-13 8 Transforms Figure 8-8: An Image (left) and the Template to Correlate (right) The correlation of the image of the letter “a” with the larger image can be computed by first rotating the image of “a” by 180o and then using the FFT-based convolution technique described above. (Note that convolution is equivalent to correlation if you rotate the convolution kernel by 180o.) To match the template to the image, you can use the fft2 and ifft2 functions. C = real(ifft2(fft2(bw) .* fft2(rot90(a,2),256,256))); figure, imshow(C,[])%Display, scaling data to appropriate range. max(C(:)) %Find max pixel value in C. ans = 51.0000 thresh = 45; %Use a threshold that s a little less than max. figure, imshow(C > thresh)%Display showing pixels over threshold. Figure 8-9: A Correlated Image (left) and its Thresholded Result (right) 8-14 Fourier Transform The left image above is the result of the correlation; bright peaks correspond to occurrences of the letter. The locations of these peaks are indicated by the white spots in the thresholded correlation image shown on the right. Note that you could also have created the template image by zooming in on the image and using the interactive version of imcrop. For example, with text.tif displayed in the current figure window, enter zoom on a = imcrop To determine the coordinates of features in an image, you can use the pixval function. 8-15 8 Transforms Discrete Cosine Transform The discrete cosine transform (DCT) represents an image as a sum of sinusoids of varying magnitudes and frequencies. The dct2 function in the Image Processing Toolbox computes the two-dimensional discrete cosine transform (DCT) of an image. The DCT has the property that, for a typical image, most of the visually significant information about the image is concentrated in just a few coefficients of the DCT. For this reason, the DCT is often used in image compression applications. For example, the DCT is at the heart of the international standard lossy image compression algorithm known as JPEG. (The name comes from the working group that developed the standard: the Joint Photographic Experts Group.) The two-dimensional DCT of an M-by-N matrix A is defined as follows. M–1 N–1 B pq = α p α q ∑ ∑ π ( 2m + 1 )p π ( 2n + 1 )q A mn cos ------------------------------- cos ----------------------------- , 2M 2N 0≤p≤M–1 0≤q≤N–1 m=0 n=0 1 ⁄ M, p = 0 αp = 2 ⁄ M, 1 ≤ p ≤ M – 1 1 ⁄ N, q = 0 αq = 2 ⁄ N, 1 ≤ q ≤ N – 1 The values B pq are called the DCT coefficients of A. (Note that matrix indices in MATLAB always start at 1 rather than 0; therefore, the MATLAB matrix elements A(1,1) and B(1,1) correspond to the mathematical quantities A 00 and B 00 , respectively.) The DCT is an invertible transform, and its inverse is given by M–1 N–1 A mn = ∑ ∑ π ( 2m + 1 )p π ( 2n + 1 )q α p α q B pq cos ------------------------------- cos ----------------------------- , 2M 2N 0≤m≤M–1 0≤n≤N–1 p=0 q=0 1 ⁄ M, p = 0 αp = 2 ⁄ M, 1 ≤ p ≤ M – 1 8-16 1 ⁄ N, q = 0 αq = 2 ⁄ N, 1 ≤ q ≤ N – 1 Discrete Cosine Transform The inverse DCT equation can be interpreted as meaning that any M-by-N matrix A can be written as a sum of MN functions of the form π ( 2m + 1 )p π ( 2n + 1 )q α p α q cos ------------------------------- cos ----------------------------- , 2M 2N 0≤p≤M–1 0≤q≤N–1 These functions are called the basis functions of the DCT. The DCT coefficients B pq , then, can be regarded as the weights applied to each basis function. For 8-by-8 matrices, the 64 basis functions are illustrated by this image. Figure 8-10: The 64 Basis Functions of an 8-by-8 Matrix Horizontal frequencies increase from left to right, and vertical frequencies increase from top to bottom. The constant-valued basis function at the upper left is often called the DC basis function, and the corresponding DCT coefficient B 00 is often called the DC coefficient. The DCT Transform Matrix The Image Processing Toolbox offers two different ways to compute the DCT. The first method is to use the function dct2. dct2 uses an FFT-based algorithm for speedy computation with large inputs. The second method is to use the DCT transform matrix, which is returned by the function dctmtx and may be more efficient for small square inputs, such as 8-by-8 or 16-by-16. The M-by-M transform matrix T is given by 8-17 8 Transforms T pq 1 -------- = M 2 π ( 2q + 1 )p ----- cos ----------------------------M 2M p = 0, 0 ≤ q ≤ M – 1 1 ≤ p ≤ M – 1, 0 ≤ q ≤ M – 1 For an M-by-M matrix A, T*A is an M-by-M matrix whose columns contain the one-dimensional DCT of the columns of A. The two-dimensional DCT of A can be computed as B=T*A*T'. Since T is a real orthonormal matrix, its inverse is the same as its transpose. Therefore, the inverse two-dimensional DCT of B is given by T'*B*T. The DCT and Image Compression In the JPEG image compression algorithm, the input image is divided into 8-by-8 or 16-by-16 blocks, and the two-dimensional DCT is computed for each block. The DCT coefficients are then quantized, coded, and transmitted. The JPEG receiver (or JPEG file reader) decodes the quantized DCT coefficients, computes the inverse two-dimensional DCT of each block, and then puts the blocks back together into a single image. For typical images, many of the DCT coefficients have values close to zero; these coefficients can be discarded without seriously affecting the quality of the reconstructed image. The example code below computes the two-dimensional DCT of 8-by-8 blocks in the input image; discards (sets to zero) all but 10 of the 64 DCT coefficients in each block; and then reconstructs the image using the two-dimensional inverse DCT of each block. The transform matrix computation method is used. I = imread('cameraman.tif'); I = im2double(I); T = dctmtx(8); B = blkproc(I,[8 8],'P1*x*P2',T,T'); mask = [1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]; 8-18 Discrete Cosine Transform B2 = blkproc(B,[8 8],'P1.*x',mask); I2 = blkproc(B2,[8 8],'P1*x*P2',T',T); imshow(I), figure, imshow(I2) Although there is some loss of quality in the reconstructed image, it is clearly recognizable, even though almost 85% of the DCT coefficients were discarded. To experiment with discarding more or fewer coefficients, and to apply this technique to other images, try running the demo function dctdemo. 8-19 8 Transforms Radon Transform The radon transform represents an image as a collection of projections along various directions. It is used in areas ranging from seismology to computer vision The radon function in the Image Processing Toolbox computes projections of an image matrix along specified directions. A projection of a two-dimensional function f(x,y) is a line integral in a certain direction. For example, the line integral of f(x,y) in the vertical direction is the projection of f(x,y) onto the x-axis; the line integral in the horizontal direction is the projection of f(x,y) onto the y-axis. Figure 8-11 shows horizontal and vertical projections for a simple two-dimensional function. f(x,y) x Projection onto the y-axis y Projection onto the x-axis Figure 8-11: Horizontal and Vertical Projections of a Simple Function Projections can be computed along any angle θ. In general, the Radon transform of f(x,y) is the line integral of f parallel to the y′ axis 8-20 Radon Transform R θ ( x′ ) = ∞ ∫–∞ f (x′ cos θ – y′ sin θ, x′ sin θ + y′ cos θ) dy′ where x′ = cos θ sin θ x y′ – sin θ cos θ y The figure below illustrates the geometry of the Radon transform. y′ y x′ f(x,y) q x x′ Rq(x′) Figure 8-12: The Geometry of the Radon Transform This command computes the Radon transform of I for the angles specified in the vector theta 8-21 8 Transforms [R,xp] = radon(I,theta); The columns of R contain the Radon transform for each angle in theta. The vector xp contains the corresponding coordinates along the x′-axis. The “center pixel” of I is defined to be floor((size(I)+1)/2); this is the pixel on the x′-axis corresponding to x′ = 0 . The commands below compute and plot the Radon transform at 0° and 45° of an image containing a single square object. I = zeros(100,100); I(25:75, 25:75) = 1; imshow(I) [R,xp] = radon(I,[0 45]); figure; plot(xp,R(:,1)); title('R_{0^o} (x\prime)') figure; plot(xp,R(:,2)); title('R_{45^o} (x\prime)') Figure 8-13: Two Radon Transforms of a Square Function 8-22 Radon Transform Note xp is the same for all projection angles. The Radon transform for a large number of angles is often displayed as an image. In this example, the Radon transform for the square image is computed at angles from 0° to 180°, in 1° increments. theta = 0:180; [R,xp] = radon(I,theta); imagesc(theta,xp,R); title('R_{\theta} (X\prime)'); xlabel('\theta (degrees)'); ylabel('X\prime'); set(gca,'XTick',0:20:180); colormap(hot); colorbar Rθ (X′) 70 60 60 40 50 20 x’ 40 0 30 20 20 40 10 60 0 20 40 60 80 100 θ (degrees) 120 140 160 180 Figure 8-14: A Radon Transform Using 180 Projections 8-23 8 Transforms Using the Radon Transform to Detect Lines The Radon transform is closely related to a common computer vision operation known as the Hough transform. You can use the radon function to implement a form of the Hough transform used to detect straight lines. The steps are: 1 Compute a binary edge image using the edge function. I = imread('ic.tif'); BW = edge(I); imshow(I) figure, imshow(BW) 2 Compute the Radon transform of the edge image. theta = 0:179; [R,xp] = radon(BW,theta); figure, imagesc(theta, xp, R); colormap(hot); xlabel('\theta (degrees)'); ylabel('X\prime'); title('R_{\theta} (X\prime)'); colorbar 8-24 Radon Transform Rθ (X′) 100 150 90 100 80 70 50 x’ 60 0 50 40 50 30 100 20 10 150 0 20 40 60 80 100 θ (degrees) 120 140 160 0 Figure 8-15: Radon Transform of an Edge Image 3 Find the locations of strong peaks in the Radon transform matrix. The locations of these peaks correspond to the location of straight lines in the original image. In this example, the strongest peak in R corresponds to θ = 94° and x′ = – 101 . The line perpendicular to that angle and located at x′ = – 101 is shown below, superimposed in red on the original image. The Radon transform geometry is shown in black. 8-25 8 Transforms x′ q = 94° x x′ = –101 Figure 8-16: The Radon Transform Geometry and the Strongest Peak (Red) Notice that the other strong lines parallel to the red line also appear as peaks at θ = 94° in the transform. Also, the lines perpendicular to this line appear as peaks at θ = 4° . The Inverse Radon Transform The iradon function performs the inverse Radon transform, which is commonly used in tomography applications. This transform inverts the Radon transform (which was introduced in the previous section), and can therefore be used to reconstruct images from projection data. As discussed in the previous section “Radon Transform” on page 8-21, given an image I and a set of angles theta, the function radon can be used to calculate the Radon transform. R = radon(I,theta); The function iradon can then be called to reconstruct the image I. IR = iradon(R,theta); In the example above, projections are calculated from the original image I. In most application areas, there is no original image from which projections are 8-26 Radon Transform formed. For example, in X-ray absorption tomography, projections are formed by measuring the attenuation of radiation that passes through a physical specimen at different angles. The original image can be thought of as a cross section through the specimen, in which intensity values represent the density of the specimen. Projections are collected using special purpose hardware, and then an internal image of the specimen is reconstructed by iradon. This allows for noninvasive imaging of the inside of a living body or another opaque object. iradon reconstructs an image from parallel beam projections. In parallel beam geometry, each projection is formed by combining a set of line integrals through an image at a specific angle. Figure 8-17 below illustrates how parallel beam geometry is applied in X-ray absorption tomography. Note that there is an equal number of n emitters and n detectors. Each detector measures the radiation emitted from its corresponding emitter, and the attenuation in the radiation gives a measure of the integrated density, or mass, of the object. This corresponds to the line integral that is calculated in the Radon transform. The parallel beam geometry used in the figure is the same as the geometry that was described under “Radon Transform” on page 8-21. f(x,y) denotes the brightness of the image and Rq(x′) is the projection at angle q. 8-27 8 Transforms x′ y Rq(x′) x′ q f(x,y) x Figure 8-17: Parallel Beam Projections Through an Object Another geometry that is commonly used is fan beam geometry, in which there is one emitter and n detectors. There are methods for resorting sets of fan beam projections into parallel beam projections, which can then be used by iradon. (For more information on these methods, see Kak & Slaney, Principles of Computerized Tomographic Imaging, IEEE Press, NY, 1988, pp. 92-93.) iradon uses the filtered backprojection algorithm to compute the inverse Radon transform. This algorithm forms an approximation to the image I based on the projections in the columns of R. A more accurate result can be obtained by using more projections in the reconstruction. As the number of projections (the length 8-28 Radon Transform of theta) increases, the reconstructed image IR more accurately approximates the original image I. The vector theta must contain monotonically increasing angular values with a constant incremental angle ∆θ. When the scalar ∆θ is known, it can be passed to iradon instead of the array of theta values. Here is an example. IR = iradon(R,Dtheta); The filtered backprojection algorithm filters the projections in R and then reconstructs the image using the filtered projections. In some cases, noise can be present in the projections. To remove high frequency noise, apply a window to the filter to attenuate the noise. Many such windowed filters are available in iradon. The example call to iradon below applies a Hamming window to the filter. See the iradon reference page for more information. IR = iradon(R,theta,'Hamming'); iradon also enables you to specify a normalized frequency, D, above which the filter has zero response. D must be a scalar in the range [0,1]. With this option, the frequency axis is rescaled, so that the whole filter is compressed to fit into the frequency range [0,D]. This can be useful in cases where the projections contain little high frequency information but there is high frequency noise. In this case, the noise can be completely suppressed without compromising the reconstruction. The following call to iradon sets a normalized frequency value of 0.85. IR = iradon(R,theta,0.85); Examples The commands below illustrate how to use radon and iradon to form projections from a sample image and then reconstruct the image from the projections. The test image is the Shepp-Logan head phantom, which can be generated by the Image Processing Toolbox function phantom. The phantom image illustrates many of the qualities that are found in real-world tomographic imaging of human heads. The bright elliptical shell along the exterior is analogous to a skull, and the many ellipses inside are analogous to brain features or tumors. 8-29 8 Transforms P = phantom(256); imshow(P) As a first step the Radon transform of the phantom brain is calculated for three different sets of theta values. R1 has 18 projections, R2 has 36 projections, and R3 had 90 projections. theta1 = 0:10:170; [R1,xp] = radon(P,theta1); theta2 = 0:5:175; [R2,xp] = radon(P,theta2); theta3 = 0:2:178; [R3,xp] = radon(P,theta3); Now the Radon transform of the Shepp-Logan Head phantom is displayed using 90 projections (R3). figure, imagesc(theta3,xp,R3); colormap(hot); colorbar xlabel('\theta'); ylabel('x\prime'); 8-30 Radon Transform Figure 8-18: Radon Transform of Head Phantom Using 90 Projections When we look at Figure 8-18, we can see some of the features of the input image. The first column in the Radon transform corresponds to a projection at 0° which is integrating in the vertical direction. The centermost column corresponds to a projection at 90°, which is integrating in the horizontal direction. The projection at 90° has a wider profile than the projection at 0° due to the larger vertical semi-axis of the outermost ellipse of the phantom. Figure 8-19 shows the inverse Radon transforms of R1, R2, and R3, which were generated above. Image I1 was reconstructed with the projections in R1, and it is the least accurate reconstruction, because it has the fewest projections. I2 was reconstructed with the 36 projections in R2, and the quality of the reconstruction is better, but it is still not clear enough to discern clearly the three small ellipses in the lower portion of the test image. I3 was reconstructed using the 90 projections in R3, and the result closely resembles the original image. Notice that when the number of projections is relatively small (as in I1 and I2), the reconstruction may include some artifacts from the back projection. To avoid this, use a larger number of angles. 8-31 8 Transforms I1 = iradon(R1,10); I2 = iradon(R2,5); I3 = iradon(R3,2); imshow(I1) figure, imshow(I2) figure, imshow(I3) I2 I1 I3 Figure 8-19: Inverse Radon Transforms of the Shepp-Logan Head Phantom 8-32 9 Morphological Operations Morphology is a technique of image processing based on shapes. The value of each pixel in the output image is based on a comparison of the corresponding pixel in the input image with its neighbors. By choosing the size and shape of the neighborhood, you can construct a morphological operation that is sensitive to specific shapes in the input image. This section describes the Image Processing toolbox morphological functions. You can use these functions to perform common image processing tasks, such as contrast enhancement, noise removal, thinning, skeletonization, filling, and segmentation. Topics covered include Terminology (p. 9-2) Provides definitions of image processing terms used in this section Dilation and Erosion (p. 9-4) Defines the two fundamental morphological operations, dilation and erosion, and some of the morphological image processing operations that are based on combinations of these operations Morphological Reconstruction (p. 9-19) Describes morphological reconstruction and the toolbox functions that use this type of processing Distance Transform (p. 9-38) Describes how to use the bwdist function to compute the distance transform of an image Example: Marker-Controlled Watershed Segmentation (p. 9-41) Steps you through a detailed example of using morphological image processing Objects, Regions, and Feature Measurement (p. 9-49) Describes functions that return information about a binary image Lookup Table Operations (p. 9-53) Describes functions that perform lookup table operations 9 Morphological Operations Terminology An understanding of the following terms will help you to use this chapter. 9-2 Terms Definitions Background In a binary image, pixels that are off, i.e., set to the value 0, are considered the background. When you view a binary image, the background pixels appear black. Connectivity The criteria that describes how pixels in an image form a connected group. For example, a connected component is “8-connected” if diagonally adjacent pixels are considered to be touching, otherwise, it is “4-connected.” The toolbox supports 2-D as well as multidimensional connectivities. See “Pixel Connectivity” on page 9-23 for more information. Foreground In a binary image, pixels that are on, i.e., set to the value 1, are considered the foreground. When you view a binary image, the foreground pixels appear white. Global maxima The highest regional maxima in the image. See the entry for regional maxima in this table for more information. Global minima The lowest regional minima in the image. See the entry for regional minima in this table for more information. Morphology A broad set of image processing operations that process images based on shapes. Morphological operations apply a structuring element to an input image, creating an output image of the same size. The most basic morphological operations are dilation and erosion. Neighborhood A set of pixels that are defined by their locations relative to the pixel of interest. A neighborhood can be defined by a structuring element or by specifying a connectivity. Object A set of pixels in a binary image that form a connected group. In the context of this chapter, “object” and “connected component” are equivalent. Terminology Terms Definitions Packed binary image A method of compressing binary images that can speed up the processing of the image. Regional maxima A connected set of pixels of constant intensity from which it is impossible to reach a point with higher intensity without first descending; that is, a connected component of pixels with the same intensity value, t, surrounded by pixels that all have a value less than t. Regional minima A connected set of constant intensity from which it is impossible to reach a point with lower intensity without first ascending; that is, a connected component of pixels with the same intensity value, t, surrounded by pixels that all have a value greater than t. Structuring element A matrix used to define a neighborhood shape and size for morphological operations, including dilation and erosion. It consists of only 0’s and 1’s and can have an arbitrary shape and size. The pixels with values of 1 define the neighborhood. 9-3 9 Morphological Operations Dilation and Erosion Dilation and erosion are two fundamental morphological operations. Dilation adds pixels to the boundaries of objects in an image, while erosion removes pixels on object boundaries. The number of pixels added or removed from the objects in an image depends on the size and shape of the structuring element used to process the image. The following sections describe the dilation and erosion functions in the toolbox: • “Understanding Dilation and Erosion” — This section provides important background information about how the dilation and erosion functions operate. • “Structuring Elements” on page 9-7 — This section describes structuring elements and how to create them. • “Dilating an Image” on page 9-11 — This section describes how to use the imdilate function. • “Eroding an Image” on page 9-12 — This section describes how to use the imerode function. • “Combining Dilation and Erosion” on page 9-14 — This section describes some of the common operations that are based on dilation and erosion. • “Dilation- and Erosion-Based Functions” on page 9-16 — This section describes toolbox functions that are based on dilation and erosion Understanding Dilation and Erosion In the morphological dilation and erosion operations, the state of any given pixel in the output image is determined by applying a rule to the corresponding pixel and its neighbors in the input image. The rule used to process the pixels defines the operation as a dilation or an erosion. This table lists the rules for both dilation and erosion. 9-4 Dilation and Erosion Table 9-1: Rules for Grayscale Dilation and Erosion Operation Rule Dilation The value of the output pixel is the maximum value of all the pixels in the input pixel’s neighborhood. In a binary image, if any of the pixels is set to the value 1, the output pixel is set to 1. Erosion The value of the output pixel is the minimum value of all the pixels in the input pixel’s neighborhood. In a binary image, if any of the pixels is set to 0, the output pixel is set to 0. The following figure illustrates the dilation of a binary image. In the figure, note how the structuring element defines the neighborhood of the pixel of interest, which is circled. (See “Structuring Elements” on page 9-7 for more information.) The dilation function applies the appropriate rule to the pixels in the neighborhood and assigns a value to the corresponding pixel in the output image. In the figure, the morphological dilation function sets the value of the output pixel to 1 because one of the elements in the neighborhood defined by the structuring element is on. Structuring Element 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 Input Image Output Image Figure 9-1: Morphological Processing of a Binary Image 9-5 9 Morphological Operations The following figure illustrates this processing for a grayscale image. The figure shows the processing of a particular pixel in the input image. Note how the function applies the rule to the input pixel’s neighborhood and uses the highest value of all the pixels in the neighborhood as the value of the corresponding pixel in the output image. Structuring Element 1 1 1 16 14 14 17 19 15 21 16 53 57 61 62 64 60 68 57 126 128 124 122 125 125 127 128 132 130 133 132 131 132 130 132 140 138 137 143 138 137 134 140 143 141 138 142 140 134 144 143 138 142 137 139 138 132 136 142 Input Image 16 17 17 19 15 21 Output Image Figure 9-2: Morphological Processing of Grayscale Images Processing Pixels at Image Borders Morphological functions position the origin of the structuring element, its center element, over the pixel of interest in the input image. For pixels at the edge of an image, parts of the neighborhood defined by the structuring element can extend past the border of the image. To process border pixels, the morphological functions assign a value to these undefined pixels, as if the functions had padded the image with additional rows and columns. The value of these “padding” pixels varies for dilation and erosion operations. The following table details the padding rules for dilation and erosion for both binary and grayscale images. 9-6 Dilation and Erosion Table 9-2: Rules for Padding Images Operation Rule Dilation Pixels beyond the image border are assigned the minimum value afforded by the data type. For binary images, these pixels are assumed to be set to 0. For grayscale images, the minimum value for uint8 images is 0. Erosion Pixels beyond the image border are assigned the maximum value afforded by the data type. For binary images, these pixels are assumed to be set to 1. For grayscale images, the maximum value for uint8 images is 255. Note By using the minimum value for dilation operations and the maximum value for erosion operations, the toolbox avoids border effects, where regions near the borders of the output image do not appear to be homogeneous with the rest of the image. For example, if erosion padded with a minimum value, eroding an image would result in a black border around the edge of the output image. Structuring Elements An essential part of the dilation and erosion operations is the structuring element used to probe the input image. Two-dimensional, or flat, structuring elements consist of a matrix of 0’s and 1’s, typically much smaller than the image being processed. The center pixel of the structuring element, called the origin, identifies the pixel of interest—the pixel being processed. The pixels in the structuring element containing 1’s define the neighborhood of the structuring element. These pixels are also considered in the dilation or erosion processing. Three dimensional, or nonflat, structuring elements use 0’s and 1’s to define the extent of the structuring element in the x- and y-plane and add height values to define the third dimension. 9-7 9 Morphological Operations The Origin of a Structuring Element The morphological functions use this code to get the coordinates of the origin of structuring elements of any size and dimension. origin = floor((size(nhood)+1)/2) (In this code, nhood is the neighborhood defining the structuring element. Because structuring elements are MATLAB objects, you cannot use the size of the STREL object itself in this calculation. You must use the STREL getnhood method to retrieve the neighborhood of the structuring element from the STREL object. For information about other STREL object methods, see the strel function reference page.) For example, the following illustrates a diamond-shaped structuring element. Structuring Element Origin 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 Figure 9-3: Origin of a Diamond-Shaped Structuring Element Creating a Structuring Element The toolbox dilation and erosion functions accept structuring element objects, called a STREL. You use the strel function to create STRELs of any arbitrary size and shape. The strel function also includes built-in support for many common shapes, such as lines, diamonds, disks, periodic lines, and balls. Note You typically choose a structuring element the same size and shape as the objects you want to process in the input image. For example, to find lines in an image, create a linear structuring element. 9-8 Dilation and Erosion For example, this code creates a flat, diamond-shaped structuring element. se = strel('diamond',3) se = Flat STREL object containing 25 neighbors. Decomposition: 3 STREL objects containing a total of 13 neighbors Neighborhood: 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 Structuring Element Decomposition To enhance performance, the strel function may break structuring elements into smaller pieces, a technique known as structuring element decomposition. For example, dilation by an 11-by-11 square structuring element can be accomplished by dilating first with a 1-by-11 structuring element, and then with an 11-by-1 structuring element. This results in a theoretical speed improvement of a factor of 5.5, although in practice the actual speed improvement is somewhat less. Structuring element decompositions used for the 'disk' and 'ball' shapes are approximations; all other decompositions are exact. Decomposition is not used with arbitrary structuring elements, unless it is a flat structuring element whose neighborhood is all 1’s. To view the sequence of structuring elements used in a decomposition, use the STREL getsequence method. The getsequence function returns an array of the structuring elements that form the decomposition. For example, here are the structuring elements created in the decomposition of a diamond shaped structuring element. sel = strel('diamond',4) sel = Flat STREL object containing 41 neighbors. Decomposition: 3 STREL objects containing a total of 13 neighbors 9-9 9 Morphological Operations Neighborhood: 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 seq = getsequence(sel) seq = 3x1 array of STREL objects seq(1) ans = Flat STREL object containing 5 neighbors. Neighborhood: 0 1 1 1 0 1 0 1 0 seq(2) ans = Flat STREL object containing 4 neighbors. Neighborhood: 0 1 1 0 0 1 0 1 0 seq(3) ans = Flat STREL object containing 4 neighbors. 9-10 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 Dilation and Erosion Neighborhood: 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 Dilating an Image To dilate an image, use the imdilate function. The imdilate function accepts two primary arguments: • The input image to be processed (grayscale, binary, or packed binary image) • A structuring element object, returned by the strel function, or a binary matrix defining the neighborhood of a structuring element imdilate also accepts two optional arguments: PADOPT and PACKOPT. The PADOPT argument affects the size of the output image. The PACKOPT argument identifies the input image as packed binary. (See the bwpack reference page for information about binary image packing.) This example dilates a simple binary image containing one rectangular object. BW = zeros(9,10); BW(4:6,4:7) = 1 BW = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 To expand all sides of the foreground component, the example uses a 3-by-3 square structuring element object. (For more information about using the strel function, see “Structuring Elements” on page 9-7.) 9-11 9 Morphological Operations SE = strel('square',3) SE = Flat STREL object containing 3 neighbors. Neighborhood: 1 1 1 1 1 1 1 1 1 To dilate the image, pass the image, BW, and the structuring element, SE, to the imdilate function. Note how dilation adds a rank of 1’s to all sides of the foreground object. BW2 = imdilate(BW,SE) BW2 = 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Eroding an Image To erode an image, use the imerode function. The imerode function accepts two primary arguments: • The input image to be processed (grayscale, binary, or packed binary image) • A structuring element object, returned by the strel function, or a binary matrix defining the neighborhood of a structuring element imerode also accepts three optional arguments: PADOPT, PACKOPT, and M. The PADOPT argument affects the size of the output image. The PACKOPT argument identifies the input image as packed binary. If the image is packed binary, M identifies the number of rows in the original image. (See the bwpack reference page for more information about binary image packing.) 9-12 Dilation and Erosion The following example erodes the binary image, circbw.tif: 1 Read the image into the MATLAB workspace. BW1 = imread('circbw.tif'); 2 Create a structuring element. The following code creates a diagonal structuring element object. (For more information about using the strel function, see “Structuring Elements” on page 9-7.) SE = strel('arbitrary ,eye(5)); SE= Flat STREL object containing 5 neighbors. Neighborhood: 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 3 Call the imerode function, passing the image, BW, and the structuring element, SE, as arguments. BW2 = imerode(BW1,SE); Notice the diagonal streaks on the right side of the output image. These are due to the shape of the structuring element. imshow(BW1) figure, imshow(BW2) 9-13 9 Morphological Operations Figure 9-4: Circbw.tif Before and After Erosion with a Diagonal Structuring Element Combining Dilation and Erosion Dilation and erosion are often used in combination to implement image processing operations. For example, the definition of a morphological opening of an image is an erosion followed by a dilation, using the same structuring element for both operations. The related operation, morphological closing of an image is the reverse: it consists of dilation followed by an erosion with the same structuring element. The following section uses imdilate and imerode to illustrate how to implement a morphological opening. Note, however, that the toolbox already includes the imopen function which performs this processing. The toolbox includes functions that perform many common morphological operations. See “Dilation- and Erosion-Based Functions” on page 9-16 for a complete list. To see some of the morphology functions used in an extended example, see “Example: Marker-Controlled Watershed Segmentation” on page 9-41. Morphological Opening You can use morphological opening to remove small objects from an image while preserving the shape and size of larger objects in the image. For example, you can use the imopen function to remove all the circuit lines from the original circuit image, circbw.tif, creating an output image that contains only the rectangular shapes of the microchips. 9-14 Dilation and Erosion To morphologically open the image, perform these steps: 1 Create a structuring element. SE = strel('rectangle ,[40 30]); The structuring element should be large enough to remove the lines when you erode the image, but not large enough to remove the rectangles. It should consist of all 1’s, so it removes everything but large continuous patches of foreground pixels. 2 Erode the image with the structuring element. BW1 = imread('circbw.tif ); BW2 = imerode(BW1,SE); imshow(BW2) This removes all of the lines, but also shrinks the rectangles. 3 To restore the rectangles to their original size, dilate the eroded image using the same structuring element, SE. BW3 = imdilate(BW2,SE); imshow(BW3) 9-15 9 Morphological Operations Dilation- and Erosion-Based Functions This section describes two common image processing operations that are based on dilation and erosion: • Skeletonization • Perimeter Determination This table lists other functions in the toolbox that perform common morphological operations that are based on dilation and erosion. For more information about these functions, see their reference pages. Table 9-3: Dilation- and Erosion-Based Functions 9-16 Function Morphological Definition bwhitmiss Logical AND of an image, eroded with one structuring element, and the image’s complement, eroded with a second structuring element. imbothat Subtracts the original image from a morphologically closed version of the image. Can be used to find intensity troughs in an image. imclose Dilates an image, and then erodes the dilated image using the same structuring element for both operations. Dilation and Erosion Table 9-3: Dilation- and Erosion-Based Functions Function Morphological Definition imopen Erodes an image and then dilates the eroded image using the same structuring element for both operations. imtophat Subtracts a morphologically opened image from the original image. Can be used to enhance contrast in an image. Skeletonization To reduce all objects in an image to lines, without changing the essential structure of the image, use the bwmorph function. This process is known as skeletonization. BW1 = imread('circbw.tif'); BW2 = bwmorph(BW1,'skel',Inf); imshow(BW1) figure, imshow(BW2) Figure 9-5: Circbw.tif Before and After Skeletonization Perimeter Determination The bwperim function determines the perimeter pixels of the objects in a binary image. A pixel is considered a perimeter pixel if it satisfies both of these criteria: • The pixel is on. 9-17 9 Morphological Operations • One (or more) of the pixels in its neighborhood is off. For example, this code finds the perimeter pixels in a binary image of a circuit board. BW1 = imread('circbw.tif'); BW2 = bwperim(BW1); imshow(BW1) figure, imshow(BW2) Figure 9-6: Circbw.tif Before and After Perimeter Determination 9-18 Morphological Reconstruction Morphological Reconstruction Morphological reconstruction is another major part of morphological image processing. Based on dilation, morphological reconstruction has these unique properties: • Processing is based on two images, a marker and a mask, rather than one image and a structuring element • Processing repeats until stability; i.e., the image no longer changes • Processing is based on the concept of connectivity, rather than a structuring element. The following sections describe the functions in the toolbox that are based on morphological reconstruction: • “Marker and Mask” – This section describes the fundamental reconstruction function, imreconstruct, and provides background information about morphological reconstruction. • “Pixel Connectivity” – This section describes how pixel connectivity affects morphological reconstruction. • “Flood-Fill Operations” – This section describes how to use the imfill function, which is based on morphological reconstruction. • “Finding Peaks and Valleys” – This section describes a group of functions, all based on morphological reconstruction, that process image extrema, i.e., the areas of high- and low-intensity in images. Marker and Mask Morphological reconstruction processes one image, called the marker, based on the characteristics of another image, called the mask. The high-points, or peaks, in the marker image specify where processing begins. The processing continues until the image values stop changing. To illustrate morphological reconstruction, consider this simple image. It contains two primary regions, the blocks of pixels containing the value 14 and 18. The background is primarily all set to 10, with some pixels set to 11. 9-19 9 Morphological Operations A = [10 10 10 10 10 10 10 10 10 10 10 14 14 14 10 11 10 10 11 10 10 14 14 14 10 10 10 11 10 10 10 14 14 14 10 10 11 10 11 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 18 18 18 10 10 10 11 10 11 10 18 18 18 10 11 10 10 11 10 10 18 18 18 10 10 10 11 10 11 10 10 10 10 10 10 10; 10; 10; 10; 10; 10; 10; 10; 10; 10]; To morphologically reconstruct this image, perform these steps: 1 Create a marker image. As with the structuring element in dilation and erosion, the characteristics of the marker image determine the processing performed in morphological reconstruction. The peaks in the marker image should identify the location of objects in the mask image that you want to emphasize. One way to create a marker image is to subtract a constant from the mask image, using imsubtract. marker = imsubtract(A,2) marker = 8 8 8 8 8 12 12 12 8 12 12 12 8 12 12 12 8 8 8 8 8 9 8 8 8 8 8 9 8 8 9 8 8 9 8 9 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 16 16 16 8 8 8 9 8 9 8 16 16 16 8 9 8 8 9 8 8 16 16 16 8 8 8 9 8 9 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 2 Call the imreconstruct function to morphologically reconstruct the image. In the output image, note how all the intensity fluctuations except the intensity peak have been removed. 9-20 Morphological Reconstruction recon = imreconstruct(marker, mask) recon = 10 10 10 10 10 10 10 12 12 12 10 10 10 12 12 12 10 10 10 12 12 12 10 10 10 10 10 10 10 10 10 10 10 10 10 16 10 10 10 10 10 16 10 10 10 10 10 16 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 16 16 16 10 10 10 10 10 10 10 16 16 16 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Understanding Morphological Reconstruction Morphological reconstruction can be thought of conceptually as repeated dilations of the marker image until the contour of the marker image fits under the mask image. In this way, the peaks in the marker image “spread out”, or dilate. This figure illustrates this processing in 1-D. Each successive dilation is constrained to lie underneath the mask. When further dilation ceases to change the image, processing stops. The final dilation is the reconstructed image. (Note: the actual implementation of this operation in the toolbox is done much more efficiently. See the imreconstruct reference page for more details.) The figure shows the successive dilations of the marker. 9-21 9 Morphological Operations Repeated dilations of marker image Mask Marker Reconstructed Image Figure 9-7: Repeated Dilations of Marker Image, Constrained by Mask 9-22 Morphological Reconstruction Pixel Connectivity Morphological processing starts at the peaks in the marker image and spreads throughout the rest of the image based on the connectivity of the pixels. Connectivity defines which pixels are connected to other pixels. For example, this binary image contains one foreground object—all the pixels that are set to 1. If the foreground is 4-connected, the image has one background object, all the pixels are set to 0. However, if the foreground is 8-connected, the foreground makes a closed loop and the image has two separate background objects: the pixels in the loop and the pixels outside the loop. 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Defining Connectivity in an Image The following table lists all the standard two- and three-dimensional connectivities supported by the toolbox. See these sections for more information: • “Choosing a Connectivity” • “Specifying Custom Connectivities” 9-23 9 Morphological Operations Table 9-4: Supported Connectivities Two-Dimensional Connectivities 4-connected Pixels are connected if their edges touch. This means that a pair of adjoining pixels are part of the same object only if they are both on and are connected along the horizontal or vertical direction. 8-connected Pixels are connected if their edges or corners touch. This means that if two adjoining pixels are on, they are part of the same object, regardless of whether they are connected along the horizontal, vertical, or diagonal direction. Three-Dimensional Connectivities 9-24 6-connected Pixels are connected if their faces touch. 6 faces 18-connected Pixels are connected if their faces or edges touch. 6 faces + 12 edges 26-connected Pixels are connected if their faces, edges, or corners touch. 6 faces + 12 edges + 8 corners Morphological Reconstruction Choosing a Connectivity The type of neighborhood you choose affects the number of objects found in an image and the boundaries of those objects. For this reason, the results of many morphology operations often differ depending upon the type connectivity you specify. For example, if you specify a 4-connected neighborhood, this binary image contains two objects; if you specify an 8-connected neighborhood, the image has one object. 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 Specifying Custom Connectivities You can also define custom neighborhoods by specifying a 3-by-3-by-...-by-3 array of 0’s and 1’s. The 1-valued elements define the connectivity of the neighborhood relative to the center element. For example, this array defines a “North/South” connectivity that has the effect of breaking up an image into independent columns. CONN = [ 0 1 0; 0 1 0; 0 1 0 ] CONN = 0 1 0 0 1 0 0 1 0 Note Connectivity arrays must be symmetric about their center element. Also, you can use 2-D connectivity array with a 3-D image; the connectivity affects each “page” in the 3-D image. 9-25 9 Morphological Operations Flood-Fill Operations The imfill function performs a flood-fill operation on binary and grayscale images. For binary images, imfill changes connected background pixels (0s) to foreground pixels (1s), stopping when it reaches object boundaries. For grayscale images, imfill brings the intensity values of dark areas that are surrounded by lighter areas up to the same intensity level as surrounding pixels. (In effect, imfill removes regional minima that are not connected to the image border. See “Finding Areas of High- or Low-Intensity” for more information.) This operation can be useful in removing irrelevant artifacts from images. This section includes information about: • Specifying the connectivity in flood-fill operations • Specifying the starting point for binary image fill operations • Filling holes in binary or grayscale images Specifying Connectivity For both binary and grayscale images, the boundary of the fill operation is determined by the connectivity you specify. Note imfill differs from the other object-based operations in that it operates on background pixels. When you specify connectivity with imfill, you are specifying the connectivity of the background, not the foreground. The implications of connectivity can be illustrated with this matrix. BW = [ 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0; 0; 0; 0; 0; 0; 0; 0]; If the background is 4-connected, this binary image contains two separate background elements (the part inside the loop and the part outside). If the 9-26 Morphological Reconstruction background is 8-connected, the pixels connect diagonally, and there is only one background element. Specifying the Starting Point For binary images, you can specify the starting point of the fill operation by passing in the location subscript or by using imfill in interactive mode, selecting starting pixels with a mouse. See the reference page for imfill for more information using imfill interactively. For example, if you call imfill, specifying the pixel BW(4,3) as the starting point, imfill only fills in the inside of the loop because, by default, the background is 4-connected. imfill(BW,[4 3]) ans = 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 If you specify the same starting point, but use an 8-connected background connectivity, imfill fills in the entire image. imfill(BW,[4 3],8) ans = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9-27 9 Morphological Operations Filling Holes A common use of the flood-fill operation is to fill “holes” in images. For example, suppose you have an image, binary or grayscale, in which the foreground objects represent spheres. In the image, these objects should appear as disks, but instead are donut shaped because of reflections in the original photograph. Before doing any further processing of the image, you may want to first fill in the “donut holes” using imfill. Because the use of flood-fill to fill holes is so common, imfill includes special syntax to support it for both binary and grayscale images. In this syntax, you just specify the argument 'holes'; you do not have to specify starting locations in each hole. To illustrate, this example fills holes in a grayscale image of blood cells. blood = imread('blood1.tif'); imshow(blood) blood2 = imcomplement(imfill(imcomplement(blood),'holes')); figure, imshow(blood2) Original 9-28 After Filling Holes Morphological Reconstruction Finding Peaks and Valleys Grayscale images can be thought of in three-dimensions: the x- and y-axes represent pixel positions and the z-axes represents the intensity of each pixel. In this interpretation, the intensity values represent elevations as in a topographical map. The areas of high-intensity and low-intensity in an image, peaks and valleys in topographical terms, can be important morphological features because they often mark relevant image objects. For example, in an image of several spherical objects, points of high intensity could represent the tops of the objects. Using morphological processing, these maxima can be used to identify objects in an image. This section covers these topics: • “Understanding the Maxima and Minima Functions” • “Finding Areas of High- or Low-Intensity” • “Suppressing Minima and Maxima” • “Imposing a Minima” Understanding the Maxima and Minima Functions An image can have multiple regional maxima or minima but only a single global maxima or minima. Determining image peaks or valleys can be used to create marker images that are used in morphological reconstruction. For a detailed example, see “Example: Marker-Controlled Watershed Segmentation” on page 9-41. This figure illustrates the concept in 1-D. 9-29 9 Morphological Operations Regional maxima Global maxima Regional minima Global minima Finding Areas of High- or Low-Intensity The toolbox includes functions that you can use to find areas of high- or low-intensity in an image: • The imregionalmax and imregionalmin functions identify all regional minima or maxima. • The imextendedmax and imextendedmin functions identify all regional minima or maxima that are greater than or less than a specified threshold. The functions accept a grayscale image as input and return a binary image as output. In the output binary image, the regional minima or maxima are set to 1; all other pixels are set to 0. 9-30 Morphological Reconstruction For example, this simple image contains two primary regional maxima, the blocks of pixels containing the value 13 and 18, and several smaller maxima, set to 11. A = [10 10 10 10 10 10 10 10 10 10 10 13 13 13 10 11 10 10 11 10 10 13 13 13 10 10 10 11 10 10 10 13 13 13 10 10 11 10 11 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 18 18 18 10 10 10 11 10 11 10 18 18 18 10 11 10 10 11 10 10 18 18 18 10 10 10 11 10 11 10 10 10 10 10 10 10; 10; 10; 10; 10; 10; 10; 10; 10; 10] The binary image returned by imregionalmax pinpoints all these regional maxima. B = imregionalmax(A) B = 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 You may want to only identify areas of the image where the change in intensity is extreme; that is, the difference between the pixel and neighboring pixels is greater than (or less than) a certain threshold. For example, to find only those regional maxima in the sample image, A, that are at least two units higher than their neighbors, use imextendedmax. 9-31 9 Morphological Operations B = imextendedmax(A,2) B = 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Suppressing Minima and Maxima In an image, every small fluctuation in intensity represents a regional minima or maxima. You may only be interested in significant minima or maxima and not in these smaller minima and maxima caused by background texture. To remove the less significant minima and maxima but retain the significant minima and maxima, use the imhmax or imhmin functions. With these functions, you can specify a contrast criteria or threshold level, h, that suppresses all maxima whose height is less than h or whose minima are greater than h. Note The imregionalmin, imregionalmax, imextendedmin and imextendedmax functions return a binary image that marks the locations of the regional minima and maxima in an image. The imhmax and imhmin functions produce an altered image. For example, this simple image contains two primary regional maxima, the blocks of pixels containing the value 14 and 18, and several smaller maxima, set to 11. 9-32 Morphological Reconstruction A = [10 10 10 10 10 10 10 10 10 10 10 14 14 14 10 11 10 10 11 10 10 14 14 14 10 10 10 11 10 10 10 14 14 14 10 10 11 10 11 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 18 18 18 10 10 10 11 10 11 10 18 18 18 10 11 10 10 11 10 10 18 18 18 10 10 10 11 10 11 10 10 10 10 10 10 10; 10; 10; 10; 10; 10; 10; 10; 10; 10]; To eliminate all regional maxima except the two significant maxima, use imhmax, specifying a threshold value of 2. Note that imhmax only affects the maxima; none of the other pixel values are changed. The two significant maxima remain, although their heights are reduced. B = imhmax(A,2) B = 10 10 10 12 10 12 10 12 10 10 10 10 10 10 10 10 10 10 10 10 10 12 12 12 10 10 10 10 10 10 10 12 12 12 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 16 16 16 10 10 10 10 10 10 10 16 16 16 10 10 10 10 10 10 10 16 16 16 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 This figure takes the second row from the sample image to illustrate in 1-D how imhmax changes the profile of the image. 9-33 9 Morphological Operations Original profile h-maxima transform Imposing a Minima You can emphasize specific minima (dark objects) in an image using the imimposemin function. The imimposemin function uses morphological reconstruction to eliminate all minima from an the image except the minima you specify. To illustrate the process of imposing a minima, this code creates a simple image containing two primary regional minima and several other regional minima. mask = uint8(10*ones(10,10)); mask(6:8,6:8) = 2; mask(2:4,2:4) = 7; mask(3,3) = 5; mask(2,9) = 9 mask(3,8) = 9 mask(9,2) = 9 mask(8,3) = 9 9-34 Morphological Reconstruction mask = 10 10 10 10 10 10 10 10 10 10 10 7 7 7 10 10 10 10 9 10 10 7 6 7 10 10 10 9 10 10 10 7 7 7 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 2 2 2 10 10 10 10 10 10 10 2 2 2 10 10 10 10 9 10 10 2 2 2 10 10 10 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Creating a Marker Image To obtain an image that emphasizes the two deepest minima and removes all others, create a marker image that pinpoints the two minima of interest. You can create the marker image by explicitly setting certain pixels to specific values or by using other morphological functions to extract the features you want to emphasize in the mask image. This example uses imextendedmin to get a binary image that shows the locations of the two deepest minima. marker = imextendedmin(mask,1) marker = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9-35 9 Morphological Operations Applying the Marker Image to the Mask Now use imimposemin to create new minima in the mask image at the points specified by the marker image. Note how imimposemin sets the values of pixels specified by the marker image to the lowest value supported by the datatype (0 for uint8 values). imimposemin also changes the values of all the other pixels in the image to eliminate the other minima. I = imimposemin(mask,marker) I = 11 11 11 11 11 11 8 8 8 11 11 8 0 8 11 11 8 8 8 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 0 0 0 11 11 11 11 11 11 11 0 0 0 11 11 11 11 11 11 11 0 0 0 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 This figure illustrates in 1-D how imimposemin changes the profile of row 2 of the image. 9-36 Morphological Reconstruction Three regional minima Original image Single minima Marker image superimposed Original image Single minima Image after minima imposition Original image Figure 9-8: Imposing a Minima 9-37 9 Morphological Operations Distance Transform The distance transform provides a metric or measure of the separation of points in the image. The Image Processing Toolbox provides a function, bwdist, that calculates the distance between each pixel that is set to off (0) and the nearest nonzero pixel for binary images. The bwdist function supports several distance metrics, listed in Table 9-5, Distance Metrics. Table 9-5: Distance Metrics Distance Metric Description Euclidean The Euclidean distance is the straight-line distance between two pixels. Illustration 0 0 0 1.41 1.0 1.41 0 1 0 1.0 0 0 0 1.41 1.0 1.41 Distance transform Image City Block 9-38 The City Block distance metric measures the path between the pixels based on a 4-connected neighborhood. Pixels whose edges touch are 1 unit apart; pixels diagonally touching are 2 units apart. 0.0 1.0 0 0 0 2 1 2 0 1 0 1 0 1 0 0 0 2 1 2 Image Distance transform Distance Transform Table 9-5: Distance Metrics (Continued) Distance Metric Description Chessboard The Chessboard distance metric measures the path between the pixels based on an 8-connected neighborhood. Pixels whose edges or corners touch are 1 unit apart. Illustration 0 0 0 1 1 1 0 1 0 1 1 1 0 0 0 1 1 1 Distance transform Image Quasi-Euclidean The Quasi-Euclidean metric measures the total Euclidean distance along a set of horizontal, vertical, and diagonal line segments. 0 0 0 0 0 2.8 2.2 2.0 2.2 2.8 0 0 0 0 0 2.2 1.4 1.0 1.4 2.2 0 0 1 0 0 2.0 1.0 0 1.0 2.0 0 0 0 0 0 2.2 1.4 1.0 1.4 2.2 0 0 0 0 0 2.8 2.2 2.0 2.2 2.8 Image Distance transform This example creates a binary image containing two overlapping circular objects. center1 = -10; center2 = -center1; dist = sqrt(2*(2*center1)^2); radius = dist/2 * 1.4; lims = [floor(center1-1.2*radius) ceil(center2+1.2*radius)]; [x,y] = meshgrid(lims(1):lims(2)); bw1 = sqrt((x-center1).^2 + (y-center1).^2) <= radius; bw2 = sqrt((x-center2).^2 + (y-center2).^2) <= radius; bw = bw1 | bw2; figure, imshow(bw), title('bw') 9-39 9 Morphological Operations To compute the distance transform of the complement of the binary image, use the bwdist function. In the image of the distance transform, note how the center of the two circular areas are white. D = bwdist(~bw); figure, imshow(D,[]), title('Distance transform of ~bw') 9-40 Example: Marker-Controlled Watershed Segmentation Example: Marker-Controlled Watershed Segmentation This example illustrates how to use many different morphology functions in combination to accomplish an image processing task: segmentation. In segmentation, objects in an image that are touching each other are divided into separate objects. (To see other extended examples, view the Image Processing Toolbox Morphology demos.) The example performs these steps: • “Step 1: Read in Images” • “Step 2: Create the Structuring Element” on page 9-42 • “Step 3: Enhance the Image Contrast” on page 9-42 • “Step 4: Exaggerate the Gaps Between Objects” on page 9-43 • “Step 5: Convert Objects of Interest” on page 9-44 • “Step 6: Detect Intensity Valleys” on page 9-45 • “Step 7: Watershed Segmentation” on page 9-46 • “Step 8: Extract Features from Label Matrix” on page 9-47 Step 1: Read in Images Read in the 'afmsurf.tif' image, which is an atomic force microscope image of a surface coating. afm = imread('afmsurf.tif'); figure, imshow(afm), title('Surface Image'); The image contains many objects of different sizes that are touching each other. Object detection in an image is an example of image segmentation. To 9-41 9 Morphological Operations segment touching objects, the Watershed transform is often used. If you view an image as a surface, with mountains (high intensity) and valleys (low intensity), the Watershed transform finds intensity valleys in an image. To get the best result, maximize the contrast of the objects of interest to minimize the number of valleys found by the Watershed transform. A common technique for contrast enhancement is the combined use of the top-hat and bottom-hat transforms. The top-hat transform is defined as the difference between the original image and its opening. The opening of an image is the collection of foreground parts of an image that fit a particular structuring element. The bottom-hat transform is defined as the difference between the closing of the original image and the original image. The closing of an image is the collection of background parts of an image that fit a particular structuring element. Step 2: Create the Structuring Element Since the objects of interest in our image look like disks, the example uses the strel function to create a disk structuring element. The size of the disk is based on an estimation of the average radius of the objects in the image. se = strel('disk', 15); Step 3: Enhance the Image Contrast The imtophat and imbothat functions return the top-hat and bottom-hat transforms, respectively, of the original image. Itop = imtophat(afm, se); Ibot = imbothat(afm, se); figure, imshow(Itop, []), title('top-hat image'); 9-42 Example: Marker-Controlled Watershed Segmentation figure, imshow(Ibot, []), title('bottom-hat image'); Step 4: Exaggerate the Gaps Between Objects The top-hat image contains the “peaks” of objects that fit the structuring element. The imbothat function shows the gaps between the objects. To maximize the contrast between the objects and the gaps that separate them from each other, the example adds the top-hat image to the original image, and then subtracts the “bottom-hat” image from the result. The example uses the toolbox image arithmetic functions, imadd and imsubtract, to perform these operations. 9-43 9 Morphological Operations Ienhance = imsubtract(imadd(Itop, afm), Ibot); figure, imshow(Ienhance), title('original + top-hat - bottom-hat'); Step 5: Convert Objects of Interest Because the watershed transform detects intensity “valleys” in an image, the example uses the imcomplement function on the enhanced image to highlight the intensity valleys. Iec = imcomplement(Ienhance); figure, imshow(Iec), title('complement of enhanced image'); 9-44 Example: Marker-Controlled Watershed Segmentation Step 6: Detect Intensity Valleys The example detects all the intensity valleys deeper than a particular threshold with the imextendedmin function. The output of the imextendedmin function is a binary image. The location rather than the size of the regions in the imextendedmin image is important. The imimposemin function modifies the image to contain only those valleys found by the imextendedmin function. The imimposemin function also changes a valley's pixel values to zero (deepest possible valley for uint8 images). All regions containing an imposed minima are detected by the watershed transform. Iemin = imextendedmin(Iec, 22); Iimpose = imimposemin(Iec, Iemin); figure, imshow(Iemin), title('extended minima image'); figure, imshow(Iimpose), title('imposed minima image'); 9-45 9 Morphological Operations Step 7: Watershed Segmentation Use the watershed function to accomplish Watershed segmentation of the imposed minima image. wat = watershed(Iimpose); The watershed function returns a label matrix containing nonnegative numbers that correspond to watershed regions. Pixels that do not fall into any watershed region are given a pixel value of 0. A good way to visualize a label matrix is to convert it to a color image, using the label2rgb function. In the color version of the image, each labeled region displays in a different color and the pixels that separate the regions display white. rgb = label2rgb(wat); figure, imshow(rgb); title('watershed segmented image'); 9-46 Example: Marker-Controlled Watershed Segmentation Step 8: Extract Features from Label Matrix Features can be extracted from the label matrix with the regionprops function. For example, we can calculate two measurements (area and orientation) and view them as a function of one another. stats = regionprops(wat, 'Area', 'Orientation'); area = [stats.Area]; orient = [stats.Orientation]; figure, plot(area, orient, 'b*'); title('Relationship of Particle Orientation to Area'); xlabel('particle area (pixels)'); ylabel('particle orientation (degrees)'); 9-47 9 Morphological Operations Particle orientation (degrees) Relationship of particle orientation to area Particle area (pixels) 9-48 Objects, Regions, and Feature Measurement Objects, Regions, and Feature Measurement The toolbox includes several functions that return information about the features in a binary image, including: • Connected-component labeling, and using the label matrix to get statistics about an image • Selecting objects in an image • Finding the area of a binary image • Finding the Euler number of a binary image Connected-Component Labeling The bwlabel and the bwlabeln functions perform connected-component labeling, which is a method for identifying each object in a binary image. The bwlabel function supports 2-D inputs only; the bwlabeln function supports inputs of any dimension. These functions return a matrix, called a label matrix. A label matrix is an image, the same size as the input image, in which the objects in the input image are distinguished by different integer values in the output matrix. For example, bwlabel can identify the objects in this binary image. BW = [0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0; 1; 1; 0; 0; 0; 0; 0]; 9-49 9 Morphological Operations X = bwlabel(BW,4) X = 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 2 2 2 0 0 0 0 0 2 2 2 0 0 3 0 0 0 0 0 0 0 3 3 0 0 0 0 0 0 3 3 0 0 0 0 0 In the output matrix, the 1’s represent one object, the 2’s a second object, and the 3’s a third. (If you had used 8-connected neighborhoods (the default), there would be only two objects, because the first and second objects would be a single object, connected along the diagonal.) Viewing a Label Matrix The label matrix returned by bwlabel or bwlabeln is of class double; it is not a binary image. One way to view it is to display it as a pseudo-color indexed image, using label2rgb. In the pseudo-color image, the number that identifies each object in the label matrix maps to a different color in the associated colormap matrix. When you view a label matrix as an RGB image, the objects in the image are easier to distinguish. To illustrate this technique, this example uses label2rgb to view the label matrix, X. The call to label2rgb specifies one of the standard MATLAB colormaps, jet. The third argument, 'k', specifies the background color (black). X = bwlabel(BW1,4); RGB = label2rgb(X, @jet, 'k'); imshow(RGB,'notruesize') 9-50 Objects, Regions, and Feature Measurement Figure 9-9: Using Color to Distinguish Objects in a Binary Image Selecting Objects in a Binary Image You can use the bwselect function to select individual objects in a binary image. You specify pixels in the input image, and bwselect returns a binary image that includes only those objects from the input image that contain one of the specified pixels. You can specify the pixels either noninteractively or with a mouse. For example, suppose you want to select objects in the image displayed in the current axes. You type BW2 = bwselect; The cursor changes to a cross-hair when it is over the image. Click on the objects you want to select; bwselect displays a small star over each pixel you click on. When you are done, press Return. bwselect returns a binary image consisting of the objects you selected, and removes the stars. See the reference page for bwselect for more information. Finding the Area of Binary Images The bwarea function returns the area of a binary image. The area is a measure of the size of the foreground of the image. Roughly speaking, the area is the number of on pixels in the image. bwarea does not simply count the number of pixels set to on, however. Rather, bwarea weights different pixel patterns unequally when computing the area. This weighting compensates for the distortion that is inherent in representing 9-51 9 Morphological Operations a continuous image with discrete pixels. For example, a diagonal line of 50 pixels is longer than a horizontal line of 50 pixels. As a result of the weighting bwarea uses, the horizontal line has area of 50, but the diagonal line has area of 62.5. This example uses bwarea to determine the percentage area increase in circbw.tif that results from a dilation operation. BW = imread('circbw.tif'); SE = ones(5); BW2 = imdilate(BW,SE); increase = (bwarea(BW2) - bwarea(BW))/bwarea(BW); increase = 0.3456 See the reference page for bwarea for more information about the weighting pattern. Finding the Euler Number of a Binary Image The bweuler function returns the Euler number for a binary image. The Euler number is a measure of the topology of an image. It is defined as the total number of objects in the image minus the number of holes in those objects. You can use either 4- or 8-connected neighborhoods. This example computes the Euler number for the circuit image, using 8-connected neighborhoods. BW1 = imread('circbw.tif'); eul = bweuler(BW1,8) eul = -85 In this example, the Euler number is negative, indicating that the number of holes is greater than the number of objects. 9-52 Lookup Table Operations Lookup Table Operations Certain binary image operations can be implemented most easily through lookup tables. A lookup table is a column vector in which each element represents the value to return for one possible combination of pixels in a neighborhood. You can use the makelut function to create lookup tables for various operations. makelut creates lookup tables for 2-by-2 and 3-by-3 neighborhoods. This figure illustrates these types of neighborhoods. Each neighborhood pixel is indicated by an x, and the center pixel is the one with a circle. x x x x x x x x x x x x x 2-by-2 neighborhood 3-by-3 neighborhood For a 2-by-2 neighborhood, there are 16 possible permutations of the pixels in the neighborhood. Therefore, the lookup table for this operation is a 16-element vector. For a 3-by-3 neighborhood, there are 512 permutations, so the lookup table is a 512-element vector. Once you create a lookup table, you can use it to perform the desired operation by using the applylut function. The example below illustrates using lookup-table operations to modify an image containing text. You begin by writing a function that returns 1 if three or more pixels in the 3-by-3 neighborhood are 1; otherwise, it returns 0. You then call makelut, passing in this function as the first argument, and using the second argument to specify a 3-by-3 lookup table. f = inline('sum(x(:)) >= 3'); lut = makelut(f,3); lut is returned as a 512-element vector of 1’s and 0’s. Each value is the output from the function for one of the 512 possible permutations. 9-53 9 Morphological Operations You then perform the operation using applylut. BW1 = imread('text.tif'); BW2 = applylut(BW1,lut); imshow(BW1) figure, imshow(BW2) Figure 9-10: Text.tif Before and After Applying a Lookup Table Operation For information about how applylut maps pixel combinations in the image to entries in the lookup table, see the reference page for applylut. Note You cannot use makelut and applylut for neighborhoods of sizes other than 2-by-2 or 3-by-3. These functions support only 2-by-2 and 3-by-3 neighborhoods, because lookup tables are not practical for neighborhoods larger than 3-by-3. For example, a lookup table for a 4-by-4 neighborhood would have 65,536 entries. 9-54 10 Analyzing and Enhancing Images This section describes the Image Processing Toolbox functions that support a range of standard image processing operations for analyzing and enhancing images. Topics covered include Terminology (p. 10-2) Provides definitions of image processing terms used in this section Pixel Values and Statistics (p. 10-3) Describes the toolbox functions that return information about the data values that make up an image Image Analysis (p. 10-10) Describes the toolbox functions that return information about the structure of an image. Image Enhancement (p. 10-14) Describes the toolbox functions used to improve an image, such as adjusting the intensity or removing noise. 10 Analyzing and Enhancing Images Terminology An understanding of the following terms will help you to use this chapter. 10-2 Terms Definitions Adaptive filter A filter whose properties vary across an image depending on the local characteristics of the image pixels. Contour A path in an image along which the image intensity values are equal to a constant. Edge A curve that follows a path of rapid change in image intensity. Edges are often associated with the boundaries of objects in a scene. Edge detection is used to identify the edges in an image. Property A quantitative measurement of an image or image region. Examples of image region properties include centroid, bounding box, and area. Histogram A graph used in image analysis that shows the distribution of intensities in an image. The information in a histogram can be used to choose an appropriate enhancement operation. For example, if an image histogram shows that the range of intensity values is small, you can use an intensity adjustment function to spread the values across a wider range. Noise Errors in the image acquisition process that result in pixel values that do not reflect the true intensities of the real scene. Profile A set of intensity values taken from regularly spaced points along a line segment or multiline path in an image. For points that do not fall on the center of a pixel, the intensity values are interpolated. Quadtree decomposition An image analysis technique that partitions an image into homogeneous blocks. Pixel Values and Statistics Pixel Values and Statistics The Image Processing Toolbox provides several functions that return information about the data values that make up an image. These functions return information about image data in various forms, including: • The data values for selected pixels (pixval, impixel) • The data values along a path in an image (improfile) • A contour plot of the image data (imcontour) • A histogram of the image data (imhist) • Summary statistics for the image data (mean2, std2, corr2) • Feature measurements for image regions (imfeature) Pixel Selection The toolbox includes two functions that provide information about the color data values of image pixels you specify: • The pixval function interactively displays the data values for pixels as you move the cursor over the image. pixval can also display the Euclidean distance between two pixels. • The impixel function returns the data values for a selected pixel or set of pixels. You can supply the coordinates of the pixels as input arguments, or you can select pixels using a mouse. To use pixval, you first display an image and then enter the pixval command. pixval installs a black bar at the bottom of the figure, which displays the (x,y) coordinates for whatever pixel the cursor is currently over, and the color data for that pixel. If you click on the image and hold down the mouse button while you move the cursor, pixval also displays the Euclidean distance between the point you clicked on and the current cursor location. pixval draws a line between these points to indicate the distance being measured. When you release the mouse button, the line and the distance display disappear. pixval gives you more immediate results than impixel, but impixel has the advantage of returning its results in a variable, and it can be called either interactively or noninteractively. If you call impixel with no input arguments, the cursor changes to a crosshair when it is over the image. You can then click 10-3 10 Analyzing and Enhancing Images on the pixels of interest; impixel displays a small star over each pixel you select. When you are done selecting pixels, press Return. impixel returns the color values for the selected pixels, and the stars disappear. In this example, you call impixel and click on three points in the displayed image, and then press Return. imshow canoe.tif vals = impixel * * * vals = 0.1294 0.5176 0.7765 0.1294 0 0.6118 0.1294 0 0.4196 Notice that the second pixel, which is part of the canoe, is pure red; its green and blue values are both 0. For indexed images, pixval and impixel both show the RGB values stored in the colormap, not the index values. Intensity Profile The improfile function calculates and plots the intensity values along a line segment or a multiline path in an image. You can supply the coordinates of the 10-4 Pixel Values and Statistics line segments as input arguments, or you can define the desired path using a mouse. In either case, improfile uses interpolation to determine the values of equally spaced points along the path. (By default, improfile uses nearest neighbor interpolation, but you can specify a different method. See Chapter 4, “Spatial Transformations”, for a discussion of interpolation.) improfile works best with intensity and RGB images. For a single line segment, improfile plots the intensity values in a two-dimensional view. For a multiline path, improfile plots the intensity values in a three-dimensional view. If you call improfile with no arguments, the cursor changes to a cross hair when it is over the image. You can then specify line segments by clicking on the endpoints; improfile draws a line between each two consecutive points you select. When you finish specifying the path, press Return. improfile displays the plot in a new figure. In this example, you call improfile and specify a single line with the mouse. The line is shown in red, and is drawn from left to right. imshow debye1.tif improfile 10-5 10 Analyzing and Enhancing Images improfile displays a plot of the data along the line. Figure 10-1: A Plot of Intensity Values Along a Line Segment in an Intensity Image Notice the peaks and valleys and how they correspond to the light and dark bands in the image. The example below shows how improfile works with an RGB image. The red line indicates where the line selection was made. Note that the line was drawn from top to bottom. imshow flowers.tif improfile 10-6 Pixel Values and Statistics The improfile function displays a plot with separate lines for the red, green, and blue intensities. Figure 10-2: A Plot of Intensity Values Along a Line Segment in an RGB Image Notice how the lines correspond to the colors in the image. For example, the central region of the plot shows high intensities of green and red, while the blue intensity is 0. These are the values for the yellow flower. 10-7 10 Analyzing and Enhancing Images Image Contours You can use the toolbox function imcontour to display a contour plot of the data in an intensity image. This function is similar to the contour function in MATLAB, but it automatically sets up the axes so their orientation and aspect ratio match the image. This example displays an intensity image of grains of rice and a contour plot of the image data. I = imread('rice.tif'); imshow(I) figure, imcontour(I) Figure 10-3: Rice.tif and Its Contour Plot You can use the clabel function to label the levels of the contours. See the description of clabel in the MATLAB Function Reference for details. Image Histogram An image histogram is a chart that shows the distribution of intensities in an indexed or intensity image. The image histogram function imhist creates this plot by making n equally spaced bins, each representing a range of data values. It then calculates the number of pixels within each range. For example, the commands below display an image of grains of rice, and a histogram based on 64 bins. 10-8 Pixel Values and Statistics I = imread('rice.tif'); imshow(I) figure, imhist(I,64) Figure 10-4: Rice.tif and Its Histogram The histogram shows a peak at around 100, due to the dark gray background in the image. For information about how to modify an image by changing the distribution of its histogram, see “Intensity Adjustment” on page 10-14. Summary Statistics You can compute standard statistics of an image using the mean2, std2, and corr2 functions. mean2 and std2 compute the mean and standard deviation of the elements of a matrix. corr2 computes the correlation coefficient between two matrices of the same size. These functions are two-dimensional versions of the mean, std, and corrcoef functions described in the MATLAB Function Reference. Region Property Measurement You can use the regionprops function to compute properties for image regions. For example, regionprops can measure such properties as the area, center of mass, and bounding box for a region you specify. See the reference page for regionprops for more information. 10-9 10 Analyzing and Enhancing Images Image Analysis Image analysis techniques return information about the structure of an image. This section describes toolbox functions that you can use for these image analysis techniques: • Edge detection • Quadtree decomposition The functions described in this section work only with intensity images. Edge Detection You can use the edge function to detect edges, which are those places in an image that correspond to object boundaries. To find edges, this function looks for places in the image where the intensity changes rapidly, using one of these two criteria: • Places where the first derivative of the intensity is larger in magnitude than some threshold • Places where the second derivative of the intensity has a zero crossing edge provides a number of derivative estimators, each of which implements one of the definitions above. For some of these estimators, you can specify whether the operation should be sensitive to horizontal or vertical edges, or both. edge returns a binary image containing 1’s where edges are found and 0’s elsewhere. The most powerful edge-detection method that edge provides is the Canny method. The Canny method differs from the other edge-detection methods in that it uses two different thresholds (to detect strong and weak edges), and includes the weak edges in the output only if they are connected to strong edges. This method is therefore less likely than the others to be “fooled” by noise, and more likely to detect true weak edges. The example below illustrates the power of the Canny edge detector. It shows the results of applying the Sobel and Canny edge detectors to the rice.tif image. I = imread('rice.tif'); BW1 = edge(I,'sobel'); BW2 = edge(I,'canny'); imshow(BW1) 10-10 Image Analysis figure, imshow(BW2) Sobel Filter Canny Filter For an interactive demonstration of edge detection, try running edgedemo. Quadtree Decomposition Quadtree decomposition is an analysis technique that involves subdividing an image into blocks that are more homogeneous than the image itself. This technique reveals information about the structure of the image. It is also useful as the first step in adaptive compression algorithms. You can perform quadtree decomposition using the qtdecomp function. This function works by dividing a square image into four equal-sized square blocks, and then testing each block to see if it meets some criterion of homogeneity (e.g., if all of the pixels in the block are within a specific dynamic range). If a block meets the criterion, it is not divided any further. If it does not meet the criterion, it is subdivided again into four blocks, and the test criterion is applied to those blocks. This process is repeated iteratively until each block meets the criterion. The result may have blocks of several different sizes. For example, suppose you want to perform quadtree decomposition on a 128-by-128 intensity image. The first step is to divide the image into four 64-by-64 blocks. You then apply the test criterion to each block; for example, the criterion might be max(block(:)) min(block(:)) <= 0.2 If one of the blocks meets this criterion, it is not divided any further; it is 64-by-64 in the final decomposition. If a block does not meet the criterion, it is 10-11 10 Analyzing and Enhancing Images then divided into four 32-by-32 blocks, and the test is then applied to each of these blocks. The blocks that fail to meet the criterion are then divided into four 16-by-16 blocks, and so on, until all blocks “pass.” Some of the blocks may be as small as 1-by-1, unless you specify otherwise. The call to qtdecomp for this example would be S = qtdecomp(I,0.2) S is returned as a sparse matrix whose nonzero elements represent the upper-left corners of the blocks; the value of each nonzero element indicates the block size. S is the same size as I. Note The threshold value is specified as a value between 0 and 1, regardless of the class of I. If I is uint8, the threshold value you supply is multiplied by 255 to determine the actual threshold to use; if I is uint16, the threshold value you supply is multiplied by 65535. The example below shows an image and a representation of its quadtree decomposition. Each black square represents a homogeneous block, and the white lines represent the boundaries between blocks. Notice how the blocks are smaller in areas corresponding to large changes in intensity in the image. Figure 10-5: An Image (left) and a Representation of its Quadtree Decomposition 10-12 Image Analysis You can also supply qtdecomp with a function (rather than a threshold value) for deciding whether to split blocks; for example, you might base the decision on the variance of the block. See the reference page for qtdecomp for more information. For an interactive demonstration of quadtree decomposition, try running qtdemo. 10-13 10 Analyzing and Enhancing Images Image Enhancement Image enhancement techniques are used to improve an image, where “improve” is sometimes defined objectively (e.g., increase the signal-to-noise ratio), and sometimes subjectively (e.g., make certain features easier to see by modifying the colors or intensities). This section discusses these image enhancement techniques: • “Intensity Adjustment” • “Noise Removal” The functions described in this section apply primarily to intensity images. However, some of these functions can be applied to color images as well. For information about how these functions work with color images, see the reference pages for the individual functions. Intensity Adjustment Intensity adjustment is a technique for mapping an image’s intensity values to a new range. For example, rice.tif. is a low contrast image. The histogram of rice.tif, shown in Figure 10-4, indicates that there are no values below 40 or above 225. If you remap the data values to fill the entire intensity range [0, 255], you can increase the contrast of the image. You can do this kind of adjustment with the imadjust function. The general syntax of imadjust is J = imadjust(I,[low_in high_in],[low_out high_out]) where low_in and high_in are the intensities in the input image which are mapped to low_out and high_out in the output image. For example, this code performs the adjustment described above. I = imread('rice.tif'); J = imadjust(I,[0.15 0.9],[0 1]); The first vector passed to imadjust, [0.15 0.9], specifies the low and high intensity values that you want to map. The second vector, [0 1], specifies the scale over which you want to map them. Thus, the example maps the intensity value 0.15 in the input image to 0 in the output image, and 0.9 to 1. 10-14 Image Enhancement Note that you must specify the intensities as values between 0 and 1 regardless of the class of I. If I is uint8, the values you supply are multiplied by 255 to determine the actual values to use; if I is uint16, the values are multiplied by 65535. To learn about an alternative way to set this limits automatically, see “Setting the Adjustment Limits Automatically” on page 10-16. This figure displays the adjusted image and its histogram. Notice the increased contrast in the image, and that the histogram now fills the entire range. imshow(J) figure, imhist(J,64) Figure 10-6: Rice.tif After an Intensity Adjustment and a Histogram of Its Adjusted Intensities Similarly, you can decrease the contrast of an image by narrowing the range of the data, as in this call. J = imadjust(I,[0 1],[0.3 0.8]); In addition to increasing or decreasing contrast, you can perform a wide variety of other image enhancements with imadjust. In the example below, the man’s coat is too dark to reveal any detail. The call to imadjust maps the range [0,51] in the uint8 input image to [128,255] in the output image. This brightens the image considerably, and also widens the dynamic range of the dark portions of the original image, making it much easier to see the details in the coat. Note, however, that because all values above 51 in the original image get mapped to 255 (white) in the adjusted image, the adjusted image appears “washed out.” 10-15 10 Analyzing and Enhancing Images I = imread('cameraman.tif'); J = imadjust(I,[0 0.2],[0.5 1]); imshow(I) figure, imshow(J) Figure 10-7: Remapping and Widening the Dynamic Range Setting the Adjustment Limits Automatically To use imadjust, you must typically perform two steps: 1 View the histogram of the image to determine the intensity value limits. 2 Specify these limits as a fraction between 0.0 and 1.0 so that you can pass them to imadjust in the [low_in high_in] vector. For a more convenient way to specify these limits, use the stretchlim function. This function calculates the histogram of the image and determines the adjustment limits automatically. The stretchlim function returns these values as fractions in a vector that you can pass as the [low_in high_in] argument to imadjust; for example, I = imread('rice.tif'); J = imadjust(I,stretchlim(I),[0 1]); By default, stretchlim uses the intensity values that represent the bottom 1% (0.01) and the top 1% (0.99) of the range as the adjustment limits. By trimming the extremes at both ends of the intensity range, stretchlim makes more room in the adjusted dynamic range for the remaining intensities. But you can 10-16 Image Enhancement specify other range limits as an argument to stretchlim. See the stretchlim reference page for more information. Gamma Correction imadjust maps low to bottom, and high to top. By default, the values between low and high are mapped linearly to values between bottom and top. For example, the value halfway between low and high corresponds to the value halfway between bottom and top. imadjust can accept an additional argument which specifies the gamma correction factor. Depending on the value of gamma, the mapping between values in the input and output images may be nonlinear. For example, the value halfway between low and high may map to a value either greater than or less than the value halfway between bottom and top. Gamma can be any value between 0 and infinity. If gamma is 1 (the default), the mapping is linear. If gamma is less than 1, the mapping is weighted toward higher (brighter) output values. If gamma is greater than 1, the mapping is weighted toward lower (darker) output values. The figure below illustrates this relationship. The three transformation curves show how values are mapped when gamma is less than, equal to, and greater than 1. (In each graph, the x-axis represents the intensity values in the input image, and the y-axis represents the intensity values in the output image.) γ<1 γ=1 γ>1 top top top bottom bottom bottom low high low high low high Figure 10-8: Plots Showing Three Different Gamma Correction Settings The example below illustrates gamma correction. Notice that in the call to imadjust, the data ranges of the input and output images are specified as empty matrices. When you specify an empty matrix, imadjust uses the default range of [0,1]. In the example, both ranges are left empty; this means that gamma correction is applied without any other adjustment of the data. 10-17 10 Analyzing and Enhancing Images [X,map] = imread('forest.tif') I = ind2gray(X,map); J = imadjust(I,[],[],0.5); imshow(I) figure, imshow(J) Figure 10-9: Forest.tif Before and After Applying Gamma Correction of 0.5 Histogram Equalization The process of adjusting intensity values can be done automatically by the histeq function. histeq performs histogram equalization, which involves transforming the intensity values so that the histogram of the output image approximately matches a specified histogram. (By default, histeq tries to match a flat histogram with 64 bins, but you can specify a different histogram instead; see the reference page for histeq.) This example illustrates using histeq to adjust an intensity image. The original image has low contrast, with most values in the middle of the intensity range. histeq produces an output image having values evenly distributed throughout the range. I = imread('pout.tif'); J = histeq(I); imshow(I) figure, imshow(J) 10-18 Image Enhancement Figure 10-10: Pout.tif Before and After Histogram Equalization The example below shows the histograms for the two images. figure, imhist(I) figure, imhist(J) Figure 10-11: Histogram Before Equalization (left) and After Equalization (right) histeq can return an additional 1-by-256 vector that shows, for each possible input value, the resulting output value. (The values in this vector are in the 10-19 10 Analyzing and Enhancing Images range [0,1], regardless of the class of the input image.) You can plot this data to get the transformation curve. For example, I = imread('pout.tif'); [J,T] = histeq(I); figure,plot((0:255)/255,T); Notice how this curve reflects the histograms in the previous figure, with the input values being mostly between 0.3 and 0.6, while the output values are distributed evenly between 0 and 1. For an interactive demonstration of intensity adjustment, try running imadjdemo. Noise Removal Digital images are prone to a variety of types of noise. There are several ways that noise can be introduced into an image, depending on how the image is created. For example: • If the image is scanned from a photograph made on film, the film grain is a source of noise. Noise can also be the result of damage to the film, or be introduced by the scanner itself. • If the image is acquired directly in a digital format, the mechanism for gathering the data (such as a CCD detector) can introduce noise. 10-20 Image Enhancement • Electronic transmission of image data can introduce noise. The toolbox provides a number of different ways to remove or reduce noise in an image. Different methods are better for different kinds of noise. The methods available include: • Linear filtering • Median filtering • Adaptive filtering Also, in order to simulate the effects of some of the problems listed above, the toolbox provides the imnoise function, which you can use to add various types of noise to an image. The examples in this section use this function. Linear Filtering You can use linear filtering to remove certain types of noise. Certain filters, such as averaging or Gaussian filters, are appropriate for this purpose. For example, an averaging filter is useful for removing grain noise from a photograph. Because each pixel gets set to the average of the pixels in its neighborhood, local variations caused by grain are reduced. See “Linear Filtering” on page 7-4 for more information. Median Filtering Median filtering is similar to using an averaging filter, in that each output pixel is set to an “average” of the pixel values in the neighborhood of the corresponding input pixel. However, with median filtering, the value of an output pixel is determined by the median of the neighborhood pixels, rather than the mean. The median is much less sensitive than the mean to extreme values (called outliers). Median filtering is therefore better able to remove these outliers without reducing the sharpness of the image. The medfilt2 function implements median filtering. The example below compares using an averaging filter and medfilt2 to remove salt and pepper noise. This type of noise consists of random pixels being set to black or white (the extremes of the data range). In both cases the size of the neighborhood used for filtering is 3-by-3. First, read in the image and add noise to it. I = imread('eight.tif'); 10-21 10 Analyzing and Enhancing Images J = imnoise(I,'salt & pepper',0.02); imshow(I) figure, imshow(J) Figure 10-12: Eight.tif Before and After Adding Salt-and-Pepper Noise Now filter the noisy image and display the results. Notice that medfilt2 does a better job of removing noise, with less blurring of edges. K = filter2(fspecial('average',3),J)/255; L = medfilt2(J,[3 3]); figure, imshow(K) figure, imshow(L) Averaging Filter Figure 10-13: Noisy Version of Eight.tif Filtered with Averaging Filter (left) and Median Filter (right) 10-22 Image Enhancement Median filtering is a specific case of order-statistic filtering, also known as rank filtering. For information about order-statistic filtering, see the reference page for the ordfilt2 function. Adaptive Filtering The wiener2 function applies a Wiener filter (a type of linear filter) to an image adaptively, tailoring itself to the local image variance. Where the variance is large, wiener2 performs little smoothing. Where the variance is small, wiener2 performs more smoothing. This approach often produces better results than linear filtering. The adaptive filter is more selective than a comparable linear filter, preserving edges and other high frequency parts of an image. In addition, there are no design tasks; the wiener2 function handles all preliminary computations, and implements the filter for an input image. wiener2, however, does require more computation time than linear filtering. wiener2 works best when the noise is constant-power (“white”) additive noise, such as Gaussian noise. The example below applies wiener2 to an image of Saturn that has had Gaussian noise added. I = imread('saturn.tif'); J = imnoise(I,'gaussian',0,0.005); K = wiener2(J,[5 5]); imshow(J) figure, imshow(K) Figure 10-14: Noisy Version of Saturn.tif Before and After Adaptive Filtering 10-23 10 Analyzing and Enhancing Images For an interactive demonstration of filtering to remove noise, try running nrfiltdemo. 10-24 11 Region-Based Processing This section describes operations that you can perform on a selected region of an image. Topics covered include Terminology (p. 11-2) Provides definitions of image processing terms used in this section Specifying a Region of Interest (p. 11-3) Describes how to specify a region of interest using the roipoly function Filtering a Region (p. 11-6) Describes how to apply a filter to a region using the roifilt2 function Filling a Region (p. 11-8) Describes how to fill a region of interest using the roifill function 11 Region-Based Processing Terminology An understanding of the following terms will help you to use this chapter. 11-2 Terms Definitions Binary mask A binary image with the same size as the image you want to process. The mask contains 1’s for all pixels that are part of the region of interest, and 0’s everywhere else. Filling a region A process that “fills” a region of interest by interpolating the pixel values from the borders of the region. This process can be used to make objects in an image seem to disappear as they are replaced with values that blend in with the background area. Filtering a region The process of applying a filter to a region of interest. For example, you can apply an intensity adjustment filter to certain regions of an image. Interpolation The process by which we estimate an image value at a location in between image pixels. Masked filtering An operation that applies filtering only to the regions of interest in an image that are identified by a binary mask. Filtered values are returned for pixels where the binary mask contains 1’s; unfiltered values are returned for pixels where the binary mask contains 0’s. Region of interest A portion of an image that you want to filter or perform some other operation on. You define a region of interest by creating a binary mask. There can be more than one region defined in an image. The regions can be “geographic” in nature, such as polygons that encompass contiguous pixels, or they can be defined by a range of intensities. In the latter case, the pixels are not necessarily contiguous. Specifying a Region of Interest Specifying a Region of Interest A region of interest is a portion of an image that you want to filter or perform some other operation on. You define a region of interest by creating a binary mask, which is a binary image with the same size as the image you want to process. The mask contains 1’s for all pixels that are part of the region of interest, and 0’s everywhere else. The following subsections discuss methods for creating binary masks: • “Selecting a Polygon” on page 11-3 • “Other Selection Methods” on page 11-4 (using any binary mask or the roicolor function) For an interactive demonstration of region-based processing, try running roidemo. Selecting a Polygon You can use the roipoly function to specify a polygonal region of interest. If you call roipoly with no input arguments, the cursor changes to a cross hair when it is over the image displayed in the current axes. You can then specify the vertices of the polygon by clicking on points in the image with the mouse. When you are done selecting vertices, press Return; roipoly returns a binary image of the same size as the input image, containing 1’s inside the specified polygon, and 0’s everywhere else. The example below illustrates using the interactive syntax of roipoly to create a binary mask. The border of the selected region in Figure 11-1, which was created using a mouse, is shown in red. I = imread('pout.tif'); imshow(I) BW = roipoly; 11-3 11 Region-Based Processing Figure 11-1: A Polygonal Region of Interest Selected Using roipoly imshow(BW) Figure 11-2: A Binary Mask Created for the Region Shown in Figure 11-1. You can also use roipoly noninteractively. See the reference page for roipoly for more information. Other Selection Methods roipoly provides an easy way to create a binary mask. However, you can use any binary image as a mask, provided that the binary image is the same size as the image being filtered. 11-4 Specifying a Region of Interest For example, suppose you want to filter the intensity image I, filtering only those pixels whose values are greater than 0.5. You can create the appropriate mask with this command. BW = (I > 0.5); You can also use the roicolor function to define the region of interest based on a color or intensity range. For more information, see the reference page for roicolor. 11-5 11 Region-Based Processing Filtering a Region You can use the roifilt2 function to process a region of interest. When you call roifilt2, you specify an intensity image, a binary mask, and a filter. roifilt2 filters the input image and returns an image that consists of filtered values for pixels where the binary mask contains 1’s, and unfiltered values for pixels where the binary mask contains 0’s. This type of operation is called masked filtering. This example uses the mask created in the example in “Selecting a Polygon” on page 11-3 to increase the contrast of the logo on the girl’s coat. h = fspecial('unsharp'); I2 = roifilt2(h,I,BW); imshow(I) figure, imshow(I2) Figure 11-3: An Image Before and After Using an Unsharp Filter on the Region of Interest. roifilt2 also enables you to specify your own function to operate on the region of interest. In the example below, the imadjust function is used to lighten parts of an image. The mask in the example is a binary image containing text. The resulting image has the text imprinted on it. BW = imread('text.tif'); I = imread('cameraman.tif'); f = inline('imadjust(x,[],[],0.3)'); I2 = roifilt2(I,BW,f); 11-6 Filtering a Region imshow(I2) Figure 11-4: An Image Brightened Using a Binary Mask Containing Text Note that roifilt2 is best suited to operations that return data in the same range as in the original image because the output image takes some of its data directly from the input image. Certain filtering operations can result in values outside the normal image data range (i.e., [0,1] for images of class double, [0,255] for images of class uint8, [0,65535] for images of class uint16). For more information, see the reference page for roifilt2. 11-7 11 Region-Based Processing Filling a Region You can use the roifill function to fill a region of interest, interpolating from the borders of the region. This function is useful for image editing, including removal of extraneous details or artifacts. roifill performs the fill operation using an interpolation method based on Laplace’s equation. This method results in the smoothest possible fill, given the values on the boundary of the region. As with roipoly, you select the region of interest with the mouse. When you complete the selection, roifill returns an image with the selected region filled in. This example uses roifill to modify the trees image. The border of the selected region is shown in red on the original image. load trees I = ind2gray(X,map); imshow(I) I2 = roifill; Figure 11-5: A Region of Interest Selected for Filling imshow(I2) 11-8 Filling a Region Figure 11-6: The Region of Interest Shown in Figure 11-5 Has Been Filled 11-9 11 Region-Based Processing 11-10 12 Image Deblurring This section describes how to deblur an image using the toolbox deblurring functions. Topics covered include Terminology (p. 12-2) Provides definitions of image processing terms used in this section Understanding Deblurring (p. 12-3) Defines deblurring and deconvolution Using the Deblurring Functions (p. 12-5) Provides step-by-step examples of using deconvwnr, deconvreg, deconvlucy, and deconvblind functions Avoiding Ringing in Deblurred Images (p. 12-20) Describes how to use the edgetaper function to avoid “ringing” in deblurred images 12 Image Deblurring Terminology An understanding of the following terms will help you to use this chapter. 12-2 Terms MATLAB Definition Deconvolution The process of reversing the effect of convolution. Distortion operator The operator that describes a process causing the acquired image to be different from the original scene. Distortion caused by a Point Spread Function (see below) is just one type of distortion. Optical transfer function (OTF) In the frequency domain, the OTF describes the response of a linear, position invariant system to an impulse. The OTF is the Fourier transform of the point spread function (PSF). Point spread function (PSF) In the spatial domain, the PSF describes the degree to which an optical system blurs (spreads) a point of light. The PSF is the inverse Fourier transform of the OTF. Understanding Deblurring Understanding Deblurring This section provides some background on deblurring techniques. The section includes these topics: • “Causes of Blurring” • “Deblurring Model” Causes of Blurring The blurring, or degradation, of an image can be caused by many factors: • Movement during the image capture process, by the camera or, when long exposure times are used, by the subject • Out-of-focus optics, use of a wide-angle lens, atmospheric turbulence, or a short exposure time, which reduced the number of photons captured • Scattered light distortion in confocal microscopy Deblurring Model A blurred or degraded image can be approximately described by this equation g = Hf + n, where: g = The blurred image H = The distortion operator, also called the point-spread function (PSF). This function, when convolved with the image, creates the distortion f = The original true image n = Additive noise, introduced during image acquisition, that corrupts the image Note The image f really doesn’t exist. This image represents what you would have if you had perfect image acquisition conditions. The Importance of the PSF Based on this model, the fundamental task of deblurring is to deconvolve the blurred image with the PSF that exactly describes the distortion. 12-3 12 Image Deblurring Note The quality of the deblurred image is mainly determined by knowledge of the PSF. To illustrate, this example takes a clear image and deliberately blurs it by convolving it with a PSF. The example uses the fspecial function to create a PSF that simulates a motion blur, specifying the length of the blur in pixels, (LEN=31), and the angle of the blur in degrees (THETA=11). Once the PSF is created, the example uses the imfilter function to convolve the PSF with the original image, I, to create the blurred image, Blurred. (To see how deblurring is the reverse of this process, using the same images, see “Deblurring with the Wiener Filter” on page 12-6.) I = imread('flowers.tif'); I = I(10+[1:256],222+[1:256],:); % crop the image figure;imshow(I);title('Original Image'); LEN = 31; THETA = 11; PSF = fspecial('motion',LEN,THETA); Blurred = imfilter(I,PSF,'circular','conv'); figure; imshow(Blurred);title('Blurred Image'); Original image 12-4 Blurred image Using the Deblurring Functions Using the Deblurring Functions The toolbox includes four deblurring functions, listed here in order of complexity: • deconvwnr — Implements deblurring using the Wiener filter • deconvreg — Implements deblurring using a regularized filter • deconvlucy — Implements deblurring using the Lucy-Richardson algorithm • deconvblind — Implements deblurring using the blind deconvolution algorithm All the functions accept a PSF and the blurred image as their primary arguments. The deconvwnr function implements a least squares solution. The deconvreg function implements a constrained least squares solution, where you can place constraints on the output image (the smoothness requirement is the default). With either of these functions, you should provide some information about the noise to reduce possible noise amplification during deblurring. The deconvlucy function implements an accelerated, damped Lucy-Richardson algorithm. This function performs multiple iterations, using optimization techniques and Poisson statistics. With this function, you do not need to provide information about the additive noise in the corrupted image. The deconvblind function implements the blind deconvolution algorithm, which performs deblurring without knowledge of the PSF. When you call deconvblind, you pass as an argument your initial guess at the PSF. The deconvblind function returns a restored PSF in addition to the restored image. The implementation uses the same damping and iterative model as the deconvlucy function. Note You may need to perform many iterations of the deblurring process, varying the parameters you specify to the deblurring functions with each iteration, until you achieve an image that, based on the limits of your information, is the best approximation of the original scene. Along the way, you must make numerous judgements about whether newly uncovered features in the image are features of the original scene, or simply artifacts of the deblurring process. 12-5 12 Image Deblurring For information about creating your own deblurring functions, see “Creating Your Own Deblurring Functions” on page 12-19. To avoid “ringing” in a deblurred image, you can use the edgetaper function to preprocess your image before passing it to the deblurring functions. See “Avoiding Ringing in Deblurred Images” on page 12-20 for more information. Deblurring with the Wiener Filter Use the deconvwnr function to deblur an image using the Wiener filter. Wiener deconvolution can be used effectively when the frequency characteristics of the image and additive noise are known, to at least some degree. In the absence of noise, the Wiener filter reduces to the ideal inverse filter. This example deblurs the blurred flower image, created in “Deblurring Model” on page 12-3, specifying the same PSF function that was used to create the blur. I = imread('flowers.tif'); I = I(10+[1:256],222+[1:256],:); % crop the image figure;imshow(I);title('Original Image'); % create PSF LEN = 31; THETA = 11; PSF = fspecial('motion',LEN,THETA); % blur the image Blurred = imfilter(I,PSF,'circular','conv'); figure; imshow(Blurred);title('Blurred Image'); % deblur the image wnr1 = deconvwnr(Blurred,PSF); figure;imshow(wnr1); title('Restored, True PSF'); 12-6 Using the Deblurring Functions Original image Blurred image Image restored by Wiener filter This example illustrates the importance of knowing the PSF, the function that caused the blur. When you know the exact PSF, as in this example, the results of deblurring can be quite effective. Refining the Result You can affect the deconvolution results by providing values for the optional arguments supported by the deconvwnr function. Using these arguments you can specify the noise-to-signal power value and/or provide autocorrelation functions to help refine the result of deblurring. To see the impact of these optional arguments, view the Image Processing Toolbox Deblurring Demos. 12-7 12 Image Deblurring Deblurring with a Regularized Filter Use the deconvreg function to deblur an image using a regularized filter. A regularized filter can be used effectively when limited information is known about the additive noise. To illustrate, this example simulates a blurred image by convolving a Gaussian filter PSF with an image (using imfilter). Additive noise in the image is simulated by adding Gaussian noise of variance V to the blurred image (using imnoise). I = imread('tissue1.tif'); I = I(350+[1:256],1:256,:); figure;imshow(I);title('Original Image'); % Create the PSF PSF = fspecial('gaussian',11,5); % Blur the image Blurred = imfilter(I,PSF,'conv'); % Add noise V = .02; BlurredNoisy = imnoise(Blurred,'gaussian',0,V); figure;imshow(BlurredNoisy);title('Blurred & Noisy'); Original image Blurred & noisy image Use deconvreg to deblur the image, specifying the PSF used to create the blur, and the noise-to-signal power ratio, NP. 12-8 Using the Deblurring Functions NP = V*prod(size(I)); % noise power [reg1 LAGRA] = deconvreg(BlurredNoisy,PSF,NP); figure,imshow(reg1),title('Restored with NP'); Restored image Refining the Result You can affect the deconvolution results by providing values for the optional arguments supported by the deconvreg function. Using these arguments you can specify the noise power value, the range over which deconvreg should iterate as it converges on the optimal solution, and the regularization operator to constrain the deconvolution. To see the impact of these optional arguments, view the Image Processing Toolbox Deblurring Demos. Deblurring with the Lucy-Richardson Algorithm Use the deconvlucy function to deblur an image using the accelerated, damped, Lucy-Richardson algorithm. The algorithm maximizes the likelihood that the resulting image, when convolved with the PSF, is an instance of the blurred image, assuming Poisson noise statistics. This function can be effective when you know the PSF, but know little about the additive noise in the image. The deconvlucy function implements several adaptations to the original Lucy-Richardson maximum likelihood algorithm which address complex image restoration tasks. Using these adaptations, you can: • Reduce the effect of noise amplification on image restoration • Account for nonuniform image quality (e.g., bad pixels, flat-field variation) • Handle camera read-out and background noise 12-9 12 Image Deblurring • Improve the restored image resolution by subsampling The following sections provide more information about each of these adaptations. Reducing the Effect of Noise Amplification Noise amplification is a common problem of maximum likelihood methods that attempt to fit data as closely as possible. After many iterations, the restored image can have a speckled appearance, especially for a smooth object observed at low signal-to-noise ratios. These speckles do not represent any real structure in the image, but are artifacts of fitting the noise in the image too closely. To control noise amplification, the deconvlucy function uses a damping parameter, DAMPAR. This parameter specifies the threshold level for the deviation of the resulting image from the original image, below which damping occurs. For pixels that deviate in the vicinity of their original value, iterations are suppressed. Damping is also used to reduce ringing, the appearance of high-frequency structures in a restored image. Ringing is not necessarily the result of noise amplification. See “Avoiding Ringing in Deblurred Images” on page 12-20 for more information. Accounting for Nonuniform Image Quality Another complication of real-life image restoration is that the data might include bad pixels, or that the quality of the receiving pixels might vary with time and position. By specifying the WEIGHT array parameter with the deconvlucy function, you can specify that certain pixels in the image be ignored. To ignore a pixel, assign a weight of zero to the element in the WEIGHT array that corresponds to the pixel in the image. The algorithm converges on predicted values for the bad pixels based on the information from neighborhood pixels. The variation in the detector response from pixel to pixel (the so-called flat-field correction) can also be accommodated by the WEIGHT array. Instead of assigning a weight of 1.0 to the good pixels, you can specify fractional values and weight the pixels according to the amount of the flat-field correction. 12-10 Using the Deblurring Functions Handling Camera Read-Out Noise Noise in charge coupled device (CCD) detectors has two primary components: • Photon counting noise with a Poisson distribution • Read-out noise with a Gaussian distribution The Lucy-Richardson iterations intrinsically account for the first type of noise. You must account for the second type of noise; otherwise, it may cause pixels with low levels of incident photons to have negative values. The deconvlucy function uses the READOUT input parameter to handle camera read-out noise. The value of this parameter is typically the sum of the read-out noise variance and the background noise (e.g., number of counts from the background radiation.) The value of the READOUT parameter specifies an offset that ensures that all values are positive. Handling Undersampled Images The restoration of undersampled data can be improved significantly if it is done on a finer grid. The deconvlucy function uses the SUBSMPL parameter to specify the subsampling rate, if the PSF is known to have a higher resolution. If the undersampled data is the result of camera pixel binning during image acquisition, the PSF observed at each pixel rate would serve as a finer grid PSF. Otherwise, the PSF could be obtained via observations taken at subpixel offsets or via optical modeling techniques. This method is especially effective for the images of stars (high signal-to-noise ratio), because the stars are effectively forced to be in the center of a pixel. If a star is centered between the pixels, it will be restored as a combination of the neighboring pixels. A finer grid would redirect the consequent spreading of the star flux back to the center of the star's image. 12-11 12 Image Deblurring Creating a Sample Blurred Image To illustrate a simple use of deconvlucy, this example simulates a blurred, noisy image by convolving a Gaussian filter PSF with an image (using imfilter) and then adding Gaussian noise of variance V to the blurred image (using imnoise). I = imread('board.tif'); I = I(50+[1:256],2+[1:256],:); figure;imshow(I);title('Original Image'); % Create the PSF PSF = fspecial('gaussian',5,5); % Simulate the blur Blurred = imfilter(I,PSF,'symmetric','conv'); % Add noise V = .002; BlurredNoisy = imnoise(Blurred,'gaussian',0,V); figure;imshow(BlurredNoisy);title('Blurred & Noisy'); Original image Blurred & noisy image Deblurring the Image Use deconvlucy to restore the blurred and noisy image, specifying the PSF used to create the blur, and limiting the number of iterations to 5 (the default is 10). 12-12 Using the Deblurring Functions Note The deconvlucy function can return values in the output image that are beyond the range of the input image. luc1 = deconvlucy(BlurredNoisy,PSF,5); figure; imshow(luc1); title('Restored Image, NUMIT = 5'); Restored image Refining the Result The deconvlucy function, by default, performs multiple iterations of the deblurring process. You can stop the processing, after a certain number of iterations, to check the result, and then restart the iterations from the point where processing stopped. To do this, pass in the input image as a cell array, for example, {BlurredNoisy}. The deconvlucy function returns the output image as a cell array which you can then pass as an input argument to deconvlucy to restart the deconvolution. The output cell array contains these four elements: • output{1} — The original input image • output{2} — The image produced by the last iteration • output{3} — The image produced by the next-to-last iteration • output{4} — Internal information used by deconvlucy to know where to restart the process 12-13 12 Image Deblurring The deconvlucy function supports several other optional arguments you can use to achieve the best possible result, such as specifying a damping parameter to handle additive noise in the blurred image. To see the impact of these optional arguments, view the Image Processing Toolbox Deblurring Demos. Deblurring with the Blind Deconvolution Algorithm Use the deconvblind function to deblur an image using the blind deconvolution algorithm. The algorithm maximizes the likelihood that the resulting image, when convolved with the resulting PSF, is an instance of the blurred image, assuming Poisson noise statistics. The blind deconvolution algorithm can be used effectively when no information about the distortion (blurring and noise) is known. The deconvblind function restores the image and the PSF simultaneously, using an iterative process similar to the accelerated, damped Lucy-Richardson algorithm. The deconvblind function, just like the deconvlucy function, implements several adaptations to the original Lucy-Richardson maximum likelihood algorithm, which address complex image restoration tasks. Using these adaptations, you can: • Reduce the effect of noise on the restoration • Account for nonuniform image quality (e.g., bad pixels) • Handle camera read-out noise For more information about these adaptations, see “Deblurring with the Lucy-Richardson Algorithm” on page 12-9. In addition, the deconvblind function supports PSF constraints that can be passed in through a user-specified function. Creating a Sample Blurred Image To illustrate blind deconvolution, this example simulates a blurred image by convolving a motion filter PSF with an image (using imfilter). I = imread('cameraman.tif'); figure; imshow(I); title('Original Image'); PSF = fspecial('motion',13,45);% Create the PSF figure; imshow(PSF,[]); title('True PSF'); Blurred = imfilter(I,PSF,'circ','conv');% Simulate the blur 12-14 Using the Deblurring Functions figure; imshow(Blurred); title('Blurred Image'); Original image Original PSF Blurred image Image Restoration: First Pass As a first pass at restoring the blurred image of the cameraman, call the deconvblind function specifying the image and an initial guess at the PSF as arguments. When you specify the PSF, you must estimate its size and the values it contains. Your guess at the size of the initial PSF is more important to the ultimate success of the restoration than the values in the PSF. To determine the size, examine the blurred image and measure the width of a blur (in pixels) around an obviously sharp object. For example, in the sample blurred image, you can measure the blur near the contour of the man’s sleeve. Because your initial guess at the values in the PSF is less important than the size, you can typically specify an array of 1’s as the initial PSF. The following example shows a restoration where the initial guess at the PSF is the same size as the true PSF that caused the blur. INITPSF = ones(size(PSF)); [J P]= deconvblind(Blurred,INITPSF,30); figure; imshow(J); title('Preliminary Restoration'); figure; imshow(P,[],'notruesize'); title('Preliminary Restoration'); 12-15 12 Image Deblurring Restored image Restored PSF This example specified an initial PSF that was the same size as the true PSF, i.e., the PSF used to create the blur. In a real application, you may need to rerun deconvblind, experimenting with PSFs of different sizes, until you achieve a satisfactory result. The restored PSF returned by each deconvolution can also provide valuable hints at the optimal PSF size. See the Image Processing Toolbox Deblurring Demos for an example. Image Restoration: Second Pass Although the first pass did succeed in deblurring the image to some extent, the ringing in the restored image around the sharp intensity contrast areas is unsatisfactory. (The example deliberately eliminated edge-related ringing by using the 'circular' option in imfilter while creating a blurred image.) This example makes a second pass at deblurring, this time achieving a better result by using both the optional WEIGHT array parameter and by refining the guess at the initial PSF, P1. Creating a WEIGHT Array. To reduce this contrast-related ringing, rerun the deconvolution, this time using the optional WEIGHT array parameter to exclude areas of high-contrast from the deblurring operation. You exclude a pixel from processing by assigning the value 0 to the corresponding element in the WEIGHT array. (See “Accounting for Nonuniform Image Quality” on page 12-10 for information about WEIGHT arrays.) This example uses edge detection and morphological processing to create a WEIGHT array. The edge, strel, and imdilate functions detect the high-contrast areas in the image. Because the blur in the image is linear, the example dilates the image twice. (For more information about using these 12-16 Using the Deblurring Functions functions, see Chapter 9, “Morphological Operations.”) To exclude the image boundary pixels (a high-contrast area) from processing, the example uses padarray to assign the value 0 to all border pixels. WEIGHT = edge(I,'sobel',.28); se1 = strel('disk',1); se2 = strel('line',13,45); WEIGHT = ~imdilate(WEIGHT,[se1 se2]); WEIGHT = padarray(WEIGHT([1:2 end-[0:1]],[1:2 end-[0:1]]),[2 2]); figure; imshow(WEIGHT); title('Weight array'); Weight array Constraining the Restored PSF. Before repeating the deconvolution with the WEIGHT array, the example refines the guess at the PSF. The reconstructed PSF, P, returned by the first pass at deconvolution shows a clear linearity (see the image of the Restored PSF in “Image Restoration: First Pass” on page 12-15). For the second pass, the example uses a new PSF, P1, which is same as the restored PSF but with the small amplitude pixels set to 0. P1 = P; P1(find(P1 < 0.01))=0; [J2 P2] = deconvblind(Blurred,P1,50,[],WEIGHT); figure; imshow(J2); title('Newly deblurred image'); figure; imshow(P2,[],'notruesize'); title('Newly reconstructed PSF'); 12-17 12 Image Deblurring Newly deblurred image Newly reconstructed PSF Refining the Result The deconvblind function, by default, performs multiple iterations of the deblurring process. You can stop the processing, after a certain number of iterations, to check the result, and then restart the iterations from the point where processing stopped. To use this feature, you must pass in both the blurred image and the PSF as cell arrays, for example, {Blurred} and {INITPSF}. The deconvblind function returns the output image and the restored PSF as cell arrays. The output image cell array contains these four elements: output{1} — The original input image output{2} — The image produced by the last iteration output{3} — The image produced by the next-to-last iteration output{4} — Internal information used by deconvblind to know where to restart the process The PSF output cell array contains similar elements. The deconvblind function supports several other optional arguments you can use to achieve the best possible result, such as specifying a damping parameter to handle additive noise in the blurred image. To see the impact of these optional arguments, as well as the functional option that allows you to place additional constraints on the PSF reconstruction, see the Image Processing Toolbox Deblurring Demos. 12-18 Using the Deblurring Functions Creating Your Own Deblurring Functions All the toolbox deblurring functions perform deconvolution in the frequency domain, where the process becomes a simple matrix multiplication. To work in the frequency domain, the deblurring functions must convert the PSF you provide into an optical transfer function (OTF), using the psf2otf function. The toolbox also provides a function to convert an OTF into a PSF, otf2psf. The toolbox makes these functions available in case you want to create your own deblurring functions. (In addition, to aid this conversion between PSFs and OTFs, the toolbox also makes the padding function, padarray, available.) 12-19 12 Image Deblurring Avoiding Ringing in Deblurred Images The Discrete Fourier Transform (DFT), used by the deblurring functions, assumes that the frequency pattern of an image is periodic. This assumption creates a high-frequency drop-off at the edges of images. In the figure, the shaded area represents the actual extent of the image; the unshaded area represents the assumed periodicity. High-frequency drop-off Image Assumed periodic repetition of the image This high-frequency drop-off can create an effect called boundary related ringing in deblurred images. In this figure, note the horizontal and vertical patterns in the image. To avoid ringing, use the edgetaper function to preprocess your images before passing them to the deblurring functions. The edgetaper function removes the high-frequency drop-off at the edge of an image by blurring the entire image and then replacing the center pixels of the blurred image with the original image. In this way, the edges of the image taper off to a lower frequency. 12-20 13 Color This section describes the toolbox functions that help you work with color image data. Note that “color” includes shades of gray; therefore much of the discussion in this chapter applies to grayscale images as well as color images. Topics covered include Terminology (p. 13-2) Provides definitions of image processing terms used in this section Working with Different Screen Bit Depths (p. 13-3) Describes how to determine the screen bit depth of your system and provides recommendations if you can change the bit depth Reducing the Number of Colors in an Image (p. 13-6) Describes how to use imapprox and rgb2ind to reduce the number of colors in an image, including information about dithering Converting to Other Color Spaces (p. 13-15) Defines the concept of image color space and describes how to convert images between color spaces 13 Color Terminology An understanding of the following terms will help you to use this chapter. 13-2 Terms Definitions Approximation The method by which the software chooses replacement colors in the event that direct matches cannot be found. The methods of approximation discussed in this chapter are colormap mapping, uniform quantization, and minimum variance quantization. Indexed image An image whose pixel values are direct indices into an RGB colormap. In MATLAB, an indexed image is represented by an array of class uint8, uint16, or double. The colormap is always an m-by-3 array of class double. We often use the variable name X to represent an indexed image in memory, and map to represent the colormap. Intensity image An image consisting of intensity (grayscale) values. In MATLAB, intensity images are represented by an array of class uint8, uint16, or double. While intensity images are not stored with colormaps, MATLAB uses a system colormap to display them. We often use the variable name I to represent an intensity image in memory. This term is synonymous with the term grayscale. RGB image An image in which each pixel is specified by three values — one each for the red, blue, and green components of the pixel’s color. In MATLAB, an RGB image is represented by an m-by-n-by-3 array of class uint8, uint16, or double. We often use the variable name RGB to represent an RGB image in memory. Screen bit depth The number of bits per screen pixel. Screen color resolution The number of distinct colors that can be produced by the screen. Working with Different Screen Bit Depths Working with Different Screen Bit Depths Most computer displays use 8, 16, or 24 bits per screen pixel. The number of bits per screen pixel determines the display’s screen bit depth. The screen bit depth determines the screen color resolution, which is how many distinct colors the display can produce. Regardless of the number of colors your system can display, MATLAB can store and process images with very high bit depths: 224 colors for uint8 RGB images, 248 colors for uint16 RGB images, and 2159 for double RGB images. These images display best on systems with 24-bit color, but usually look fine on 16-bit systems as well. (For additional information about how MATLAB handles color, see the MATLAB graphics documentation.) This section: • Describes how to determine your system’s screen bit depth • Provides guidelines for choosing a screen bit depth Determining Your Systems Screen Bit Depth To determine your system’s screen bit depth, enter this command at the MATLAB prompt. get(0,'ScreenDepth') ans = 16 13-3 13 Color The integer MATLAB returns represents the number of bits per screen pixel: Value Screen Bit Depth 8 8-bit displays supports 256 colors. An 8-bit display can produce any of the colors available on a 24-bit display, but only 256 distinct colors can appear at one time. (There are 256 shades of gray available, but if all 256 shades of gray are used, they take up all of the available color slots.) 16 16-bit displays usually use 5 bits for each color component, resulting in 32 (i.e., 25) levels each of red, green, and blue. This supports 32,768 (i.e., 215) distinct colors (of which 32 are shades of gray). Some systems use the extra bit to increase the number of levels of green that can be displayed. In this case, the number of different colors supported by a 16-bit display is actually 64,536 (i.e. 216). 24 24-bit displays use 8 bits for each of the three color components, resulting in 256 (i.e., 28) levels each of red, green, and blue. This supports 16,777,216 (i.e., 224) different colors. (Of these colors, 256 are shades of gray. Shades of gray occur where R=G=B.) The 16 million possible colors supported by 24-bit display can render a life-like image. 32 32-bit displays use 24 bits to store color information and use the remaining 8 bits to store transparency data (alpha channel). For information about how MATLAB supports the alpha channel, see Transparency. Choosing a Screen Bit Depth Depending on your system, you may be able to choose the screen bit depth you want to use. (There may be trade-offs between screen bit depth and screen color resolution.) In general, 24-bit display mode produces the best results. If you need to use a lower screen bit depth, 16-bit is generally preferable to 8-bit. However, keep in mind that a 16-bit display has certain limitations, such as: • An image may have finer gradations of color than a 16-bit display can represent. If a color is unavailable, MATLAB uses the closest approximation. 13-4 Working with Different Screen Bit Depths • There are only 32 shades of gray available. If you are working primarily with grayscale images, you may get better display results using 8-bit display mode, which provides up to 256 shades of gray. For information about reducing the number of colors used by an image, see “Reducing the Number of Colors in an Image” on page 13-6. 13-5 13 Color Reducing the Number of Colors in an Image This section describes how to reduce the number of colors in an indexed or RGB image. A discussion is also included about dithering, which is used by the toolbox’s color-reduction functions (see below.) Dithering is used to increase the apparent number of colors in an image. The table below summarizes the Image Processing Toolbox functions for color reduction. Function Purpose imapprox Reduces the number of colors used by an indexed image, enabling you specify the number of colors in the new colormap. rgb2ind Converts an RGB image to an indexed image, enabling you to specify the number of colors to store in the new colormap. On systems with 24-bit color displays, RGB (truecolor) images can display up to 16,777,216 (i.e., 224) colors. On systems with lower screen bit depths, RGB images still displays reasonably well, because MATLAB automatically uses color approximation and dithering if needed. Indexed images, however, may cause problems if they have a large number of colors. In general, you should limit indexed images to 256 colors for the following reasons: • On systems with 8-bit display, indexed images with more than 256 colors will need to be dithered or mapped and, therefore, may not display well. • On some platforms, colormaps cannot exceed 256 entries. • If an indexed image has more than 256 colors, MATLAB cannot store the image data in a uint8 array, but generally uses an array of class double instead, making the storage size of the image much larger (each pixel uses 64 bits). • Most image file formats limit indexed images to 256 colors. If you write an indexed image with more than 256 colors (using imwrite) to a format that does not support more than 256 colors, you will receive an error. 13-6 Reducing the Number of Colors in an Image Using rgb2ind rgb2ind converts an RGB image to an indexed image, reducing the number of colors in the process. This function provides the following methods for approximating the colors in the original image: • Quantization - Uniform quantization - Minimum variance quantization • Colormap mapping The quality of the resulting image depends on the approximation method you use, the range of colors in the input image, and whether or not you use dithering. Note that different methods work better for different images. See “Dithering” on page 13-13 for a description of dithering and how to enable or disable it. Quantization Reducing the number of colors in an image involves quantization. The function rgb2ind uses quantization as part of its color reduction algorithm. rgb2ind supports two quantization methods: uniform quantization and minimum variance quantization. An important term in discussions of image quantization is RGB color cube, which is used frequently throughout this section. The RGB color cube is a three-dimensional array of all of the colors that are defined for a particular data type. Since RGB images in MATLAB can be of type uint8, uint16, or double, three possible color cube definitions exist. For example, if an RGB image is of class uint8, 256 values are defined for each color plane (red, blue, and green), and, in total, there will be 224 (or 16,777,216) colors defined by the color cube. This color cube is the same for all uint8 RGB images, regardless of which colors they actually use. The uint8, uint16, and double color cubes all have the same range of colors. In other words, the brightest red in an uint8 RGB image displays the same as the brightest red in a double RGB image. The difference is that the double RGB color cube has many more shades of red (and many more shades of all colors). Figure 13-1, below, shows an RGB color cube for a uint8 image. 13-7 13 Color B 255 White (255,255,255) 255 G 255 R Figure 13-1: RGB Color Cube for uint8 Images Quantization involves dividing the RGB color cube into a number of smaller boxes, and then mapping all colors that fall within each box to the color value at the center of that box. Uniform quantization and minimum variance quantization differ in the approach used to divide up the RGB color cube. With uniform quantization, the color cube is cut up into equal-sized boxes (smaller cubes). With minimum variance quantization, the color cube is cut up into boxes (not necessarily cubes) of different sizes; the sizes of the boxes depend on how the colors are distributed in the image. Uniform Quantization. To perform uniform quantization, call rgb2ind and specify a tolerance. The tolerance determines the size of the cube-shaped boxes into which the RGB color cube is divided. The allowable range for a tolerance setting is [0,1]. For example, if you specify a tolerance of 0.1, the edges of the boxes are one-tenth the length of the RGB color cube and the maximum total number of boxes is n = (floor(1/tol)+1)^3 13-8 Reducing the Number of Colors in an Image The commands below perform uniform quantization with a tolerance of 0.1. RGB = imread('flowers.tif'); [x,map] = rgb2ind(RGB, 0.1); Figure 13-2 illustrates uniform quantization of a uint8 image. For clarity, the figure shows a two-dimensional slice (or color plane) from the color cube where Red=0, and Green and Blue range from 0 to 255. The actual pixel values are denoted by the centers of the x’s. B 255 x x x x x center pixel value 255 G Figure 13-2: Uniform Quantization on a Slice of the RGB Color Cube After the color cube has been divided, all empty boxes are thrown out. Therefore, only one of the boxes in Figure 13-2 is used to produce a color for the colormap. As shown earlier, the maximum length of a colormap created by uniform quantization can be predicted, but the colormap can be smaller than the prediction because rgb2ind removes any colors that do not appear in the input image. Minimum Variance Quantization. To perform minimum variance quantization, call rgb2ind and specify the maximum number of colors in the output image’s colormap. The number you specify determines the number of boxes into which 13-9 13 Color the RGB color cube is divided. These commands use minimum variance quantization to create an indexed image with 185 colors. RGB = imread('flowers.tif'); [X,map] = rgb2ind(RGB,185); Minimum variance quantization works by associating pixels into groups based on the variance between their pixel values. For example, a set of blue pixel values may be grouped together because none of their values is greater than 5 from the center pixel of the group. In minimum variance quantization, the boxes that divide the color cube vary in size, and do not necessarily fill the color cube. If some areas of the color cube do not have pixels, there are no boxes in these areas. While you set the number of boxes, n, to be used by rgb2ind, the placement is determined by the algorithm as it analyzes the color data in your image. Once the image is divided into n optimally located boxes, the pixels within each box are mapped to the pixel value at the center of the box, as in uniform quantization. The resulting colormap usually has the number of entries you specify. This is because the color cube is divided so that each region contains at least one color that appears in the input image. If the input image uses fewer colors than the number you specify, the output colormap will have fewer than n colors, and the output image will contain all of the colors of the input image. Figure 13-3 shows the same two-dimensional slice of the color cube as was used in Figure 13-2 (for demonstrating uniform quantization). Eleven boxes have been created using minimum variance quantization. 13-10 Reducing the Number of Colors in an Image B 255 x x x x x x x x x x x x x x x x x x x x xx x x x x x center pixel value x x x x x x x x 255 G Figure 13-3: Minimum Variance Quantization on a Slice of the RGB Color Cube For a given number of colors, minimum variance quantization produces better results than uniform quantization, because it takes into account the actual data. Minimum variance quantization allocates more of the colormap entries to colors that appear frequently in the input image. It allocates fewer entries to colors that appear infrequently. As a result, the accuracy of the colors is higher than with uniform quantization. For example, if the input image has many shades of green and few shades of red, there will be more greens than reds in the output colormap. Note that the computation for minimum variance quantization takes longer than that for uniform quantization. Colormap Mapping If you specify an actual colormap to use, rgb2ind uses colormap mapping (instead of quantization) to find the colors in the specified colormap that best match the colors in the RGB image. This method is useful if you need to create images that use a fixed colormap. For example, if you want to display multiple indexed images on an 8-bit display, you can avoid color problems by mapping them all to the same colormap. Colormap mapping produces a good approximation if the specified colormap has similar colors to those in the RGB 13-11 13 Color image. If the colormap does not have similar colors to those in the RGB image, this method produces poor results. This example illustrates mapping two images to the same colormap. The colormap used for the two images is created on the fly using the MATLAB function colorcube, which creates an RGB colormap containing the number of colors that you specify. (colorcube always creates the same colormap for a given number of colors.) Because the colormap includes colors all throughout the RGB color cube, the output images can reasonably approximate the input images. RGB1 RGB2 X1 = X2 = = imread('autumn.tif'); = imread('flowers.tif'); rgb2ind(RGB1,colorcube(128)); rgb2ind(RGB2,colorcube(128)); Note The function subimage is also helpful for displaying multiple indexed images. For more information see “Displaying Multiple Images in the Same Figure” on page 3-20 or the reference page for subimage. Reducing Colors in an Indexed Image Use imapprox when you need to reduce the number of colors in an indexed image. imapprox is based on rgb2ind and uses the same approximation methods. Essentially, imapprox first calls ind2rgb to convert the image to RGB format, and then calls rgb2ind to return a new indexed image with fewer colors. For example, these commands create a version of the trees image with 64 colors, rather than the original 128. load trees [Y,newmap] = imapprox(X,map,64); imshow(Y, newmap); The quality of the resulting image depends on the approximation method you use, the range of colors in the input image, and whether or not you use dithering. Note that different methods work better for different images. See “Dithering” on page 13-13 for a description of dithering and how to enable or disable it. 13-12 Reducing the Number of Colors in an Image Dithering When you use rgb2ind or imapprox to reduce the number of colors in an image, the resulting image may look inferior to the original, because some of the colors are lost. rgb2ind and imapprox both perform dithering to increase the apparent number of colors in the output image. Dithering changes the colors of pixels in a neighborhood so that the average color in each neighborhood approximates the original RGB color. For an example of how dithering works, consider an image that contains a number of dark pink pixels for which there is no exact match in the colormap. To create the appearance of this shade of pink, the Image Processing Toolbox selects a combination of colors from the colormap, that, taken together as a six-pixel group, approximate the desired shade of pink. From a distance, the pixels appear to be correct shade, but if you look up close at the image, you can see a blend of other shades, perhaps red and pale pink pixels. The commands below load a 24-bit image, and then use rgb2ind to create two indexed images with just eight colors each. rgb=imread('lily.tif'); imshow(rgb); [X_no_dither,map]=rgb2ind(rgb,8,'nodither'); [X_dither,map]=rgb2ind(rgb,8,'dither'); figure, imshow(X_no_dither,map); figure, imshow(X_dither,map); Original image Without dithering With dithering Figure 13-4: Examples of Color Reduction with and Without Dithering 13-13 13 Color Notice that the dithered image has a larger number of apparent colors but is somewhat fuzzy-looking. The image produced without dithering has fewer apparent colors, but an improved spatial resolution when compared to the dithered image. One risk in doing color reduction without dithering is that the new image my contain false contours (see the rose in the upper-right corner). 13-14 Converting to Other Color Spaces Converting to Other Color Spaces The Image Processing Toolbox represents colors as RGB values, either directly (in an RGB image) or indirectly (in an indexed image, where the colormap is stored in RGB format). However, there are other models besides RGB for representing colors numerically. For example, a color can be represented by its hue, saturation, and value components (HSV) instead. The various models for color data are called color spaces. The functions in the Image Processing Toolbox that work with color assume that images use the RGB color space. However, the toolbox provides support for other color spaces though a set of conversion functions. You can use these functions to convert between RGB and the following color spaces: • National Television Systems Committee (NTSC) • YCbCr • Hue, saturation, value (HSV) These section describes these color spaces and the conversion routines for working with them: • “NTSC Color Space” • “YCbCr Color Space” on page 13-16 • “HSV Color Space” on page 13-16 NTSC Color Space The NTSC color space is used in televisions in the United States. One of the main advantages of this format is that grayscale information is separated from color data, so the same signal can be used for both color and black and white sets. In the NTSC format, image data consists of three components: luminance (Y), hue (I), and saturation (Q). The first component, luminance, represents grayscale information, while the last two components make up chrominance (color information). The function rgb2ntsc converts colormaps or RGB images to the NTSC color space. ntsc2rgb performs the reverse operation. For example, these commands convert the flowers image to NTSC format. RGB = imread('flowers.tif'); YIQ = rgb2ntsc(RGB); 13-15 13 Color Because luminance is one of the components of the NTSC format, the RGB to NTSC conversion is also useful for isolating the gray level information in an image. In fact, the toolbox functions rgb2gray and ind2gray use the rgb2ntsc function to extract the grayscale information from a color image. For example, these commands are equivalent to calling rgb2gray. YIQ = rgb2ntsc(RGB); I = YIQ(:,:,1); Note In YIQ color space, I is one of the two color components, not the grayscale component. YCbCr Color Space The YCbCr color space is widely used for digital video. In this format, luminance information is stored as a single component (Y), and chrominance information is stored as two color-difference components (Cb and Cr). Cb represents the difference between the blue component and a reference value. Cr represents the difference between the red component and a reference value. YCbCr data can be double precision, but the color space is particularly well suited to uint8 data. For uint8 images, the data range for Y is [16, 235], and the range for Cb and Cr is [16, 240]. YCbCr leaves room at the top and bottom of the full uint8 range so that additional (nonimage) information can be included in a video stream. The function rgb2ycbcr converts colormaps or RGB images to the YCbCr color space. ycbcr2rgb performs the reverse operation. For example, these commands convert the flowers image to YCbCr format. RGB = imread('flowers.tif'); YCBCR = rgb2ycbcr(RGB); HSV Color Space The HSV color space (hue, saturation, value) is often used by people who are selecting colors (e.g., of paints or inks) from a color wheel or palette, because it corresponds better to how people experience color than the RGB color space 13-16 Converting to Other Color Spaces does. The functions rgb2hsv and hsv2rgb convert images between the RGB and HSV color spaces. As hue varies from 0 to 1.0, the corresponding colors vary from red, through yellow, green, cyan, blue, and magenta, back to red, so that there are actually red values both at 0 and 1.0. As saturation varies from 0 to 1.0, the corresponding colors (hues) vary from unsaturated (shades of gray) to fully saturated (no white component). As value, or brightness, varies from 0 to 1.0, the corresponding colors become increasingly brighter. Figure 13-5 illustrates the HSV color space. Hue Hue 1 0.8 1 0 0.6 1 0.4 Saturation Value 0.2 0 1 0 1 0.5 0.5 0 05 Figure 13-5: Illustration of the HSV Color Space 0 The function rgb2hsv converts colormaps or RGB images to the HSV color space. hsv2rgb performs the reverse operation. These commands convert an RGB image to HSV color space. 13-17 13 Color RGB = imread('flowers.tif'); HSV = rgb2hsv(RGB); For closer inspection of the HSV color space, the next block of code displays the separate color planes (hue, saturation, and value) of an HSV image. RGB=reshape(ones(64,1)*reshape(jet(64),1,192),[64,64,3]); HSV=rgb2hsv(RGB); H=HSV(:,:,1); S=HSV(:,:,2); V=HSV(:,:,3); imshow(H) figure, imshow(S); figure, imshow(V); figure, imshow(RGB); Hue plane Saturation plane Value plane Original image Figure 13-6: The Separated Color Planes of an HSV Image 13-18 Converting to Other Color Spaces The images in Figure 13-6 can be scrutinized for a better understanding of how the HSV color space works. As you can see by looking at the hue plane image, hue values make a nice linear transition from high to low. If you compare the hue plane image against the original image, you can see that shades of deep blue have the highest values, and shades of deep red have the lowest values. (In actuality, there are values of red on both ends of the hue scale, which you can see if you look back at the model of the HSV color space in Figure 13-5. To avoid confusion, our sample image uses only the red values from the beginning of the hue range.) Saturation can be thought of as the purity of a color. As the saturation plane image shows, the colors with the highest saturation have the highest values and are represented as white. In the center of the saturation image, notice the various shades of gray. These correspond to a mixture of colors; the cyans, greens, and yellow shades are mixtures of true colors. Value is roughly equivalent to brightness, and you will notice that the brightest areas of the value plane correspond to the brightest colors in the original image. 13-19 13 Color 13-20 14 Function Reference This section describes the Image Processing Toolbox functions. Functions – By Category (p. 14-2) Contains a group of tables that organize the toolbox functions into category groups Functions – Alphabetical List (p. 14-16) Contains separate reference pages for each toolbox function 14 Function Reference Functions – By Category This section provides brief descriptions of all the functions in the Image Processing Toolbox. The functions are listed in tables in the following broad categories If you know the name of a function, go directly to the “Functions – Alphabetical List” section to view its reference page. “Image Input, Output, and Display” Functions for importing, exporting, and displaying images and converting between image formats “Spatial Transformation and Registration” Functions for performing spatial transformations and image registration “Image Analysis and Statistics” Functions for performing image analysis and getting pixel values and statistics “Image Enhancement and Restoration” Functions for image enhancement and restoration, such as deblurring. “Linear Filtering and Transforms” Functions for creating and using linear filters and transform “Morphological Operations” Functions for performing morphological image processing “Region-Based, Neighborhood, and Block Processing” Functions to define regions of interest and operate on these regions “Colormap and Color Space Functions” Functions for working with image color “Miscellaneous Functions” functions that perform image arithmetic, array operations, and set and get Image Processing Toolbox preferences Image Input, Output, and Display • “Image Display” • “Image File I/O” 14-2 Functions – By Category • “Image Types and Type Conversions” Image Display colorbar Display colorbar. (This is a MATLAB function.) getimage Get image data from axes image Create and display image object. (This is a MATLAB function. See the online MATLAB Function Reference for its reference page.) imagesc Scale data and display as image. (This is a MATLAB function. See the online MATLAB Function Reference for its reference page.) immovie Make movie from multiframe indexed image imshow Display image montage Display multiple image frames as rectangular montage subimage Display multiple images in single figure truesize Adjust display size of image warp Display image as texture-mapped surface zoom Zoom in and out of image or 2-D plot. (This is a MATLAB function.) 14-3 14 Function Reference Image File I/O dicominfo Read metadata from a DICOM message dicomread Read a DICOM image imfinfo Return information about image file. (This is a MATLAB function.) imread Read image file. (This is a MATLAB function.) imwrite Write image file. (This is a MATLAB function.) Image Types and Type Conversions 14-4 dither Convert image using dithering double Convert data to double precision. (This is a MATLAB function.) gray2ind Convert intensity image to indexed image grayslice Create indexed image from intensity image by thresholding graythresh Compute global image threshold using Otsu's method im2bw Convert image to binary image by thresholding im2double Convert image array to double precision im2java Convert image to instance of Java image object im2uint16 Convert image array to 16-bit unsigned integers im2uint8 Convert image array to 8-bit unsigned integers ind2gray Convert indexed image to intensity image ind2rgb Convert indexed image to RGB image isbw Return true for binary image isgray Return true for intensity image isind Return true for indexed image isrgb Return true for RGB image label2rgb Convert a label matrix to an RGB image Functions – By Category mat2gray Convert matrix to intensity image rgb2gray Convert RGB image or colormap to grayscale rgb2ind Convert RGB image to indexed image uint16 Convert data to unsigned 16-bit integers. (This is a MATLAB function.) uint8 Convert data to unsigned 8-bit integers. (This is a MATLAB function.) Spatial Transformation and Registration • “Spatial Transformations” • “Image Registration” 14-5 14 Function Reference Spatial Transformations checkerboard Create checkerboard image findbounds Find output bounds for spatial transformation fliptform Flip the input and output roles of a TFORM structure imcrop Crop image imresize Resize image imrotate Rotate image interp2 2-D data interpolation. (This is a MATLAB function. See the online MATLAB Function Reference for its reference page.) imtransform Apply 2-D spatial transformation to image makeresampler Create resampling structure maketform Create geometric transformation structure tformarray Geometric transformation of a multi-dimensional array tformfwd Apply forward geometric transformation tforminv Apply inverse geometric transformation Image Registration cpcorr Tune control point locations using cross-correlation cp2tform Infer geometric transformation from control point pairs cpselect Control point selection tool cpstruct2pairs Convert CPSTRUCT to valid pairs of control points normxcorr2 Normalized two-dimensional cross-correlation Image Analysis and Statistics • “Image Analysis” • “Pixel Values and Statistics” 14-6 Functions – By Category Image Analysis edge Find edges in intensity image qtdecomp Perform quadtree decomposition qtgetblk Get block values in quadtree decomposition qtsetblk Set block values in quadtree decomposition Pixel Values and Statistics corr2 Compute 2-D correlation coefficient imcontour Create contour plot of image data imfeature Compute feature measurements for image regions imhist Display histogram of image data impixel Determine pixel color values improfile Compute pixel-value cross-sections along line segments mean2 Compute mean of matrix elements pixval Display information about image pixels regionprops Measure properties of image regions std2 Compute standard deviation of matrix elements Image Enhancement and Restoration • “Image Enhancement” • “Image Restoration (Deblurring)” 14-7 14 Function Reference Image Enhancement histeq Enhance contrast using histogram equalization imadjust Adjust image intensity values or colormap imnoise Add noise to an image medfilt2 Perform 2-D median filtering ordfilt2 Perform 2-D order-statistic filtering stretchlim Find limits to contrast stretch an image wiener2 Perform 2-D adaptive noise-removal filtering Image Restoration (Deblurring) deconvblind Restore image using blind deconvolution deconvlucy Restore image using accelerated Richardson-Lucy algorithm deconvreg Restore image using Regularized filter deconvwnr Restore image using Wiener filter edgetaper Taper the discontinuities along the image edges otf2psf Convert optical transfer function to point-spread function psf2otf Convert point-spread function to optical transfer function Linear Filtering and Transforms • “Linear Filtering” • “Linear 2-D Filter Design” • “Image Transforms” 14-8 Functions – By Category Linear Filtering conv2 Perform 2-D convolution. (This is a MATLAB function.) convmtx2 Compute 2-D convolution matrix convn Perform N-D convolution. (This is a MATLAB function.) filter2 Perform 2-D filtering. (This is a MATLAB function.) fspecial Create predefined filters imfilter Multidimensional image filtering Linear 2-D Filter Design freqspace Determine 2-D frequency response spacing. (This is a MATLAB function.) freqz2 Compute 2-D frequency response fsamp2 Design 2-D FIR filter using frequency sampling ftrans2 Design 2-D FIR filter using frequency transformation fwind1 Design 2-D FIR filter using 1-D window method fwind2 Design 2-D FIR filter using 2-D window method Image Transforms dct2 Compute 2-D discrete cosine transform dctmtx Compute discrete cosine transform matrix fft2 Compute 2-D fast Fourier transform. (This is a MATLAB function.) fftn Compute N-D fast Fourier transform. (This is a MATLAB function.) fftshift Reverse quadrants of output of FFT. (This is a MATLAB function.) idct2 Compute 2-D inverse discrete cosine transform ifft2 Compute 2-D inverse fast Fourier transform. (This is a MATLAB function.) 14-9 14 Function Reference ifftn Compute N-D inverse fast Fourier transform. (This is a MATLAB function.) iradon Compute inverse Radon transform phantom Generate a head phantom image radon Compute Radon transform Morphological Operations • “Intensity and Binary Images” • “Binary Images” • “Structuring Element (STREL) Creation and Manipulation” 14-10 Functions – By Category Intensity and Binary Images conndef Default connectivity array imbothat Perform bottom-hat filtering imclearborder Suppress light structures connected to image border imclose Close image imdilate Dilate image imerode Erode image imextendedmax Extended-maxima transform imextendedmin Extended-minima transform imfill Fill image regions imhmax H-maxima transform imhmin H-minima transform imimposemin Impose minima imopen Open image imreconstruct Perform morphological reconstruction imregionalmax Regional maxima of image imregionalmin Regional minima of image imtophat Perform tophat filtering watershed Find image watershed regions Binary Images applylut Perform neighborhood operations using lookup tables bwarea Area of objects in binary image bwareaopen Binary area open; remove small objects bwdist Distance transform bweuler Euler number of binary image bwfill Fill background regions in binary image 14-11 14 Function Reference bwhitmiss Binary hit-miss operation bwlabel Label connected components in 2-D binary image bwlabeln Label connected components in N-D binary image. bwmorph Perform morphological operations on binary image bwpack Pack binary image bwperim Find perimeter of objects in binary image bwselect Select objects in binary image bwulterode Ultimate erosion bwunpack Unpack a packed binary image imregionalmin Regional minima of image imtophat Perform tophat filtering makelut Construct lookup table for use with applylut Structuring Element (STREL) Creation and Manipulation getheight Get the height of a structuring element getneighbors Get structuring element neighbor locations and heights getnhood Get structuring element neighborhood getsequence Extract sequence of decomposed structuring elements isflat Return true for flat structuring element reflect Reflect structuring element strel Create morphological structuring element translate Translate structuring element Region-Based, Neighborhood, and Block Processing • “Region-Based Processing” • “Neighborhood and Block Processing” 14-12 Functions – By Category Region-Based Processing roicolor Select region of interest, based on color roifill Smoothly interpolate within arbitrary region roifilt2 Filter a region of interest roipoly Select polygonal region of interest Neighborhood and Block Processing bestblk Choose block size for block processing blkproc Implement distinct block processing for image col2im Rearrange matrix columns into blocks colfilt Perform neighborhood operations using columnwise functions im2col Rearrange image blocks into columns nlfilter Perform general sliding-neighborhood operations Colormap and Color Space Functions • “Colormap Manipulation” • “Color Space Conversions” 14-13 14 Function Reference Colormap Manipulation brighten Brighten or darken colormap. (This is a MATLAB function.) cmpermute Rearrange colors in colormap cmunique Find unique colormap colors and corresponding image colormap Set or get color lookup table. (This is a MATLAB function.) imapprox Approximate indexed image by one with fewer colors rgbplot Plot RGB colormap components. (This is a MATLAB function.) Color Space Conversions hsv2rgb Convert HSV values to RGB color space. (This is a MATLAB function.) ntsc2rgb Convert NTSC values to RGB color space rgb2hsv Convert RGB values to HSV color space. (This is a MATLAB function.) rgb2ntsc Convert RGB values to NTSC color space rgb2ycbcr Convert RGB values to YCbCr color space ycbcr2rgb Convert YCbCr values to RGB color space Miscellaneous Functions • “Image Arithmetic” • “Toolbox Preferences” • “Array Operations” 14-14 Functions – By Category Image Arithmetic imabsdiff Compute absolute difference of two images imadd Add two images, or add constant to image imcomplement Complement image imdivide Divide two images, or divide image by constant. imlincomb Compute linear combination of images immultiply Multiply two images, or multiply image by constant imsubtract Subtract two images, or subtract constant from image Toolbox Preferences iptgetpref Get value of Image Processing Toolbox preference iptsetpref Set value of Image Processing Toolbox preference Array Operations padarray Pad an array 14-15 14 Function Reference Functions – Alphabetical List This section contains detailed descriptions of all toolbox functions. Each function reference page contains some or all of this information: • The function name • The purpose of the function • The function syntax All valid input argument and output argument combinations are shown. In some cases, an ellipsis (. . .) is used for the input arguments. This means that all preceding input argument combinations are valid for the specified output argument(s). • A description of each argument • A description of the function • Additional remarks about usage • An example of usage • Related functions 14-16 applylut Purpose 14applylut Perform neighborhood operations on binary images using lookup tables Syntax A = applylut(BW,LUT) Description A = applylut(BW,LUT) performs a 2-by-2 or 3-by-3 neighborhood operation on binary image BW by using a lookup table (LUT). LUT is either a 16-element or 512-element vector returned by makelut. The vector consists of the output values for all possible 2-by-2 or 3-by-3 neighborhoods. Class Support BW can be numeric or logical, and it must be real, two-dimensional, and nonsparse. LUT can be numeric or logical, and it must be a real vector with 16 or 512 elements. If all the elements of LUT are 0 or 1, then A is logical. If all the elements of LUT are integers between 0 and 255, then A is uint8. For all other cases, A is double. Algorithm applylut performs a neighborhood operation on a binary image by producing a matrix of indices into lut, and then replacing the indices with the actual values in lut. The specific algorithm used depends on whether you use 2-by-2 or 3-by-3 neighborhoods. 2-by-2 Neighborhoods For 2-by-2 neighborhoods, length(lut) is 16. There are four pixels in each neighborhood, and two possible states for each pixel, so the total number of permutations is 24 = 16. To produce the matrix of indices, applylut convolves the binary image BW with this matrix. 8 4 2 1 The resulting convolution contains integer values in the range [0,15]. applylut uses the central part of the convolution, of the same size as BW, and adds 1 to each value to shift the range to [1,16]. It then constructs A by replacing the values in the cells of the index matrix with the values in lut that the indices point to. 14-21 applylut 3-by-3 Neighborhoods For 3-by-3 neighborhoods, length(lut) is 512. There are nine pixels in each neighborhood, and 2 possible states for each pixel, so the total number of permutations is 29 = 512. To produce the matrix of indices, applylut convolves the binary image BW with this matrix. 256 128 64 32 16 8 4 2 1 The resulting convolution contains integer values in the range [0,511]. applylut uses the central part of the convolution, of the same size as BW, and adds 1 to each value to shift the range to [1,512]. It then constructs A by replacing the values in the cells of the index matrix with the values in lut that the indices point to. Example In this example, you perform erosion using a 2-by-2 neighborhood. An output pixel is on only if all four of the input pixel’s neighborhood pixels are on. lut = makelut('sum(x(:)) == 4',2); BW1 = imread('text.tif'); BW2 = applylut(BW1,lut); imshow(BW1) figure, imshow(BW2) See Also 14-22 makelut bestblk Purpose 14bestblk Determine block size for block processing Syntax siz = bestblk([m n],k) [mb,nb] = bestblk([m n],k) Description siz = bestblk([m n],k) returns, for an m-by-n image, the optimal block size for block processing. k is a scalar specifying the maximum row and column dimensions for the block; if the argument is omitted, it defaults to 100. The return value, siz, is a 1-by-2 vector containing the row and column dimensions for the block. [mb,nb] = bestblk([m n],k) returns the row and column dimensions for the block in mb and nb, respectively. Algorithm bestblk returns the optimal block size given m, n, and k. The algorithm for determining siz is: • If m is less than or equal to k, return m. • If m is greater than k, consider all values between min(m/10,k/2) and k. Return the value that minimizes the padding required. The same algorithm is then repeated for n. Example siz = bestblk([640 800],72) siz = 64 See Also 50 blkproc 14-23 blkproc Purpose 14blkproc Implement distinct block processing for an image Syntax B B B B Description B = blkproc(A,[m n],fun) processes the image A by applying the function fun to each distinct m-by-n block of A, padding A with zeros if necessary. fun is a function that accepts an m-by-n matrix, x, and returns a matrix, vector, or scalar y. = = = = blkproc(A,[m n],fun) blkproc(A,[m n],fun,P1,P2,...) blkproc(A,[m n],[mborder nborder],fun,...) blkproc(A,'indexed',...) y = fun(x) blkproc does not require that y be the same size as x. However, B is the same size as A only if y is the same size as x. B = blkproc(A,[m n],fun,P1,P2,...) passes the additional parameters P1,P2,..., to fun. B = blkproc(A,[m n],[mborder nborder],fun,...) defines an overlapping border around the blocks. blkproc extends the original m-by-n blocks by mborder on the top and bottom, and nborder on the left and right, resulting in blocks of size (m+2*mborder)-by-(n+2*nborder). The blkproc function pads the border with zeros, if necessary, on the edges of A. The function fun should operate on the extended block. The line below processes an image matrix as 4-by-6 blocks, each having a row border of 2 and a column border of 3. Because each 4-by-6 block has this 2-by-3 border, fun actually operates on blocks of size 8-by-12. B = blkproc(A,[4 6],[2 3],fun,...) B = blkproc(A,'indexed',...) processes A as an indexed image, padding with zeros if the class of A is uint8 or uint16, or ones if the class of A is double. Class Support 14-24 The input image A can be of any class supported by fun. The class of B depends on the class of the output from fun. blkproc Example fun can be a function_handle created using @. This example uses blkproc to compute the 2-D DCT of each 8-by-8 block to the standard deviation of the elements in that block. I = imread('cameraman.tif'); fun = @dct2; J = blkproc(I,[8 8],fun); imagesc(J), colormap(hot) 50 100 150 200 250 50 100 150 200 250 fun can also be an inline object. This example uses blkproc to set the pixels in each 8-by-8 block to the standard deviation of the elements in that block. I = imread('alumgrns.tif'); fun = inline('std2(s)*ones(size(x))'); I2 = blkproc(I,[8 8],'std2(x)*ones(size(x))'); imshow(I) figure, imshow(I2,[]); 14-25 blkproc See Also 14-26 colfilt, nlfilter, inline brighten Purpose 14brighten Brighten or darken a colormap brighten is a MATLAB function. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-27 bwarea Purpose 14bwarea Compute the area of the objects in a binary image Syntax total = bwarea(BW) Description total = bwarea(BW) estimates the area of the objects in binary image BW. total is a scalar whose value corresponds roughly to the total number of on pixels in the image, but may not be exactly the same because different patterns of pixels are weighted differently. Class Support BW can be numeric or logical. For numeric input, any non-zero pixels are considered to be "on". The return value, total, is of class double. Algorithm bwarea estimates the area of all of the on pixels in an image by summing the areas of each pixel in the image. The area of an individual pixel is determined by looking at its 2-by-2 neighborhood. There are six different patterns, each representing a different area: • Patterns with zero on pixels (area = 0) • Patterns with one on pixel (area = 1/4) • Patterns with two adjacent on pixels (area = 1/2) • Patterns with two diagonal on pixels (area = 3/4) • Patterns with three on pixels (area = 7/8) • Patterns with all four on pixels (area = 1) Keep in mind that each pixel is part of four different 2-by-2 neighborhoods. This means, for example, that a single on pixel surrounded by off pixels has a total area of 1. Example This example computes the area in the objects of a 256-by-256 binary image. BW = imread('circles.tif'); imshow(BW); 14-28 bwarea bwarea(BW) ans = 15799 See Also bweuler, bwperim References Pratt, William K. Digital Image Processing. New York: John Wiley & Sons, Inc., 1991. p. 634. 14-29 bwareaopen Purpose 14bwareaopen Binary area open; remove small objects Syntax BW2 = bwareaopen(BW,P) BW2 = bwareaopen(BW,P,CONN) Description BW2 = bwareaopen(BW,P) removes from a binary image all connected components (objects) that have fewer than P pixels, producing another binary image BW2. The default connectivity is 8 for two dimensions, 26 for three dimensions, and conndef(ndims(BW),'maximal') for higher dimensions. BW2 = bwareaopen(BW,P,CONN) specifies the desired connectivity. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may be defined in a more general way for any dimension by using for CONN a 3-by-3-by-...-by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. Class Support BW can be a logical or numeric array of any dimension, and it must be nonsparse. The return value, BW2, is of class logical. Algorithm The basic steps are: 1 Determine the connected components. L = bwlabeln(BW, CONN); 14-30 bwareaopen 2 Compute the area of each component. S = regionprops(L, 'Area'); 3 Remove small objects. bw2 = ismember(L, find([S.Area] >= P)); Example Remove all objects containing fewer than 40 pixels in an image. 1 Read in the image and display it. bw = imread('text.tif'); imshow(bw) 2 Remove all objects smaller than 40 pixels. Note the missing letters. bw2 = bwareaopen(bw,40); figure, imshow(bw2) 14-31 bwareaopen See Also 14-32 bwlabel, bwlabeln, conndef, regionprops. bwdist Purpose 14bwdist Distance transform Syntax D = bwdist(BW) [D,L] = bwdist(BW) [D,L] = bwdist(BW,METHOD) Description D = bwdist(BW) computes the Euclidean distance transform of the binary image BW. For each pixel in BW, the distance transform assigns a number that is the distance between that pixel and the nearest nonzero pixel of BW. bwdist uses the Euclidean distance metric by default. BW can have any dimension. D is the same size as BW. [D,L] = bwdist(BW) also computes the nearest-neighbor transform and returns it as a label matrix, L, which has the same size as BW and D. Each element of L contains the linear index of the nearest nonzero pixel of BW. [D,L] = bwdist(BW,METHOD) computes the distance transform, where METHOD specifies an alternate distance metric. METHOD can take any of these values. 'chessboard' In 2-D, the chessboard distance between (x1,y1) and (x2,y2) is: max ( x 1 – x 2 , y 1 – y 2 ) 'cityblock' In 2-D, the cityblock distance between (x1,y1) and (x2,y2) is: x 1 – x 2 + y1 – y2 14-33 bwdist In 2-D, the Euclidean distance between (x1,y1) and (x2,y2) is: 'euclidean' 2 ( x1 – x2 ) + ( y1 – y2 ) 2 This is the default method. In 2-D, the quasi-Euclidean distance between (x1,y1) and (x2,y2) is: 'quasi-euclidean' x 1 – x 2 + ( 2 – 1 ) y 1 – y 2 , x 1 – x2 > y1 – y 2 ( 2 – 1 ) x 1 – x 2 + y 1 – y 2 , otherwise The METHOD string may be abbreviated. Note bwdist uses fast algorithms to compute the true Euclidean distance transform, especially in the 2-D case. The other methods are provided primarily for pedagogical reasons. However, the alternative distance transforms are sometimes significantly faster for multidimensional input images, particularly those that have many nonzero elements. Class support BW can be numeric or logical, and it must be nonsparse. D and L are double matrices with the same size as BW. Example Here is a simple example of the Euclidean distance transform. bw = zeros(5,5); bw = 0 0 0 1 0 0 0 0 0 0 bw(2,2) = 1; bw(4,4) = 1 0 0 0 0 0 [D,L] = bwdist(bw) 14-34 0 0 0 1 0 0 0 0 0 0 bwdist D = 1.4142 1.0000 1.4142 2.2361 3.1623 1.0000 0 1.0000 2.0000 2.2361 1.4142 1.0000 1.4142 1.0000 1.4142 2.2361 2.0000 1.0000 0 1.0000 3.1623 2.2361 1.4142 1.0000 1.4142 L = 7 7 7 7 7 7 7 7 7 19 7 7 7 19 19 7 7 19 19 19 7 19 19 19 19 In the nearest neighbor matrix, L, the values 7 and 19 represent the position of the nonzero elements using linear matrix indexing. If a pixel contains a 7, its closest nonzero neighbor is at linear position 7. This example compares the 2-D distance transforms for the each of the supported distance methods. In the figure, note how the quasi-euclidean distance transform best approximates the circular shape achieved by the euclidean distance method. bw = zeros(200,200); bw(50,50) = 1; bw(50,150) = 1; bw(150,100) = 1; D1 = bwdist(bw,'euclidean'); D2 = bwdist(bw,'cityblock'); D3 = bwdist(bw,'chessboard'); D4 = bwdist(bw,'quasi-euclidean'); figure subplot(2,2,1), subimage(mat2gray(D1)), title('Euclidean') hold on, imcontour(D1) subplot(2,2,2), subimage(mat2gray(D2)), title('City block') hold on, imcontour(D2) subplot(2,2,3), subimage(mat2gray(D3)), title('Chessboard') hold on, imcontour(D3) subplot(2,2,4), subimage(mat2gray(D4)), title('Quasi-Euclidean') hold on, imcontour(D4) 14-35 bwdist This example compares isosurface plots for the distance transforms of a 3-D image containing a single nonzero pixel in the center. bw = zeros(50,50,50); bw(25,25,25) = 1; D1 = bwdist(bw); D2 = bwdist(bw,'cityblock'); D3 = bwdist(bw,'chessboard'); D4 = bwdist(bw,'quasi-euclidean'); figure subplot(2,2,1), isosurface(D1,15), axis equal, view(3) camlight, lighting gouraud, title('Euclidean') subplot(2,2,2), isosurface(D2,15), axis equal, view(3) camlight, lighting gouraud, title('City block') subplot(2,2,3), isosurface(D3,15), axis equal, view(3) camlight, lighting gouraud, title('Chessboard') subplot(2,2,4), isosurface(D4,15), axis equal, view(3) camlight, lighting gouraud, title('Quasi-Euclidean') 14-36 bwdist Algorithm For two-dimensional Euclidean distance transforms, bwdist uses the second algorithm described in: Heinz Breu, Joseph Gil, David Kirkpatrick, and Michael Werman, “Linear Time Euclidean Distance Transform Algorithms,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 5 May 1995, pp. 529-533. For higher dimensional Euclidean distance transforms, bwdist uses a nearest neighbor search on an optimized kd-tree, as described in: Jerome H. Friedman, Jon Louis Bentley, and Raphael Ari Finkel, “An Algorithm for Finding Best Matches in Logarithmic Expected Time,” ACM Transactions on Mathematics Software, vol. 3, no. 3, September 1997, pp. 209-226. For cityblock, chessboard, and quasi-Euclidean distance transforms, bwdist uses the two-pass, sequential scanning algorithm described in: A. Rosenfeld and J. Pfaltz, “Sequential operations in digital picture processing,” Journal of the Association for Computing Machinery, vol. 13, no. 4, 1966, pp. 471-494. 14-37 bwdist The different distance measures are achieved by using different sets of weights in the scans, as described in: David Paglieroni, “Distance Transforms: Properties and Machine Vision Applications,” Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing, vol. 54, no. 1, January 1992, pp. 57-58. See Also 14-38 watershed bweuler Purpose 14bweuler Compute the Euler number of a binary image Syntax eul = bweuler(BW,n) Description eul = bweuler(BW,n) returns the Euler number for the binary image BW. The return value, eul, is a scalar whose value is the total number of objects in the image minus the total number of holes in those objects. The argument n can have a value of either 4 or 8, where 4 specifies 4-connected objects and 8 specifies 8-connected objects; if the argument is omitted, it defaults to 8. Class Support BW can be numeric or logical and it must be real, nonsparse and two-dimensional. The return value, eul, is of class double. Example BW = imread('circles.tif'); imshow(BW); bweuler(BW) ans = 2 Algorithm bweuler computes the Euler number by considering patterns of convexity and concavity in local 2-by-2 neighborhoods. See [2] for a discussion of the algorithm used. See Also bwmorph, bwperim 14-39 bweuler References [1] Horn, Berthold P. K., Robot Vision. New York: McGraw-Hill, 1986. pp. 73-77. [2] Pratt, William K. Digital Image Processing. New York: John Wiley & Sons, Inc., 1991. p. 633. 14-40 bwfill Purpose 14bwfill Fill background regions in a binary image Note This function is obsolete and may be removed in future versions. Use imfill instead. Syntax BW2 = bwfill(BW1,c,r,n) BW2 = bwfill(BW1,n) [BW2,idx] = bwfill(...) BW2 = bwfill(x,y,BW1,xi,yi,n) [x,y,BW2,idx,xi,yi] = bwfill(...) BW2 = bwfill(BW1,'holes',n) [BW2,idx] = bwfill(BW1,'holes',n) Description BW2 = bwfill(BW1,c,r,n) performs a flood-fill operation on the input binary image BW1, starting from the pixel (r,c). If r and c are equal-length vectors, the fill is performed in parallel from the starting pixels (r(k),c(k)). n can have a value of either 4 or 8 (the default), where 4 specifies 4-connected foreground and 8 specifies 8-connected foreground. The foreground of BW1 comprises the on pixels (i.e., having value of 1). BW2 = bwfill(BW1,n) displays the image BW1 on the screen and lets you select the starting points using the mouse. If you omit BW1, bwfill operates on the image in the current axes. Use normal button clicks to add points. Press Backspace or Delete to remove the previously selected point. A shift-click, right-click, or double-click selects a final point and then starts the fill; pressing Return finishes the selection without adding a point. [BW2,idx] = bwfill(...) returns the linear indices of all pixels filled by bwfill. BW2 = bwfill(x,y,BW1,xi,yi,n) uses the vectors x and y to establish a nondefault spatial coordinate system for BW1. xi and yi are scalars or equal-length vectors that specify locations in this coordinate system. 14-41 bwfill [x,y,BW2,idx,xi,yi] = bwfill(...) returns the XData and YData in x and y; the output image in BW2; linear indices of all filled pixels in idx; and the fill starting points in xi and yi. BW2 = bwfill(BW1,'holes',n) fills the holes in the binary image BW1. bwfill automatically determines which pixels are in object holes, and then changes the value of those pixels from 0 to 1. n defaults to 8 if you omit the argument. [BW2,idx] = bwfill(BW1,'holes',n) returns the linear indices of all pixels filled in by bwfill. If bwfill is used with no output arguments, the resulting image is displayed in a new figure. Remarks bwfill differs from many other binary image operations in that it operates on background pixels, rather than foreground pixels. If the foreground is 8-connected, the background is 4-connected, and vice versa. Note, however, that you specify the connectedness of the foreground when you call bwfill. Class Support The input image, BW1, must be a numeric or logical matrix. The output image, BW2, is logical. Example BW1 = [1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 BW2 = bwfill(BW1,3,3,8) BW2 = 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 0 1 0 0 0 14-42 0 1 0 0 1 1 0 0 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0] 0 0 0 0 1 0 0 0 bwfill I = imread('blood1.tif'); BW3 = ~im2bw(I); BW4 = bwfill(BW3,'holes'); imshow(BW3) figure, imshow(BW4) See Also bwselect, roifill 14-43 bwhitmiss Purpose 14bwhitmiss Binary hit-and-miss operation Syntax BW2 = bwhitmiss(BW1,SE1,SE2) BW2 = bwhitmiss(BW1,INTERVAL) Description BW2 = bwhitmiss(BW1,SE1,SE2) performs the hit-and-miss operation defined by the structuring elements SE1 and SE2. The hit-and-miss operation preserves pixels whose neighborhoods match the shape of SE1 and don't match the shape of SE2. SE1 and SE2 may be flat structuring element objects, created by strel, or neighborhood arrays. The neighborhoods of SE1 and SE2 should not have any overlapping elements. The syntax bwhitmiss(BW1,SE1,SE2) is equivalent to imerode(BW1,SE1) & imerode(~BW1,SE2). BW2 = bwhitmiss(BW1,INTERVAL) performs the hit-and-miss operation defined in terms of a single array, called an interval. An interval is an array whose elements can contain either 1, 0, or -1. The 1-valued elements make up the domain of SE1; the -1-valued elements make up the domain of SE2; and the 0-valued elements are ignored. The syntax bwhitmiss(INTERVAL) is equivalent to bwhitmiss(BW1,INTERVAL == 1, INTERVAL == -1). Class support BW1 can be a logical or numeric array of any dimension, and it must be nonsparse. BW2 is always a logical array with the same size as BW1. SE1 and SE2 must be flat STREL objects or they must be logical or numeric arrays containing 1s and 0s. INTERVAL must be an array containing 1s, 0s, and -1s. Example This example performs the hit-and-miss operation on a binary image using an interval. bw = [0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0] interval = [0 -1 -1 1 1 -1 0 1 0]; 14-44 bwhitmiss bw2 = bwhitmiss(bw,interval) bw2 = 0 0 0 0 0 0 See Also 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 imdilate, imerode, strel 14-45 bwlabel Purpose 14bwlabel Label connected components in a binary image Syntax L = bwlabel(BW,n) [L,num] = bwlabel(BW,n) Description L = bwlabel(BW,n) returns a matrix L, of the same size as BW, containing labels for the connected objects in BW. n can have a value of either 4 or 8, where 4 specifies 4-connected objects and 8 specifies 8-connected objects; if the argument is omitted, it defaults to 8. The elements of L are integer values greater than or equal to 0. The pixels labeled 0 are the background. The pixels labeled 1 make up one object, the pixels labeled 2 make up a second object, and so on. [L,num] = bwlabel(BW,n) returns in num the number of connected objects found in BW. Remarks bwlabel supports 2-D inputs only; bwlabeln supports inputs of any dimension. In some cases, you might prefer to use bwlabeln even for 2-D problems because it can be faster. If you have a 2-D input whose objects are relatively "thick" in the vertical direction, bwlabel will probably be faster; otherwise bwlabeln will probably be faster. Class Support BW can be logical or numeric, and it must be real, 2-D, and nonsparse. L is of class double. Remarks You can use the MATLAB find function in conjunction with bwlabel to return vectors of indices for the pixels that make up a specific object. For example, to return the coordinates for the pixels in object 2 [r,c] = find(bwlabel(BW)==2) You can display the output matrix as a pseudo-color indexed image. Each object appears in a different color, so the objects are easier to distinguish than in the original image. See label2rgb for more information. Example 14-46 This example illustrates using 4-connected objects. Notice objects 2 and 3; with 8-connected labeling, bwlabel would consider these a single object rather than two separate objects. bwlabel BW = [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0]; L = bwlabel(BW,4) L = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 2 2 0 0 0 3 0 0 0 0 3 3 3 3 0 0 0 0 0 0 0 0 0 [r,c] = find(L==2); rc = [r c] rc = 2 3 2 3 Algorithm 5 5 6 6 bwlabel uses the general procedure outlined in reference [1], pp. 40-48: 1 Run-length encode the input image. 2 Scan the runs, assigning preliminary labels and recording label equivalences in a local equivalence table. 3 Resolve the equivalence classes. 4 Relabel the runs based on the resolved equivalence classes. 14-47 bwlabel See Also bweuler, bwlabeln, bwselect, label2rgb Reference [1] Haralick, Robert M., and Linda G. Shapiro. Computer and Robot Vision, Volume I. Addison-Wesley, 1992. pp. 28-48. 14-48 bwlabeln Purpose 14bwlabeln Label connected components in N-D binary image Syntax L = bwlabeln(BW) [L,NUM] = bwlabeln(BW) [L,NUM] = bwlabeln(BW,CONN) Description L = bwlabeln(BW) returns a label matrix, L, containing labels for the connected components in BW. BW can have any dimension; L is the same size as BW. The elements of L are integer values greater than or equal to 0. The pixels labeled 0 are the background. The pixels labeled 1 make up one object, the pixels labeled 2 make up a second object, and so on. The default connectivity is 8 for two dimensions, 26 for three dimensions, and conndef(ndims(BW), 'maximal') for higher dimensions. [L,NUM] = bwlabeln(BW) returns in NUM the number of connected objects found in BW. [L,NUM] = bwlabeln(BW,CONN) specifies the desired connectivity. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may also be defined in a more general way for any dimension by using for CONN a 3-by-3-by- ...-by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. 14-49 bwlabeln Remarks bwlabel supports 2-D inputs only; bwlabeln supports inputs of any dimension. In some cases, you might prefer to use bwlabeln even for 2-D problems because it can be faster. If you have a 2-D input whose objects are relatively "thick" in the vertical direction, bwlabel will probably be faster; otherwise bwlabeln will probably be faster. Class Support BW can be numeric or logical, and it must be real and nonsparse. L is of class double. Example BW = cat(3,[1 1 0; 0 0 0; 1 0 0],... [0 1 0; 0 0 0; 0 1 0],... [0 1 1; 0 0 0; 0 0 1]) bwlabeln(BW) ans(:,:,1) = 1 0 2 1 0 0 0 0 0 ans(:,:,2) = 0 0 0 1 0 2 0 0 0 ans(:,:,3) = 0 0 0 Algorithm 1 0 0 1 0 2 bwlabeln uses the following general procedure: 1 Scan all image pixels, assigning preliminary labels to nonzero pixels and recording label equivalences in a union-find table. 14-50 bwlabeln 2 Resolve the equivalence classes using the union-find algorithm [1]. 3 Relabel the pixels based on the resolved equivalence classes. See Also bwlabel, label2rgb Reference [1] Robert Sedgewick, Algorithms in C, 3rd ed., Addison-Wesley, 1998, pp. 11-20. 14-51 bwmorph Purpose Syntax Description 14bwmorph Perform morphological operations on binary images BW2 = bwmorph(BW1,operation) BW2 = bwmorph(BW1,operation,n) BW2 = bwmorph(BW1,operation) applies a specific morphological operation to the binary image BW1. BW2 = bwmorph(BW1,operation,n) applies the operation n times. n can be Inf, in which case the operation is repeated until the image no longer changes. operation is a string that can have one of the values listed below. 'bothat' 'erode' 'shrink' 'bridge' 'fill' 'skel' 'clean' 'hbreak' 'spur' 'close' 'majority' 'thicken' 'diag' 'open' 'thin' 'dilate' 'remove' 'tophat' 'bothat' (“bottom hat”) performs morphological closing (dilation followed by erosion) and subtracts the original image. 'bridge' bridges previously unconnected pixels. For example, 1 1 0 0 0 0 0 1 1 becomes 1 1 0 0 1 0 0 1 1 'clean' removes isolated pixels (individual 1’s that are surrounded by 0’s), such as the center pixel in this pattern. 0 0 0 0 1 0 0 0 0 'close' performs morphological closing (dilation followed by erosion). 14-52 bwmorph 'diag' uses diagonal fill to eliminate 8-connectivity of the background. For example, 0 1 0 1 0 0 0 0 0 becomes 0 1 0 1 1 0 0 0 0 'dilate' performs dilation using the structuring element ones(3). 'erode' performs erosion using the structuring element ones(3). 'fill' fills isolated interior pixels (individual 0’s that are surrounded by 1’s), such as the center pixel in this pattern. 1 1 1 1 0 1 1 1 1 'hbreak' removes H-connected pixels. For example, 1 0 1 1 1 1 1 0 1 becomes 1 0 1 1 0 1 1 0 1 'majority' sets a pixel to 1 if five or more pixels in its 3-by-3 neighborhood are 1’s; otherwise, it sets the pixel to 0. 'open' implements morphological opening (erosion followed by dilation). 'remove' removes interior pixels. This option sets a pixel to 0 if all of its 4-connected neighbors are 1, thus leaving only the boundary pixels on. 'shrink', with n = Inf, shrinks objects to points. It removes pixels so that objects without holes shrink to a point, and objects with holes shrink to a connected ring halfway between each hole and the outer boundary. This option preserves the Euler number. 'skel', with n = Inf, removes pixels on the boundaries of objects but does not allow objects to break apart. The pixels remaining make up the image skeleton. This option preserves the Euler number. 'spur' removes spur pixels. For example, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14-53 bwmorph 0 0 1 0 1 1 1 0 0 0 0 0 becomes 0 0 1 0 1 1 0 0 0 0 0 0 'thicken', with n = Inf, thickens objects by adding pixels to the exterior of objects until doing so would result in previously unconnected objects being 8-connected. This option preserves the Euler number. 'thin', with n = Inf, thins objects to lines. It removes pixels so that an object without holes shrinks to a minimally connected stroke, and an object with holes shrinks to a connected ring halfway between each hole and the outer boundary. This option preserves the Euler number. 'tophat' (“top hat”) returns the image minus the morphological opening of the image. Class Support Example The input image BW1 can be numeric or logical. It must be 2-D, real and nonsparse. The output image BW2 is of class logical. BW1 = imread('circles.tif'); imshow(BW1); BW2 = bwmorph(BW1,'remove'); BW3 = bwmorph(BW1,'skel',Inf); imshow(BW2) figure, imshow(BW3) 14-54 bwmorph See Also bweuler, bwperim, dilate, erode References [1] Haralick, Robert M., and Linda G. Shapiro. Computer and Robot Vision, Volume I. Addison-Wesley, 1992. [2] Pratt, William K. Digital Image Processing. John Wiley & Sons, Inc., 1991. 14-55 bwpack Purpose 14bwpack Pack binary image Syntax BWP = bwpack(BW) Description BWP = bwpack(BW) packs the uint8 binary image BW into the uint32 array BWP, which is known as a packed binary image. Because each 8-bit pixel value in the binary image has only two possible values, 1 and 0, bwpack can map each pixel to a single bit in the packed output image. bwpack processes the image pixels by column, mapping groups of 32 pixels into the bits of a uint32 value. The first pixel in the first row corresponds to the least significant bit of the first uint32 element of the output array. The first pixel in the 32nd input row corresponds to the most significant bit of this same element. The first pixel of the 33rd row corresponds to the least significant bit of the second output element, and so on. If BW is M-by-N, then BWP is ceil(M/32)-by-N. This figure illustrates how bwpack maps the pixels in a binary image to the bits in a packed binary image. 32 rows Input Image (each pixel is uint8 value) 0 1 1 0 1 1 0 0 1 1. 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 1 0 0 Output Array 32 bits 0 0 1 1 0 1 1 0 0 ... 0 14-56 .. ... bwpack Binary image packing is used to accelerate some binary morphological operations, such as dilation and erosion. If the input to imdilate or imerode is a packed binary image, the functions use a specialized routine to perform the operation faster. bwunpack is used to unpack packed binary images. Class Support BW can be logical or numeric, and it must be 2-D, real, and nonsparse. BWP is of class uint32. Example Pack, dilate, and unpack a binary image: bw = imread('text.tif'); bwp = bwpack(bw); bwp_dilated = imdilate(bwp,ones(3,3),'ispacked ); bw_dilated = bwunpack(bwp_dilated, size(bw,1)); See Also bwunpack, imdilate, imerode 14-57 bwperim Purpose 14bwperim Find perimeter pixels in binary image Syntax BW2 = bwperim(BW1) BW2 = bwperim(BW1,CONN) Description BW2 = bwperim(BW1) returns a binary image containing only the perimeter pixels of objects in the input image BW1. A pixel is part of the perimeter if it is nonzero and it is connected to at least one zero-valued pixel. The default connectivity is 4 for two dimensions, 6 for three dimensions, and conndef(ndims(BW), 'minimal') for higher dimensions. BW2 = bwperim(BW1,CONN) specifies the desired connectivity. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may also be defined in a more general way for any dimension by using for CONN a 3-by-3-by-...-by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. Class Support Example 14-58 BW1 must be logical or numeric, and it must be nonsparse. BW2 is of class logical. BW1 = imread('circbw.tif'); BW2 = bwperim(BW1,8); imshow(BW1) bwperim figure, imshow(BW2) See Also bwarea, bweuler, bwfill, conndef 14-59 bwselect Purpose Syntax 14bwselect Select objects in a binary image BW2 = bwselect(BW1,c,r,n) BW2 = bwselect(BW1,n) [BW2,idx] = bwselect(...) BW2 = bwselect(x,y,BW1,xi,yi,n) [x,y,BW2,idx,xi,yi] = bwselect(...) Description BW2 = bwselect(BW1,c,r,n) returns a binary image containing the objects that overlap the pixel (r,c). r and c can be scalars or equal-length vectors. If r and c are vectors, BW2 contains the sets of objects overlapping with any of the pixels (r(k),c(k)). n can have a value of either 4 or 8 (the default), where 4 specifies 4-connected objects and 8 specifies 8-connected objects. Objects are connected sets of on pixels (i.e., pixels having a value of 1). BW2 = bwselect(BW1,n) displays the image BW1 on the screen and lets you select the (r,c) coordinates using the mouse. If you omit BW1, bwselect operates on the image in the current axes. Use normal button clicks to add points. Pressing Backspace or Delete removes the previously selected point. A shift-click, right-click, or double-click selects the final point; pressing Return finishes the selection without adding a point. [BW2,idx] = bwselect(...) returns the linear indices of the pixels belonging to the selected objects. BW2 = bwselect(x,y,BW1,xi,yi,n) uses the vectors x and y to establish a nondefault spatial coordinate system for BW1. xi and yi are scalars or equal-length vectors that specify locations in this coordinate system. [x,y,BW2,idx,xi,yi] = bwselect(...) returns the XData and YData in x and y; the output image in BW2; linear indices of all the pixels belonging to the selected objects in idx; and the specified spatial coordinates in xi and yi. If bwselect is called with no output arguments, the resulting image is displayed in a new figure. Example 14-60 BW1 = imread('text.tif'); c = [16 90 144]; r = [85 197 247]; bwselect BW2 = bwselect(BW1,c,r,4); imshow(BW1) figure, imshow(BW2) Class Support The input image, BW1, can be logical or numeric and must be 2-D and nonsparse. The output image, BW2, is of class logical. See Also bwfill, bwlabel, impixel, roipoly, roifill 14-61 bwulterode Purpose 14bwulterode Ultimate erosion Syntax BW2 = bwulterode(BW) BW2 = bwulterode(BW,METHOD,CONN) Description BW2 = bwulterode(BW) computes the ultimate erosion of the binary image BW. The ultimate erosion of BW consists of the regional maxima of the Euclidean distance transform of the complement of BW. The default connectivity for computing the regional maxima is 8 for two dimensions, 26 for three dimensions, and conndef(ndims(BW), 'maximal') for higher dimensions. BW2 = bwulterode(BW,METHOD,CONN) specifies the distance transform method and the regional maxima connectivity. METHOD can be one of the strings 'euclidean', 'cityblock', 'chessboard', or 'quasi-euclidean'. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may be defined in a more general way for any dimension by using for CONN a 3-by-3-by... - by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. Class Support 14-62 BW can be numeric or logical and it must be nonsparse. It can have any dimension. The return value, BW2, is always a logical array. bwulterode Example See Also bw = imread('circles.tif'); imshow(bw), title('Original') bw2 = bwulterode(bw); figure, imshow(bw2), title('Ultimate erosion') bwdist, conndef, imregionalmax 14-63 bwunpack Purpose 14bwunpack Unpack binary image Syntax BW = bwunpack(BWP,M) Description BW = bwunpack(BWP,M) unpacks the packed binary image BWP. BWP is a uint32 array. When it unpacks BWP, bwunpack maps the least significant bit of the first row of BWP to the first pixel in the first row of BW. The most significant bit of the first element of BWP maps to the first pixel in the 32nd row of BW, and so on. BW is M-by-N, where N is the number of columns of BWP. If M is omitted, its default value is 32*size(BWP,1). Binary image packing is used to accelerate some binary morphological operations, such as dilation and erosion. If the input to imdilate or imerode is a packed binary image, the functions use a specialized routine to perform the operation faster. bwpack is used to create packed binary images. Class Support BWP is of class uint32 and must be real, 2-D, and nonsparse. The return value, BW, is of class uint8. Example Pack, dilate, and unpack a binary image. bw = imread('text.tif'); bwp = bwpack(bw); bwp_dilated = imdilate(bwp,ones(3,3),'ispacked'); bw_dilated = bwunpack(bwp_dilated, size(bw,1)); See Also 14-64 bwpack, imdilate, imerode checkerboard Purpose 14checkerboard Create checkerboard image Syntax I = checkerboard I = checkerboard(N) I = checkerboard(N,P,Q) Description I = checkerboard creates a 8-by-8 square checkerboard image that has four identifiable corners. Each square has 10 pixels per side. The light squares on the left half of the checkerboard are white. The light squares on the right half of the checkerboard are gray. I = checkerboard(N) creates a checkerboard image where each square has N pixels per side. I = checkerboard(N,P,Q) creates a rectangular checkerboard, where P specifies the number of rows and Q specifies the number of columns. Each row and column is made up of tiles. Each tile contains four squares, N pixels per side, defined as: TILE = [DARK LIGHT; LIGHT DARK] If you omit Q, it defaults to P and the checkerboard is square Example Create a checkerboard where the side of every square is 20 pixels in length. I = checkerboard(20); imshow(I) Create a 2-by-3 rectangular checkerboard. J = checkerboard(20,2,3); figure, imshow(J) 14-65 checkerboard Create a black and white checkerboard. K = (checkerboard > 0.5); figure, imshow(K) See Also 14-66 cp2tform, imtransform, maketform cmpermute Purpose 14cmpermute Rearrange the colors in a colormap Syntax [Y,newmap] = cmpermute(X,map) [Y,newmap] = cmpermute(X,map,index) Description [Y,newmap] = cmpermute(X,map) randomly reorders the colors in map to produce a new colormap newmap. The cmpermute function also modifies the values in X to maintain correspondence between the indices and the colormap, and returns the result in Y. The image Y and associated colormap newmap produce the same image as X and map. [Y,newmap] = cmpermute(X,map,index) uses an ordering matrix (such as the second output of sort) to define the order of colors in the new colormap. Class Support The input image X can be of class uint8 or double. Y is returned as an array of the same class as X. Example To arrange a colormap in order by luminance, use ntsc = rgb2ntsc(map); [dum,index] = sort(ntsc(:,1)); [Y,newmap] = cmpermute(X,map,index); See Also randperm, sort in the MATLAB Function Reference 14-67 cmunique Purpose 14cmunique Find unique colormap colors and the corresponding image Syntax [Y,newmap] = cmunique(X,map) [Y,newmap] = cmunique(RGB) [Y,newmap] = cmunique(I) Description [Y,newmap] = cmunique(X,map) returns the indexed image Y and associated colormap newmap that produce the same image as (X,map) but with the smallest possible colormap. The cmunique function removes duplicate rows from the colormap and adjusts the indices in the image matrix accordingly. [Y,newmap] = cmunique(RGB) converts the truecolor image RGB to the indexed image Y and its associated colormap newmap. The return value, newmap, is the smallest possible colormap for the image, containing one entry for each unique color in RGB. (Note that newmap may be very large, because the number of entries can be as many as the number of pixels in RGB.) [Y,newmap] = cmunique(I) converts the intensity image I to an indexed image Y and its associated colormap newmap. The return value, newmap, is the smallest possible colormap for the image, containing one entry for each unique intensity level in I. Class Support The input image can be of class uint8, uint16, or double. The class of the output image Y is uint8 if the length of newmap is less than or equal to 256. If the length of newmap is greater than 256, Y is of class double. See Also gray2ind, rgb2ind 14-68 col2im Purpose 14col2im Rearrange matrix columns into blocks Syntax A = col2im(B,[m n],[mm nn], block_type) A = col2im(B,[m n],[mm nn]) Description col2im rearranges matrix columns into blocks. block_type is a string with one of these values: • 'distinct' for m-by-n distinct blocks • 'sliding' for m-by-n sliding blocks (default) A = col2im(B,[m n],[mm nn],'distinct') rearranges each column of B into a distinct m-by-n block to create the matrix A of size mm-by-nn. If B = [A11(:) A12(:) A21(:) A22(:)], where each column has length m*n, then A = [A11 A12;A21 A22] where each Aij is m-by-n. A = col2im(B,[m n],[mm nn],'sliding') rearranges the row vector B into a matrix of size (mm-m+1)-by-(nn-n+1). B must be a vector of size 1-by-(mm-m+1)*(nn-n+1). B is usually the result of processing the output of im2col(...,'sliding') using a column compression function (such as sum). A = col2im(B,[m n],[mm nn]) uses the default block_type of 'sliding'. Class Support B can be logical or numeric. The return value, A, is of the same class as B. See Also blkproc, colfilt, im2col, nlfilter 14-69 colfilt Purpose 14colfilt Perform neighborhood operations using column-wise functions Syntax B B B B Description colfilt processes distinct or sliding blocks as columns. colfilt can perform similar operations to blkproc and nlfilter, but often executes much faster. = = = = colfilt(A,[m n],block_type,fun) colfilt(A,[m n],block_type,fun,P1,P2,...) colfilt(A,[m n],[mblock nblock],block_type,fun,...) colfilt(A,'indexed',...) B = colfilt(A,[m n],block_type,fun) processes the image A by rearranging each m-by-n block of A into a column of a temporary matrix, and then applying the function fun to this matrix. fun can be a function_handle, created using @, or an inline object. colfilt zero pads A, if necessary. Before calling fun, colfilt calls im2col to create the temporary matrix. After calling fun, colfilt rearranges the columns of the matrix back into m-by-n blocks using col2im. block_type is a string with one of these values: • 'distinct' for m-by-n distinct blocks • 'sliding' for m-by-n sliding neighborhoods B = colfilt(A,[m n],'distinct',fun) rearranges each m-by-n distinct block of A into a column in a temporary matrix, and then applies the function fun to this matrix. fun must return a matrix of the same size as the temporary matrix. colfilt then rearranges the columns of the matrix returned by fun into m-by-n distinct blocks. B = colfilt(A,[m n],'sliding',fun) rearranges each m-by-n sliding neighborhood of A into a column in a temporary matrix, and then applies the function fun to this matrix. fun must return a row vector containing a single value for each column in the temporary matrix. (Column compression functions such as sum return the appropriate type of output.) colfilt then rearranges the vector returned by fun into a matrix of the same size as A. B = colfilt(A,[m n],block_type,fun,P1,P2,...) passes the additional parameters P1,P2,..., to fun. The colfilt function calls fun using, y = fun(x,P1,P2,...) 14-70 colfilt where x is the temporary matrix before processing, and y is the temporary matrix after processing. B = colfilt(A,[m n],[mblock nblock],block_type,fun,...) processes the matrix A as above, but in blocks of size mblock-by-nblock to save memory. Note that using the [mblock nblock] argument does not change the result of the operation. B = colfilt(A,'indexed',...) processes A as an indexed image, padding with zeros if the class of A is uint8 or uint16, or ones if the class of A is double. Class Support The input image A can be of any class supported by fun. The class of B depends on the class of the output from fun. Example This example sets each output pixel to the mean value of the input pixel’s 5-by-5 neighborhood. I = imread('tire.tif') imshow(I) I2 = uint8(colfilt(I,[5 5],'sliding',@mean)); figure, imshow(I2) See Also blkproc, col2im, im2col, nlfilter 14-71 colorbar Purpose 14colorbar Display a colorbar colorbar is a MATLAB function. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-72 conndef Purpose 14conndef Create connectivity array Syntax CONN = conndef(NUM_DIMS,TYPE) Description CONN = conndef(NUM_DIMS,TYPE) returns the connectivity array defined by TYPE for NUM_DIMS dimensions. TYPE can have either of the values listed in this table. 'minimal' Defines a neighborhood whose neighbors are touching the central element on an (N-1)-dimensional surface, for the N-dimensional case. See Examples for an illustration. 'maximal' Defines a neighborhood including neighbors that touch the central element in any way; it is ones(repmat(3,1,NUM_DIMS)). See Examples for an illustration. Several Image Processing Toolbox functions use conndef to create the default connectivity input argument. Examples The minimal connectivity array for two dimensions includes the neighbors touching the central element along a line. conn1 = conndef(2,'minimal') conn1 = 0 1 0 1 1 1 0 1 0 The minimal connectivity array for three dimensions includes all the neighbors touching the central element along a face. conndef(3,'minimal') ans(:,:,1) 0 0 0 = 0 1 0 0 0 0 14-73 conndef ans(:,:,2) 0 1 0 = 1 1 1 0 1 0 ans(:,:,3) 0 0 0 = 0 1 0 0 0 0 The maximal connectivity array for two dimensions includes all the neighbors touching the central element in any way. conn2 = conndef(2,'maximal') conn2 = 1 1 1 14-74 1 1 1 1 1 1 conv2 Purpose 14conv2 Perform two-dimensional convolution conv2 is a function in MATLAB. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-75 convmtx2 Purpose 14convmtx2 Compute two-dimensional convolution matrix Syntax T = convmtx2(H,m,n) T = convmtx2(H,[m n]) Description T = convmtx2(H,m,n) or T = convmtx2(H,[m n]) returns the convolution matrix T for the matrix H. If X is an m-by-n matrix, then reshape(T*X(:),size(H)+[m n]-1) is the same as conv2(X,H). Class Support The inputs are all of class double. The output matrix T is of class sparse. The number of nonzero elements in T is no larger than prod(size(H))*m*n. See Also conv2 convmtx in the Signal Processing Toolbox User’s Guide 14-76 convn Purpose 14convn Perform N-dimensional convolution convn is a MATLAB function. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-77 corr2 Purpose 14corr2 Compute the two-dimensional correlation coefficient between two matrices Syntax r = corr2(A,B) Description r = corr2(A,B) computes the correlation coefficient between A and B, where A and B are matrices or vectors of the same size. Class Support A and B can be numeric or logical. The return value, r, is a scalar double. Algorithm corr2 computes the correlation coefficient using ∑ ∑ ( Amn – A ) ( B mn –B ) m n r = ----------------------------------------------------------------------------------------------------- 2 2 ( A mn – A ) ( B mn – B ) ∑∑ ∑∑ m n m n where A = mean2(A), and B = mean2(B). See Also std2 corrcoef in the MATLAB Function Reference 14-78 cp2tform Purpose 14cp2tform Infer geometric transformation from control point pairs Syntax TFORM = cp2tform(input_points,base_points,transformtype) TFORM = cp2tform(CPSTRUCT,transformtype) TFORM = cp2tform(input_points,base_points,transformtype,parameter) TFORM = cp2tform(CPSTRUCT,transformtype,parameter) [TFORM,input_points,base_points] = cp2tform(CPSTRUCT,...) [TFORM,input_points,base_points,input_points_bad,base_points_bad] = cp2tform(...,'piecewise linear') Description TFORM = cp2tform(input_points,base_points,transformtype) takes pairs of control points and uses them to infer a spatial transformation. The function returns a TFORM structure containing the spatial transformation. input_points is an m-by-2 double matrix containing the x and y coordinates of control points in the image you want to transform. base_points is an m-by-2 double matrix containing the x and y coordinates of control points specified in the base image. The transformtype argument specifies the type of transformation you want to infer. For more information about supported types, see Transform Type. TFORM = cp2tform(CPSTRUCT,transformtype) passes a CPSTRUCT structure that contains the control point matrices for the input and base images. The Control Point Selection Tool, cpselect, creates the CPSTRUCT. [TFORM,input_points,base_points] = cp2tform(CPSTRUCT,...) returns the control points that were actually used in input_points, and base_points. Unmatched and predicted points are not used. For more information, see cpstruct2pairs. Transform Type transformtype specifies the type of spatial transformation to infer. This table lists all of the transformation types supported by cp2tform, in order of complexity. (See Algorithms for detailed information about each transform type.) The table includes the minimum number of control point pairs you must select for each type. The 'lwm' and 'polynomial' transform types can each 14-79 cp2tform take an optional, additional parameter. See the syntax descriptions that follow for details. Transformation Type Description Minimum Control Points 'linear conformal' Use this transformation when shapes in the input image are unchanged, but the image is distorted by some combination of translation, rotation, and scaling. Straight lines remain straight, and parallel lines are still parallel. 2 pairs 'affine' Use this transformation when shapes in the input image exhibit shearing. Straight lines remain straight, and parallel lines remain parallel, but rectangles become parallelograms. 3 pairs 'projective' Use this transformation when the scene appears “tilted.” Straight lines remain straight, but parallel lines converge toward “vanishing points” which may or may not fall within the image. 4 pairs 14-80 Example cp2tform Use this transformation when objects in the image are curved. The higher the order of the polynomial, the better the fit, but the result can contain more curves than the base image. 6 pairs (order 2) 'piecewise linear' Use this transformation when parts of the image appear distorted differently. 4 pairs 'lwm' Use this transformation (local weighted mean), when the distortion varies locally and piecewise linear is not sufficient. 6 pairs (12 pairs recommended) 'polynomial' 10 pairs (order 3) 16 pairs (order 4) Note When transformtype is 'linear conformal', 'affine', 'projective', or 'polynomial', and input_points and base_points (or CPSTRUCT) have the minimum number of control points needed for a particular transformation, cp2tform finds the coefficients exactly. If input_points and base_points include more than the minimum number of points, cp2tform uses a least squares solution. For more information, see mldivide. TFORM = cp2tform(input_points,base_points,'polynomial',ORDER) TFORM = cp2tform(CPSTRUCT,'polynomial',order) When 'polynomial' is the transform type, you can optionally specify the order of the polynomial to use. order can be the scalar value 2, 3, or 4. If you omit order, it defaults to 3. TFORM = cp2tform(input_points,base_points,'lwm',N) TFORM = cp2tform(CPSTRUCT,'lwm',N) When 'lwm' is the transform type, you can optionally specify the number of points, N, used to infer each polynomial. The radius of influence extends out to the furthest control point used to infer that polynomial. The N closest points are used to infer a polynomial of order 2 for each control point pair. If you omit N, 14-81 cp2tform it defaults to 12. N can be as small as 6, but making N small risks generating ill-conditioned polynomials. [TFORM,input_points,base_points,input_points_bad,base_points_bad]= cp2tform(input_points,base_points,'piecewise linear') [TFORM,input_points,base_points,input_points_bad,base_points_bad]= cp2tform(CPSTRUCT,'piecewise linear') When 'piecewise linear' is the transform type, cp2tform can optionally return the control points that were actually used in input_points and base_points, and return two arrays, input_points_bad and base_points_bad, that contain control points that were eliminated because they were middle vertices of degenerate fold-over triangles. Algorithms cp2tform uses the following general procedure: 1 Use valid pairs of control points to infer a spatial transformation or an inverse-mapping from output space (x,y) to input space (u,v) according to transformtype. 2 Return TFORM structure containing spatial transformation. The procedure varies depending on the transformtype. Linear Conformal Linear conformal transformations may include a rotation, a scaling, and a translation. Shapes and angles are preserved. Parallel lines remain parallel. Straight lines remain straight. Let: sc = scale*cos(angle) ss = scale*sin(angle) [u v] = [x y 1] * [ sc -ss ss sc tx ty] Solve for sc, ss, tx, ty. t_lc = cp2tform(input_points,base_points,'linear conformal'); The coefficients of the inverse mapping are stored in t_lc.tdata.Tinv. 14-82 cp2tform Since linear conformal transformations are a subset of affine transformations, t_lc.forward_fcn is @affine_fwd and t_lc.inverse_fcn is @affine_inv. At least two control-point pairs are needed to solve for the four unknown coefficients. Affine In an affine transformation, the x and y dimensions can be scaled or sheared independently and there may be a translation. Parallel lines remain parallel. Straight lines remain straight. Linear conformal transformations are a subset of affine transformations. For an affine transformation: [u v] = [x y 1] * Tinv Tinv is a 3-by-2 matrix. Solve for the 6 elements of Tinv. t_affine = cp2tform(input_points,base_points,'affine'); The coefficients of the inverse mapping are stored in t_affine.tdata.Tinv. At least 3 control-point pairs are needed to solve for the 6 unknown coefficients. Projective In a projective transformation, quadrilaterals map to quadrilaterals. Straight lines remain straight. Affine transformations are a subset of projective transformations. For a projective transformation: [up vp wp] = [x y w] * Tinv where u = up/wp v = vp/wp Tinv is a 3-by-3 matrix. Assuming Tinv = [ A D G; B E H; 14-83 cp2tform C F I ]; u = (Ax + By + C)/(Gx + Hy + I) v = (Dx + Ey + F)/(Gx + Hy + I) Solve for the 9 elements of Tinv. t_proj = cp2tform(input_points,base_points,'projective'); The coefficients of the inverse mapping are stored in t_proj.tdata.Tinv. At least 4 control-point pairs are needed to solve for the 9 unknown coefficients. Polynomial In a polynomial transformation, polynomial functions of x and y determine the mapping. Second-Order Polynomials For a second-order polynomial transformation: [u v] = [1 x y x*y x^2 y^2] * Tinv Both u and v are second-order polynomials of x and y. Each second-order polynomial has six terms. To specify all coefficients, Tinv has size 6-by-2. t_poly_ord2 = cp2tform(input_points,base_points,'polynomial'); The coefficients of the inverse mapping are stored in t_poly_ord2.tdata. At least 6 control-point pairs are needed to solve for the 12 unknown coefficients. 14-84 cp2tform Third-Order Polynomials For a third-order polynomial transformation: [u v] = [1 x y x*y x^2 y^2 y*x^2 x*y^2 x^3 y^3] * Tinv Both u and v are third-order polynomials of x and y. Each third-order polynomial has ten terms. To specify all coefficients, Tinv has size 10-by-2. t_poly_ord3 = cp2tform(input_points, base_points, 'polynomial',3); The coefficients of the inverse mapping are stored in t_poly_ord3.tdata. At least 10 control-point pairs are needed to solve for the 20 unknown coefficients. Fourth-Order Polynomials For a fourth-order polynomial transformation: [u v] = [1 x y x*y x^2 y^2 y*x^2 x*y^2 x^3 y^3] * Tinv Both u and v are fourth-order polynomials of x and y. Each fourth-order polynomial has fifteen terms. To specify all coefficients, Tinv has size 15-by-2. t_poly_ord4 = cp2tform(input_points, base_points, 'polynomial',4); The coefficients of the inverse mapping are stored in t_poly_ord4.tdata. At least 15 control-point pairs are needed to solve for the 30 unknown coefficients. Piecewise Linear In a piecewise linear transformation, linear (affine) transformations are applied separately to each triangular region of the image [1]. 14-85 cp2tform 1 Find a Delaunay triangulation of the base control points. 2 Using the 3 vertices of each triangle, infer an affine mapping from base to input coordinates. Note At least 4 control-point pairs are needed. Four pairs result in two triangles with distinct mappings. Local Weighted Mean For each control point in base_points: 1 Find the N closest control points. 2 Use these N points and their corresponding points in input_points to infer a second order polynomial. 3 Calculate the radius of influence of this polynomial as the distance from the center control point to the furthest point used to infer the polynomial (using base_points). [2] Note At least 6 control-point pairs are needed to solve for the second-order polynomial. Ill-conditioned polynomials may result if too few pairs are used. Example I = checkerboard; J = imrotate(I,30); base_points = [11 11; 41 71]; input_points = [14 44; 70 81]; cpselect(J,I,input_points,base_points); t = cp2tform(input_points,base_points,'linear conformal'); To recover angle and scale: ss = t.tdata.Tinv(2,1); % ss = scale * sin(angle) sc = t.tdata.Tinv(1,1); % sc = scale * cos(angle) angle = atan2(ss,sc)*180/pi scale = sqrt(ss*ss + sc*sc) 14-86 cp2tform See Also cpcorr, cpselect, cpstruct2pairs, imtransform References [1] Ardeshir Goshtasby, Piecewise linear mapping functions for image registration, Pattern Recognition, Vol 19, pp. 459-466, 1986. [2] Ardeshir Goshtasby, Image registration by local approximation methods, Image and Vision Computing, Vol 6, p. 255-261, 1988. 14-87 cpcorr Purpose Syntax Description 14cpcorr Tune control point locations using cross correlation input_points = cpcorr(input_points_in,base_points_in,input,base) input_points = cpcorr(input_points_in,base_points_in,input,base) uses normalized cross-correlation to adjust each pair of control points specified in input_points_in and base_points_in. input_points_in must be an M-by-2 double matrix containing the coordinates of control points in the input image. base_points_in is an M-by-2 double matrix containing the coordinates of control points in the base image. cpcorr returns the adjusted control points in input_points, a double matrix the same size as input_points_in. If cpcorr cannot correlate a pair of control points, input_points contains the same coordinates as input_points_in for that pair. cpcorr only moves the position of a control point by up to 4 pixels. Adjusted coordinates are accurate to one tenth of a pixel. cpcorr is designed to get subpixel accuracy from the image content and coarse control point selection. Note input and base images must have the same scale for cpcorr to be effective. cpcorr cannot adjust a point if any of the following occur: • Points are too near the edge of either image • Regions of images around points contain Inf or NaN • Region around a point in input image has zero standard deviation • Regions of images around points are poorly correlated Class Support The images, input and base, can be of class logical, uint8, uint16, or double and must contain finite values. The control point pairs are of class double. Algorithm cpcorr uses the following general procedure. For each control point pair: 14-88 cpcorr 1 Extract an 11-by-11 template around the input control point and a 21-by-21 region around the base control point. 2 Calculate the normalized cross-correlation of the template with the region. 3 Find the absolute peak of the cross-correlation matrix. 4 Use the position of the peak to adjust the coordinates of the input control point. Example This example uses cpcorr to fine-tune control points selected in an image. Note the difference in the values of the input_points matrix and the input_points_adj matrix. input = imread('lily.tif'); base = imread('flowers.tif'); input_points = [127 93; 74 59]; base_points = [323 195; 269 161]; input_points_adj = cpcorr(input_points,base_points,... input(:,:,1),base(:,:,1)) input_points_adj = 126.0000 72.1000 See Also 94.0000 60.0000 cp2tform, cpselect, imtransform, normxcorr2 14-89 cpselect Purpose 14cpselect Control Point Selection Tool Syntax cpselect(input,base) cpselect(input,base,CPSTRUCT_IN ) cpselect(input,base,XYINPUT_IN,XYBASE_IN) H = cpselect(input,base,...) Description cpselect(input,base) starts the Control Point Selection Tool, a graphical user interface that enables you to select control points in two related images. input is the image that needs to be warped to bring it into the coordinate system of the base image. input and base can be either variables that contain images or strings that identify files containing grayscale images. The Control Point Selection Tool returns the control points in a CPSTRUCT structure. (For more information, see “Using the Control Point Selection Tool” in Chapter 5.) cpselect(input,base,CPSTRUCT_IN) starts cpselect with an initial set of control points that are stored in CPSTRUCT_IN. This syntax allows you to restart cpselect with the state of control points previously saved in CPSTRUCT_IN. cpselect(input,base,xyinput_in,xybase_in) starts cpselect with a set of initial pairs of control points. xyinput_in and xybase_in are m-by-2 matrices that store the input and base coordinates, respectively. H = cpselect(input,base,...) returns a handle H to the tool. You can use the dispose(H) or H.dispose syntaxes to close the tool from the command line. Class Support The input images can be of class uint8, uint16, double, or logical. Algorithm cpselect uses the following general procedure for control point prediction. 1 Find all valid pairs of control points. 2 Infer a spatial transformation between input and base control points using method that depends on the number of valid pairs as follows: 14-90 2 pairs Linear conformal 3 pairs Affine 4 or more pairs Projective cpselect 3 Apply spatial transformation to the new point to generate the predicted point. 4 Display predicted point. Example Start tool with saved images. aerial = imread('westconcordaerial.png'); cpselect(aerial(:,:,1),'westconcordorthophoto.png') Start tool with workspace images and points. I = checkerboard; J = imrotate(I,30); base_points = [11 11; 41 71]; input_points = [14 44; 70 81]; cpselect(J,I,input_points,base_points); See Also cpcorr, cp2tform, cpstruct2pairs, imtransform 14-91 cpstruct2pairs Purpose 14cpstruct2pairs Convert CPSTRUCT to valid pairs of control points Syntax [input_points, base_points] = cpstruct2pairs(CPSTRUCT) Description [input_points, base_points] = cpstruct2pairs(CPSTRUCT) takes a CPSTRUCT (produced by cpselect) and returns the arrays of coordinates of valid control point pairs in input_points and base_points. cpstruct2pairs eliminates unmatched points and predicted points. Example Start the Control Point Selection Tool, cpselect. cpselect('lily.tif','flowers.tif') Using cpselect, pick control points in the images. Select Save To Workspace from the File menu to save the points to the workspace. On the Save dialog box, check the Structure with all points checkbox and uncheck Input points and Base points. Click OK. Use cpstruct2pairs to extract the input and base points from the CPSTRUCT. [input_points,base_points] = cpstruct2pairs(cpstruct); See Also 14-92 cp2tform, cpselect, imtransform dct2 Purpose 14dct2 Compute two-dimensional discrete cosine transform Syntax B = dct2(A) B = dct2(A,m,n) B = dct2(A,[m n]) Description B = dct2(A) returns the two-dimensional discrete cosine transform of A. The matrix B is the same size as A and contains the discrete cosine transform coefficients B(k1,k2). B = dct2(A,m,n) or B = dct2(A,[m n]) pads the matrix A with zeros to size m-by-n before transforming. If m or n is smaller than the corresponding dimension of A, dct2 truncates A. Class Support A can be numeric or logical. The returned matrix, B, is of class double. Algorithm The discrete cosine transform (DCT) is closely related to the discrete Fourier transform. It is a separable, linear transformation; that is, the two-dimensional transform is equivalent to a one-dimensional DCT performed along a single dimension followed by a one-dimensional DCT in the other dimension. The definition of the two-dimensional DCT for an input image A and output image B is M–1 N–1 B pq = α p α q ∑ ∑ π ( 2m + 1 )p π ( 2n + 1 )q A mn cos ------------------------------- cos ----------------------------- , 2M 2N 0≤p≤M–1 0≤q≤N–1 m=0 n=0 1 ⁄ M, p = 0 αp = 2 ⁄ M, 1 ≤ p ≤ M – 1 1 ⁄ N, q = 0 αq = 2 ⁄ N, 1 ≤ q ≤ N – 1 where M and N are the row and column size of A, respectively. If you apply the DCT to real data, the result is also real. The DCT tends to concentrate information, making it useful for image compression applications. This transform can be inverted using idct2. Example The commands below compute the discrete cosine transform for the autumn image. Notice that most of the energy is in the upper left corner. 14-93 dct2 RGB = imread('autumn.tif'); I = rgb2gray(RGB); J = dct2(I); imshow(log(abs(J)),[]), colormap(jet(64)), colorbar Now set values less than magnitude 10 in the DCT matrix to zero, and then reconstruct the image using the inverse DCT function idct2. J(abs(J) < 10) = 0; K = idct2(J)/255; imshow(K) 14-94 dct2 See Also fft2, idct2, ifft2 References [1] Jain, Anil K. Fundamentals of Digital Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1989. pp. 150-153. [2] Pennebaker, William B., and Joan L. Mitchell. JPEG: Still Image Data Compression Standard. Van Nostrand Reinhold, 1993. 14-95 dctmtx Purpose 14dctmtx Compute discrete cosine transform matrix Syntax D = dctmtx(n) Description D = dctmtx(n) returns the n-by-n DCT (discrete cosine transform) matrix. D*A is the DCT of the columns of A and D'*A is the inverse DCT of the columns of A (when A is n-by-n). Class Support n is an integer scalar of class double. D is returned as a matrix of class double. Remarks If A is square, the two-dimensional DCT of A can be computed as D*A*D'. This computation is sometimes faster than using dct2, especially if you are computing a large number of small DCTs, because D needs to be determined only once. For example, in JPEG compression, the DCT of each 8-by-8 block is computed. To perform this computation, use dctmtx to determine D, and then calculate each DCT using D*A*D' (where A is each 8-by-8 block). This is faster than calling dct2 for each individual block. See Also 14-96 dct2 deconvblind Purpose 14deconvblind Restore image using the blind deconvolution algorithm Syntax [J,PSF] [J,PSF] [J,PSF] [J,PSF] [J,PSF] [J,PSF] Description [J,PSF] = deconvblind(I,INITPSF) deconvolves image I using the maximum likelihood algorithm, returning both the deblurred image, J, and a restored point-spread function, PSF. The input array, I, and your initial guess at the PSF, INITPSF, can be numeric arrays or cell arrays. (Use cell arrays when you want to be able to perform additional deconvolutions that start where your initial deconvolution finished. See Resuming Deconvolution for more information.) The restored PSF is a positive array that is the same size as INITPSF, normalized so its sum adds up to 1. = = = = = = deconvblind(I,INITPSF) deconvblind(I,INITPSF,NUMIT) deconvblind(I,INITPSF,NUMIT,DAMPAR) deconvblind(I,INITPSF,NUMIT,DAMPAR,WEIGHT) deconvblind(I,INITPSF,NUMIT,DAMPAR,WEIGHT,READOUT) deconvblind(...,FUN,P1,P2,...,PN) Note The PSF restoration is affected strongly by the size of the initial guess, INITPSF, and less by the values it contains. For this reason, specify an array of ones as your INITPSF. To improve the restoration, deconvblind supports several optional parameters, described below. Use [] as a place holder if you do not specify an intermediate parameter. [J,PSF] = deconvblind(I,INITPSF,NUMIT) specifies the number of iterations (default is 10). [J,PSF] = deconvblind(I,INITPSF,NUMIT,DAMPAR) specifies the threshold deviation of the resulting image from the input image I (in terms of the standard deviation of Poisson noise) below which damping occurs. The iterations are suppressed for the pixels that deviate within the DAMPAR value from their original value. This suppresses the noise generation in such pixels, preserving necessary image details elsewhere. The default value is 0 (no damping). 14-97 deconvblind [J,PSF] = deconvblind(I,INITPSF,NUMIT,DAMPAR,WEIGHT) specifies which pixels in the input image, I, are considered in the restoration. By default, WEIGHT is a unit array, the same size as the input image. You can assign a value between 0.0 and 1.0 to elements in the WEIGHT array. The value of an element in the WEIGHT array determines how much the pixel at the corresponding position in the input image is considered. For example, to exclude a pixel from consideration, assign it a value of 0 in the WEIGHT array. You can adjust the weight value assigned to each pixel according to the amount of flat-field correction. [J,PSF] = deconvblind(I,INITPSF,NUMIT,DAMPAR,WEIGHT,READOUT), where READOUT is an array (or a value) corresponding to the additive noise (e.g., background, foreground noise) and the variance of the read-out camera noise. READOUT has to be in the units of the image. The default value is 0. [J,PSF] = deconvblind(...,FUN,P1,P2,...,PN), where FUN is a function describing additional constraints on the PSF. There are four ways to specify FUN: • Function handle (@) • Inline object • String containing function name • String containing a MATLAB expression FUN is called at the end of each iteration. FUN must accept the PSF as its first argument and can accept additional parameters P1, P2,..., PN. The FUN function should return one argument, PSF, that is the same size as the original PSF and which satisfies the positivity and normalization constraints. Note The output image J could exhibit ringing introduced by the discrete Fourier transform used in the algorithm. To reduce the ringing use I = edgetaper(I,PSF) prior calling deconvblind. Resuming Deconvolution 14-98 You can use deconvblind to perform a deconvolution that starts where a previous deconvolution stopped. To use this feature, pass the input image, I, and the initial guess at the PSF, INITPSF, as cell arrays: {I} and {INITPSF}. When you do, the deconvblind function returns the output image J and the deconvblind restored point spread function, PSF, as cell arrays, which can then be passed as the input arrays into the next deconvblind call. The output cell array, J, contains four elements: J{1} contains I, the original image J{2} contains the result of the last iteration J{3} contains the result of the next-to-last iteration J{4} is an array generated by the iterative algorithm Class Support Example I can be of class uint8, uint16, or double. The DAMPAR and READOUT arguments have to be of the same class as the input image. Other inputs have to be of class double. The output image, J, or the first array of the output cell array, is of the same class as the input image. I = checkerboard(8); PSF = fspecial('gaussian',7,10); V = .0001; BlurredNoisy = imnoise(imfilter(I,PSF),'gaussian',0,V); WT = zeros(size(I)); WT(5:end-4,5:end-4) = 1; INITPSF = ones(size(PSF)); FUN = inline('PSF + P1','PSF','P1'); [J P]= deconvblind(BlurredNoisy,INITPSF,20,10*sqrt(V),WT,FUN,0); subplot(221);imshow(BlurredNoisy); title('A = Blurred and Noisy'); subplot(222);imshow(PSF,[]); title('True PSF'); subplot(223);imshow(J); title('Deblured Image'); subplot(224);imshow(P,[]); title('Recovered PSF'); See Also deconvlucy, deconvreg, deconvwnr, otf2psf, padarray, psf2otf 14-99 deconvlucy Purpose 14deconvlucy Restore image using the Lucy-Richardson algorithm Syntax J = deconvlucy(I,PSF) J = deconvlucy(I,PSF,NUMIT) J = deconvlucy(I,PSF,NUMIT,DAMPAR) J = deconvlucy(I,PSF,NUMIT,DAMPAR,READOUT) J = deconvlucy(I,PSF,NUMIT,DAMPAR,READOUT,WEIGHT) deconvlucy(I,PSF,NUMIT,DAMPAR,READOUT,WEIGHT,SUBSMPL) Description J = deconvlucy(I,PSF) restores image I, degraded by convolution with a point-spread function, PSF, and possibly by additive noise. The algorithm is based on maximizing the likelihood of the resulting image J being an instance of the original image I under Poisson statistics. The input array, I, can be a numeric array or cell array. (Use a cell array when you want to be able to perform additional deconvolutions that start where your initial deconvolution finished. See Resuming Deconvolution for more information.) To improve the restoration, deconvlucy supports several optional parameters. Use [] as a place holder if you do not specify an intermediate parameter. J = deconvlucy(I,PSF,NUMIT) specifies the number of iterations the deconvlucy function performs. If this value is not specified, the default is 10. J = deconvlucy(I,PSF,NUMIT,DAMPAR) specifies the threshold deviation of the resulting image from the image I (in terms of the standard deviation of Poisson noise), below which damping occurs. Iterations are suppressed for pixels that deviate beyond the DAMPAR value from their original value. This suppresses the noise generation in such pixels, preserving necessary image details elsewhere. The default value is 0 (no damping). J = deconvlucy(I,PSF,NUMIT,DAMPAR,WEIGHT) specifies the weight to be assigned to each pixel to reflect its recording quality in the camera. A bad pixel is excluded from the solution by assigning it zero weight value. Instead of giving a weight of unity for good pixels, one could adjust their weight according to the amount of flat-field correction. The default is a unit array of the same size as input image I. J = deconvlucy(I,PSF,NUMIT,DAMPAR,WEIGHT,READOUT) specifies a value corresponding to the additive noise (e.g., background, foreground noise) and 14-100 deconvlucy the variance of the read-out camera noise. READOUT has to be in the units of the image. The default value is 0. J = deconvlucy(I,PSF,NUMIT,DAMPAR,WEIGHT,READOUT,SUBSMPL), where SUBSMPL denotes subsampling and is used when the PSF is given on a grid that is SUBSMPL times finer than the image. The default value is 1. Note The output image J could exhibit ringing introduced by the discrete Fourier transform used in the algorithm. To reduce the ringing use I = edgetaper(I,PSF) prior calling deconvlucy. Resuming Deconvolution If input I is a cell array, the output J becomes a cell array of size 1-by-4, where J{1} contains I, the original image J{2} contains the result of the last iteration J{3} contains the result of the next-to-last iteration J{4} is an array generated by the iterative algorithm The input cell array could contain one numerical array (the blurred image), or four numerical arrays if it was the output from the previous run of deconvlucy. Class Support I can be of class uint8, uint16, or double. The DAMPAR and READOUT arguments have to be of the same class as the input image. Other inputs have to be of class double. Output image (or the first array of the output cell) is of the same class as the input image. Example I = checkerboard(8); PSF = fspecial('gaussian',7,10); V = .0001; BlurredNoisy = imnoise(imfilter(I,PSF),'gaussian',0,V); WT = zeros(size(I)); WT(5:end-4,5:end-4) = 1; J1 = deconvlucy(BlurredNoisy,PSF); J2 = deconvlucy(BlurredNoisy,PSF,20,sqrt(V)); J3 = deconvlucy(BlurredNoisy,PSF,20,sqrt(V),[],WT); subplot(221);imshow(BlurredNoisy); 14-101 deconvlucy title('A = Blurred and Noisy'); subplot(222);imshow(J1); title('deconvlucy(A,PSF)'); subplot(223);imshow(J2); title('deconvlucy(A,PSF,NI,DP)'); subplot(224);imshow(J3); title('deconvlucy(A,PSF,NI,DP,[],WT)'); See Also 14-102 deconvblind, deconvreg, deconvwnr, otf2psf, padarray, psf2otf deconvreg Purpose 14deconvreg Restore image using a regularized filter Syntax J = J = J = J = [J, Description J = deconvreg(I,PSF) restores image I that was degraded by convolution with a point-spread function PSF and possibly by additive noise. The algorithm is a constrained optimum in a sense of least square error between the estimated and the true images under requirement of preserving image smoothness. deconvreg(I,PSF) deconvreg(I,PSF,NOISEPOWER) deconvreg(I,PSF,NOISEPOWER,LRANGE) deconvreg(I,PSF,NOISEPOWER,LRANGE,REGOP) LAGRA] = deconvreg(I,PSF,...) J = deconvreg(I,PSF,NOISEPOWER), where NOISEPOWER is the additive noise power. The default value is 0. J = deconvreg(I,PSF,NOISEPOWER,LRANGE), where LRANGE is a vector specifying range where the search for the optimal solution is performed. The algorithm finds an optimal Lagrange multiplier, LAGRA, within the LRANGE range. If LRANGE is a scalar, the algorithm assumes that LAGRA is given and equal to LRANGE; the NP value is then ignored. The default range is between [1e-9 and 1e9]. J = deconvreg(I,PSF,NOISEPOWER,LRANGE,REGOP), where REGOP is the regularization operator to constrain the deconvolution. The default regularization operator is the Laplacian operator, to retain the image smoothness. The REGOP array dimensions must not exceed the image dimensions, any nonsingleton dimensions must correspond to the nonsingleton dimensions of PSF. [J, LAGRA] = deconvreg(I,PSF,...) outputs the value of the Lagrange multiplier, LAGRA, in addition to the restored image J. Note The output image J could exhibit ringing introduced by the discrete Fourier transform used in the algorithm. To reduce the ringing, process the image with the edgetaper function prior to calling the deconvreg function; for example, I = edgetaper(I,PSF). 14-103 deconvreg Class Support Example I can be of class uint8, uint16, or double. Other inputs have to be of class double. J is of the same class as I. I = checkerboard(8); PSF = fspecial('gaussian',7,10); V = .01; BlurredNoisy = imnoise(imfilter(I,PSF),'gaussian',0,V); NOISEPOWER = V*prod(size(I)); [J LAGRA] = deconvreg(BlurredNoisy,PSF,NOISEPOWER); subplot(221); imshow(BlurredNoisy); title('A = Blurred and Noisy'); subplot(222); imshow(J); title('[J LAGRA] = deconvreg(A,PSF,NP)'); subplot(223); imshow(deconvreg(BlurredNoisy,PSF,[],LAGRA/10)); title('deconvreg(A,PSF,[],0.1*LAGRA)'); subplot(224); imshow(deconvreg(BlurredNoisy,PSF,[],LAGRA*10)); title('deconvreg(A,PSF,[],10*LAGRA)'); See Also 14-104 deconvblind, deconvlucy, deconvwnr, otf2psf, padarray, psf2otf deconvwnr Purpose 14deconvwnr Restore image using the Wiener filter Syntax J = deconvwnr(I,PSF) J = deconvwnr(I,PSF,NSR) J = deconvwnr(I,PSF,NCORR,ICORR) Description J = deconvwnr(I,PSF) restores image I that was degraded by convolution with a point-spread function, PSF, and possibly by additive noise. The algorithm is optimal in a sense of least mean square error between the estimated and the true image, and uses the correlation matrixes of image and noise. In the absence of noise, the Weiner filter reduces to the ideal inverse filter. J = deconvwnr(I,PSF,NSR), where NSR is the noise-to-signal power ratio. NSR could be a scalar or an array of the same size as I. The default value is 0. J = deconvwnr(I,PSF,NCORR,ICORR), where NCORR and ICORR are the autocorrelation functions of the noise and the original image, respectively. NCORR and ICORR could be of any size or dimension not exceeding the original image. An N-dimensional NCORR or ICORR array corresponds to the autocorrelation within each dimension. A vector NCORR or ICORR represents an autocorrelation function in first dimension if PSF is a vector. If PSF is an array, the 1-D autocorrelation function is extrapolated by symmetry to all nonsingleton dimensions of PSF. A scalar NCORR or ICORR represents the power of the noise or the image. Note The output image J could exhibit ringing introduced by the discrete Fourier transform used in the algorithm. To reduce the ringing, process the image with the edgetaper function prior to calling the deconvwnr function; for example, I = edgetaper(I,PSF) Class Support Example I can be of class uint8, uint16, or double. Other inputs have to be of class double. J is of the same class as I. I = checkerboard(8); noise = 0.1*randn(size(I)); PSF = fspecial('motion',21,11); Blurred = imfilter(I,PSF,'circular'); 14-105 deconvwnr BlurredNoisy = im2uint8(Blurred + noise); NSR = sum(noise(:).^2)/sum(I(:).^2);% noise-to-power ratio NP = abs(fftn(noise)).^2;% noise power NPOW = sum(NP(:))/prod(size(noise)); NCORR = fftshift(real(ifftn(NP)));% noise autocorrelation function, centered IP = abs(fftn(I)).^2;% original image power IPOW = sum(IP(:))/prod(size(I)); ICORR = fftshift(real(ifftn(IP)));% image autocorrelation function, centered ICORR1 = ICORR(:,ceil(size(I,1)/2)); NSR = NPOW/IPOW; subplot(221);imshow(BlurredNoisy,[]); title('A = Blurred and Noisy'); subplot(222);imshow(deconvwnr(BlurredNoisy,PSF,NSR),[]); title('deconvwnr(A,PSF,NSR)'); subplot(223);imshow(deconvwnr(BlurredNoisy,PSF,NCORR,ICORR),[]); title('deconvwnr(A,PSF,NCORR,ICORR)'); subplot(224);imshow(deconvwnr(BlurredNoisy,PSF,NPOW,ICORR1),[]); title('deconvwnr(A,PSF,NPOW,ICORR_1_D)'); See Also 14-106 deconvblind, deconvlucy, deconvreg, otf2psf, padarray, psf2otf dicominfo Purpose 14dicominfo Read metadata from a DICOM message Syntax info = dicominfo(filename) info = dicominfo(filename,'dictionary', D) Description info = dicominfo(filename) reads the metadata from the compliant Digital Imaging and Communications in Medicine (DICOM) file specified in the string, filename. info = dicominfo(filename,'dictionary', D) uses the data dictionary file given in the string D to read the DICOM message. The file in D must be on the MATLAB search path. The default dictionary file is dicom-dict.txt. Examples info = dicominfo('CT-MONO2-16-ankle.dcm') info = Filename: FileModDate: FileSize: Format: FormatVersion: Width: Height: BitDepth: ColorType: . . . See Also [1x47 char] '24-Dec-2000 19:54:47' 525436 'DICOM' 3 512 512 16 'grayscale' dicomread, dicomwrite 14-107 dicomread Purpose 14dicomread Read a DICOM image Syntax X = dicomread(filename) X = dicomread(info) [X,map] = dicomread(...) [X,map,alpha] = dicomread(...) [X,map,alpha,overlays] = dicomread(...) Description X = dicomread(filename) reads the image data from the compliant Digital Imaging and Communications in Medicine (DICOM) file, filename. For single-frame grayscale images, X is an M-by-N array. For single-frame true-color images, X is an M-by-N-by-3 array. Multiframe images are always 4-D arrays. X = dicomread(info) reads the image data from the message referenced in the DICOM metadata structure info. The info structure is produced by the dicominfo function. [X,map] = dicomread(...) returns the image X and the colormap MAP. If X is a grayscale or true-color image, MAP is empty. [X,map,alpha] = dicomread(...) returns the image X, the colormap map, and an alpha channel matrix for X. The values of alpha are 0 if the pixel is opaque; otherwise they are row indices into map. The RGB value in map should be substituted for the value in X to use alpha. alpha has the same height and width as X and is 4-D for a multiframe image. [X,map,alpha,overlays] = dicomread(...) also returns the image X, the colormap map, an alpha channel matrix for X, and any overlays from the DICOM file. Each overlay is a 1-bit black and white image with the same height and width as X. If multiple overlays are present in the file, overlays is a 4-D multiframe image. If no overlays are in the file, overlays is empty. The first input argument, either filename or info, can be followed by a set of parameter name/value pairs. [...] = dicomread(filename,param1, value1, param2, value2, ...) [...] = dicomread(info, param1, value1, param2, value2, ...) 14-108 dicomread Supported parameters names and values include the following. 'Frames', V dicomread reads only the frames in the vector V from the image. V must be an integer scalar, a vector of integers, or the string 'all'. The default value is 'all'. 'Dictionary', D dicomread uses the data dictionary file whose filename is in the string D. The default value is 'dicom-dict.txt'. 'Raw', TF dicomread performs pixel-level transformations depending on whether TF is 1 or 0. If TF is 1 (the default), dicomread reads the exact pixels from the image and no pixel-level transformations are performed. If TF is 0, images are rescaled to use the full dynamic range, and color images are automatically converted to the RGB colorspace. Note 1: Because the HSV colorspace is inadequately defined in the DICOM standard, dicomread does not automatically convert them to RGB. Note 2: dicomread never rescales or changes the color spaces of images containing signed data. Note 3: Rescaling values and applying colorspace conversions does not change the metadata in any way. Consequently, metadata values that refer to pixel values (such as window center/width or LUTs) may not be correct when pixels are scaled or converted. Examples Example 1 Use dicomread to retrieve the data matrix, X, and colormap matrix, map, needed to create a montage. [X, map] = dicomread('US-PAL-8-10x-echo.dcm'); montage(X, map); 14-109 dicomread Example 2 Call dicomread with the information retrieved from the DICOM file using dicominfo. Display the image with imshow. info = dicominfo('CT-MONO2-16-ankle.dcm'); Y = dicomread(info); imshow(Y, []); See Also 14-110 dicominfo, dicomwrite dicomwrite Purpose 14dicomwrite Write images as DICOM file Syntax dicomwrite(X, filename) dicomwrite(X, map, filename) dicomwrite(...,param1,value1,param2,value2,...) dicomwrite(...,meta_struct,...) dicomwrite(...,info,...) status = dicomwrite(...) Description dicomwrite(X,filename) writes the binary, grayscale, or truecolor image, X, to the file, filename, where filename is a string specifying the name of the Digital Imaging and Communications in Medicine (DICOM) file to create. dicomwrite(X,map,filename) writes the indexed image, X, with colormap, map. dicomwrite(...,param1,value1,param2,value2,...) specifies additional metadata to write to the DICOM file. The parameters (param1, param2, etc.) are either names of DICOM file attributes or options that affect how the file is written. Each attribute or option has a corresponding value (value1, value2, etc.). To find a list of the names of DICOM attributes, see the data dictionary file, dicom-dict.txt, included with the Image Processing Toolbox. This table lists the options supported by the dicomwrite function. Option Name Description Values 'Endian' Specifies the byte-ordering for the file. 'Little' [Default] 'Big' 'VR' Specifies whether the two-letter value representation (VR) code should be written to the file ('explicit') or inferred from the data dictionary ('implicit'). 'Implicit' [Default] 'Explicit' Note: If you specify the 'Endian' value 'Big', you can only specify the 'VR' value of 'Explicit'. 14-111 dicomwrite Option Name Description Values 'CompressionMode' Specifies the type of compression to use when storing the image. 'None' [Default] 'JPEG lossy' 'RLE' 'TransferSyntax' A DICOM UID specifying the DICOM Transfer Syntax. Note: If you specify the 'TransferSyntax' option, dicomwrite ignores the other three options, if they are specified. The 'TransferSyntax' option encodes the settings for the 'Endian', 'VR' and 'CompressionMode' options in a single value. A DICOM Transfer Syntax that specifies the default values for 'Endian', 'VR', and 'CompressionMode' options. dicomwrite(...,meta_struct,...) specifies metadata in a structure, meta_struct. The structure's field names must be the names of DICOM file attributes or options. The field's value is the value of that attribute or option. dicomwrite(...,info,...) specifies metadata in the metadata structure, info, which is produced by the dicominfo function. For more information about this structure, see dicominfo. status = dicomwrite(...) returns a structure that lists three types of metadata that were passed to dicomwrite: • Metadata that does not affect how the DICOM file is written • Metadata that does not pertain to the type of image being written • Metadata that is not modifiable by a user This syntax can be useful when you specify an info structure that was created by dicominfo to the dicomwrite function. An info structure can contain many fields. If no metadata was specified, dicomwrite returns an empty matrix ([]). 14-112 dicomwrite The structure returned by dicomwrite contains these three fields. Example Field Description 'dicominfo_fields' A cell array containing the names of metadata passed to dicomwrite that does not affect how the file is written. 'wrong_IOD' A cell array containing the names of attributes passed to dicomwrite that do not pertain to the type of image being written. (IOD=Information Object Definition) 'not_modifiable' A cell array containing the names of valid metadata fields for the image that cannot be modified by the user. This example uses dicominfo to retrieve information about the contents of the sample DICOM file included with the Image Processing Toolbox. The example uses dicomread to read the data from the file and then writes the data into a new DICOM file, including the metadata from the original file. info = dicominfo('CT-MONO2-16-ankle.dcm'); Y = dicomread(info); status = dicomwrite(Y,'my_dicomfile.dcm',info); status = dicominfo_fields: {12x1 cell} wrong_IOD: {21x1 cell} not_modifiable: {23x1 cell} status.dicominfo_fields ans = 'BitDepth' 'ColorType' 'FileModDate' 'FileSize' 'FileStruct' 'Filename' 14-113 dicomwrite 'Format' 'FormatVersion' 'Height' 'SelectedFrames' 'StartOfPixelData' 'Width' See Also 14-114 dicomread, dicominfo dilate Purpose 14dilate Perform dilation on a binary image Note This function is obsolete and may be removed in future versions. Use imdilate instead. Syntax BW2 = dilate(BW1,SE) BW2 = dilate(BW1,SE,alg) BW2 = dilate(BW1,SE,...,n) Description BW2 = dilate(BW1,SE) performs dilation on the binary image BW1, using the binary structuring element SE. SE is a matrix containing only 1’s and 0’s. BW2 = dilate(BW1,SE,alg) performs dilation using the specified algorithm. alg is a string that can have one of these values: • 'spatial' (default) – processes the image in the spatial domain. • 'frequency' – processes the image in the frequency domain. Both algorithms produce the same result, but they make different trade-offs between speed and memory use. The frequency algorithm is faster for large images and structuring elements than the spatial algorithm, but uses much more memory. BW2 = dilate(BW1,SE,...,n) performs the dilation operation n times. Class Support The input image BW1 can be of class double or uint8. The output image BW2 is of class uint8. Remarks You should use the frequency algorithm only if you have a large amount of memory on your system. If you use this algorithm with insufficient memory, it may actually be slower than the spatial algorithm, due to virtual memory paging. If the frequency algorithm slows down your system excessively, or if you receive “out of memory” messages, use the spatial algorithm instead. Example BW1 = imread('text.tif'); SE = ones(6,2); BW2 = dilate(BW1,SE); 14-115 dilate imshow(BW1) figure, imshow(BW2) See Also bwmorph, imdilate, imerode References [1] Gonzalez, Rafael C., and Richard E. Woods. Digital Image Processing. Addison-Wesley, 1992. p. 518. [2] Haralick, Robert M., and Linda G. Shapiro. Computer and Robot Vision, Volume I. Addison-Wesley, 1992. p. 158. 14-116 dither Purpose 14dither Convert an image, increasing apparent color resolution by dithering Syntax X = dither(RGB,map) BW = dither(I) Description X = dither(RGB,map) creates an indexed image approximation of the RGB image in the array RGB by dithering the colors in colormap map. map can not have more than 65,536 colors. X = dither(RGB,map,Qm,Qe) creates an indexed image from RGB, specifying the parameters Qm and Qe. Qm specifies the number of quantization bits to use along each color axis for the inverse color map, and Qe specifies the number of quantization bits to use for the color space error calculations. If Qe < Qm, dithering cannot be performed and an undithered indexed image is returned in X. If you omit these parameters, dither uses the default values Qm = 5, Qe = 8. BW = dither(I) converts the intensity image in the matrix I to the binary (black and white) image BW by dithering. Class Support The input image, RGB or I, can be of class uint8, uint16, or double. All other input arguments must be of class double. The output indexed image, X, is of class uint8 if it is an indexed image with 256 or fewer colors; otherwise its class is uint16. The output binary image, BW, is of class logical. Algorithm dither increases the apparent color resolution of an image by applying Floyd-Steinberg’s error diffusion dither algorithm. References [1] Floyd, R. W. and L. Steinberg. “An Adaptive Algorithm for Spatial Gray Scale,” International Symposium Digest of Technical Papers. Society for Information Displays, 1975. p. 36. [2] Lim, Jae S. Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1990. pp. 469-476. See Also rgb2ind 14-117 double Purpose 14double Convert data to double precision double is a MATLAB built-in function. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-118 edge Purpose Syntax 14edge Find edges in an intensity image BW = edge(I,'sobel') BW = edge(I,'sobel',thresh) BW = edge(I,'sobel',thresh,direction) [BW,thresh] = edge(I,'sobel',...) BW = edge(I,'prewitt') BW = edge(I,'prewitt',thresh) BW = edge(I,'prewitt',thresh,direction) [BW,thresh] = edge(I,'prewitt',...) BW = edge(I,'roberts') BW = edge(I,'roberts',thresh) [BW,thresh] = edge(I,'roberts',...) BW = edge(I,'log') BW = edge(I,'log',thresh) BW = edge(I,'log',thresh,sigma) [BW,threshold] = edge(I,'log',...) BW = edge(I,'zerocross',thresh,h) [BW,thresh] = edge(I,'zerocross',...) BW = edge(I,'canny') BW = edge(I,'canny',thresh) BW = edge(I,'canny',thresh,sigma) [BW,threshold] = edge(I,'canny',...) Description edge takes an intensity image I as its input, and returns a binary image BW of the same size as I, with 1’s where the function finds edges in I and 0’s elsewhere. edge supports six different edge-finding methods: • The Sobel method finds edges using the Sobel approximation to the derivative. It returns edges at those points where the gradient of I is maximum. 14-119 edge • The Prewitt method finds edges using the Prewitt approximation to the derivative. It returns edges at those points where the gradient of I is maximum. • The Roberts method finds edges using the Roberts approximation to the derivative. It returns edges at those points where the gradient of I is maximum. • The Laplacian of Gaussian method finds edges by looking for zero crossings after filtering I with a Laplacian of Gaussian filter. • The zero-cross method finds edges by looking for zero crossings after filtering I with a filter you specify. • The Canny method finds edges by looking for local maxima of the gradient of I. The gradient is calculated using the derivative of a Gaussian filter. The method uses two thresholds, to detect strong and weak edges, and includes the weak edges in the output only if they are connected to strong edges. This method is therefore less likely than the others to be “fooled” by noise, and more likely to detect true weak edges. The parameters you can supply differ depending on the method you specify. If you do not specify a method, edge uses the Sobel method. Sobel Method BW = edge(I,'sobel') specifies the Sobel method. BW = edge(I,'sobel',thresh) specifies the sensitivity threshold for the Sobel method. edge ignores all edges that are not stronger than thresh. If you do not specify thresh, or if thresh is empty ([]), edge chooses the value automatically. BW = edge(I,'sobel',thresh,direction) specifies direction of detection for the Sobel method. direction is a string specifying whether to look for 'horizontal' or 'vertical' edges, or 'both' (the default). [BW,thresh] = edge(I,'sobel',...) returns the threshold value. Prewitt Method BW = edge(I,'prewitt') specifies the Prewitt method. BW = edge(I,'prewitt',thresh) specifies the sensitivity threshold for the Prewitt method. edge ignores all edges that are not stronger than thresh. If 14-120 edge you do not specify thresh, or if thresh is empty ([]), edge chooses the value automatically. BW = edge(I,'prewitt',thresh,direction) specifies direction of detection for the Prewitt method. direction is a string specifying whether to look for 'horizontal' or 'vertical' edges, or 'both' (the default). [BW,thresh] = edge(I,'prewitt',...) returns the threshold value. Roberts Method BW = edge(I,method) specifies the Roberts method. BW = edge(I,method,thresh) specifies the sensitivity threshold for the Roberts method. edge ignores all edges that are not stronger than thresh. If you do not specify thresh, or if thresh is empty ([]), edge chooses the value automatically. [BW,thresh] = edge(I,method,...) returns the threshold value. Laplacian of Gaussian Method BW = edge(I,'log') specifies the Laplacian of Gaussian method. BW = edge(I,'log',thresh) specifies the sensitivity threshold for the Laplacian of Gaussian method. edge ignores all edges that are not stronger than thresh. If you do not specify thresh, or if thresh is empty ([]), edge chooses the value automatically. BW = edge(I,'log',thresh,sigma) specifies the Laplacian of Gaussian method, using sigma as the standard deviation of the LoG filter. The default sigma is 2; the size of the filter is n-by-n, where n = ceil(sigma*3)*2+1. [BW,thresh] = edge(I,'log',...) returns the threshold value. Zero-cross Method BW = edge(I,'zerocross',thresh,h) specifies the zero-cross method, using the filter h. thresh is the sensitivity threshold; if the argument is empty ([]), edge chooses the sensitivity threshold automatically. [BW,thresh] = edge(I,'zerocross',...) returns the threshold value. 14-121 edge Canny Method BW = edge(I,'canny') specifies the Canny method. BW = edge(I,'canny',thresh) specifies sensitivity thresholds for the Canny method. thresh is a two-element vector in which the first element is the low threshold, and the second element is the high threshold. If you specify a scalar for thresh, this value is used for the high threshold and 0.4*thresh is used for the low threshold. If you do not specify thresh, or if thresh is empty ([]), edge chooses low and high values automatically. BW = edge(I,'canny',thresh,sigma) specifies the Canny method, using sigma as the standard deviation of the Gaussian filter. The default sigma is 1; the size of the filter is chosen automatically, based on sigma. [BW,thresh] = edge(I,'canny',...) returns the threshold values as a two-element vector. Class Support I can be of class uint8, uint16, or double. BW is of class logical. Remarks For the 'log' and 'zerocross' methods, if you specify a threshold of 0, the output image has closed contours, because it includes all of the zero crossings in the input image. Example Find the edges of the rice.tif image using the Prewitt and Canny methods. I = imread('rice.tif'); BW1 = edge(I,'prewitt'); BW2 = edge(I,'canny'); imshow(BW1); figure, imshow(BW2) 14-122 edge See Also fspecial References [1] Canny, John. “A Computational Approach to Edge Detection,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 1986. Vol. PAMI-8, No. 6, pp. 679-698. [2] Lim, Jae S. Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1990. pp. 478-488. [3] Parker, James R. Algorithms for Image Processing and Computer Vision. New York: John Wiley & Sons, Inc., 1997. pp. 23-29. 14-123 edgetaper Purpose 14edgetaper Taper the discontinuities along the image edges Syntax J = edgetaper(I,PSF) Description J = edgetaper(I,PSF) blurs the edges of the input image, I, using the point spread function, PSF. The output image, J, is the weighted sum of the original image, I, and its blurred version. The weighting array, determined by the autocorrelation function of PSF, makes J equal to I in its central region, and equal to the blurred version of I near the edges. The edgetaper function reduces the ringing effect in image deblurring methods that use the discrete Fourier transform, such as deconvwnr, deconvreg, and deconvlucy. Note The size of the PSF cannot exceed half of the image size in any dimension. Class Support Example See Also 14-124 I and PSF can be of class uint8, uint16, or double. J is of the same class as I. I = imread('cameraman.tif'); PSF = fspecial('gaussian',60,10); J = edgetaper(I,PSF); subplot(1,2,1);imshow(I,[]);title('original image'); subplot(1,2,2);imshow(J,[]);title('edges tapered'); deconvlucy, deconvreg, deconvwnr, otf2psf, padarray, psf2otf erode Purpose 14erode Perform erosion on a binary image Note This function is obsolete and may be removed in future versions. Use imerode instead. Syntax BW2 = erode(BW1,SE) BW2 = erode(BW1,SE,alg) BW2 = erode(BW1,SE,...,n) Description BW2 = erode(BW1,SE) performs erosion on the binary image BW1, using the binary structuring element SE. SE is a matrix containing only 1’s and 0’s. BW2 = erode(BW1,SE,alg) performs erosion using the specified algorithm. alg is a string that can have one of these values: • 'spatial' (default) – processes the image in the spatial domain • 'frequency' – processes the image in the frequency domain Both algorithms produce the same result, but they make different tradeoffs between speed and memory use. The frequency algorithm is faster for large images and structuring elements than the spatial algorithm, but uses much more memory. BW2 = erode(BW1,SE,...,n) performs the erosion operation n times. Class Support The input image BW1 can be of class double or uint8. The output image BW2 is of class uint8. Remarks You should use the frequency algorithm only if you have a large amount of memory on your system. If you use this algorithm with insufficient memory, it may actually be slower than the spatial algorithm, due to virtual memory paging. If the frequency algorithm slows down your system excessively, or if you receive “out of memory” messages, use the spatial algorithm instead. Example BW1 = imread('text.tif'); SE = ones(3,1); BW2 = erode(BW1,SE); 14-125 erode imshow(BW1) figure, imshow(BW2) See Also bwmorph, imdilate, imerode References [1] Gonzalez, Rafael C., and Richard E. Woods. Digital Image Processing. Addison-Wesley, 1992. p. 518. [2] Haralick, Robert M., and Linda G. Shapiro. Computer and Robot Vision, Volume I. Addison-Wesley, 1992. p. 158. 14-126 fft2 Purpose 14fft2 Compute two-dimensional fast Fourier transform fft2 is a function in MATLAB. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-127 fftn Purpose 14fftn Compute N-dimensional fast Fourier transform fftn is a function in MATLAB. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-128 fftshift Purpose 14fftshift Shift zero-frequency component of fast Fourier transform to center of spectrum fftshift is a function in MATLAB. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-129 filter2 Purpose 14filter2 Perform two-dimensional linear filtering filter2 is a function in MATLAB. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-130 findbounds Purpose 14findbounds Find output bounds for spatial transformation Syntax outbounds = findbounds(TFORM,inbounds) Description outbounds = findbounds(TFORM,inbounds) estimates the output bounds corresponding to a given spatial transformation and a set of input bounds. TFORM is a spatial transformation structure as returned by maketform. inbounds is 2-by-NUM_DIMS matrix. The first row of inbounds specifies the lower bounds for each dimension, and the second row specifies the upper bounds. NUM_DIMS has to be consistent with the ndims_in field of TFORM. outbounds has the same form as inbounds. It is an estimate of the smallest rectangular region completely containing the transformed rectangle represented by the input bounds. Since outbounds is only an estimate, it may not completely contain the transformed input rectangle. Notes imtransform uses findbounds to compute the 'OutputBounds' parameter if the user does not provide it. If TFORM contains a forward transformation (a nonempty forward_fcn field), then findbounds works by transforming the vertices of the input bounds rectangle and then taking minimum and maximum values of the result. If TFORM does not contain a forward transformation, then findbounds estimates the output bounds using the Nelder-Mead optimization function fminsearch. If the optimization procedure fails, findbounds issues a warning and returns outbounds=inbounds. Example See Also inbounds = [0 0; 1 1] tform = maketform('affine',[2 0 0; .5 3 0; 0 0 1]) outbounds = findbounds(tform, inbounds) cp2tform, imtransform, maketform, tformarray, tformfwd, tforminv 14-131 fliptform Purpose 14fliptform Flip the input and output roles of a TFORM structure Syntax TFLIP = fliptform(T) Description TFLIP = fliptform(T) creates a new spatial transformation structure, a TFORM struct, by flipping the roles of the inputs and outputs in an existing TFORM struct. Example T = maketform('affine', [.5 0 0; .5 2 0; 0 0 1]); T2 = fliptform(T) The following are equivalent: x = tformfwd([-3 7],T) x = tforminv([-3 7],T2) See Also 14-132 maketform, tformfwd, tforminv freqspace Purpose 14freqspace Determine frequency spacing for two-dimensional frequency response freqspace is a function in MATLAB. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-133 freqz2 Purpose 14freqz2 Compute two-dimensional frequency response Syntax [H,f1,f2] = freqz2(h,n1,n2) [H,f1,f2] = freqz2(h,[n2 n1]) [H,f1,f2] = freqz2(h,f1,f2) [H,f1,f2] = freqz2(h) [...] = freqz2(h,...,[dx dy]) [...] = freqz2(h,...,dx) freqz2(...) Description [H,f1,f2] = freqz2(h,n1,n2) returns H, the n2-by-n1 frequency response of h, and the frequency vectors f1 (of length n1) and f2 (of length n2). h is a two-dimensional FIR filter, in the form of a computational molecule. f1 and f2 are returned as normalized frequencies in the range -1.0 to 1.0, where 1.0 corresponds to half the sampling frequency, or π radians. [H,f1,f2] = freqz2(h,[n2 n1]) returns the same result returned by [H,f1,f2] = freqz2(h,n1,n2). [H,f1,f2] = freqz2(h) uses [n2 n1] = [64 64]. [H,f1,f2] = freqz2(h,f1,f2) returns the frequency response for the FIR filter h at frequency values in f1 and f2. These frequency values must be in the range -1.0 to 1.0, where 1.0 corresponds to half the sampling frequency, or π radians. [...] = freqz2(h,...,[dx dy]) uses [dx dy] to override the intersample spacing in h. dx determines the spacing for the x-dimension and dy determines the spacing for the y-dimension. The default spacing is 0.5, which corresponds to a sampling frequency of 2.0. [...] = freqz2(h,...,dx) uses dx to determine the intersample spacing in both dimensions. With no output arguments, freqz2(...) produces a mesh plot of the two-dimensional magnitude frequency response. Class Support 14-134 The input matrix h can be of class double or of any integer class. All other inputs to freqz2 must be of class double. All outputs are of class double. freqz2 Example Use the window method to create a 16-by-16 filter, then view its frequency response using freqz2. Hd = zeros(16,16); Hd(5:12,5:12) = 1; Hd(7:10,7:10) = 0; h = fwind1(Hd,bartlett(16)); colormap(jet(64)) freqz2(h,[32 32]); axis ([-1 1 -1 1 0 1]) See Also freqz in the Signal Processing Toolbox User’s Guide 14-135 fsamp2 Purpose 14fsamp2 Design two-dimensional FIR filter using frequency sampling Syntax h = fsamp2(Hd) h = fsamp2(f1,f2,Hd,[m n]) Description fsamp2 designs two-dimensional FIR filters based on a desired two-dimensional frequency response sampled at points on the Cartesian plane. h = fsamp2(Hd) designs a two-dimensional FIR filter with frequency response Hd, and returns the filter coefficients in matrix h. (fsamp2 returns h as a computational molecule, which is the appropriate form to use with filter2.) The filter h has a frequency response that passes through points in Hd. If Hd is m-by-n, then h is also m-by-n. Hd is a matrix containing the desired frequency response sampled at equally spaced points between -1.0 and 1.0 along the x and y frequency axes, where 1.0 corresponds to half the sampling frequency, or π radians. H d ( f 1, f 2 ) = H d ( ω 1, ω 2 ) ω 1 = πf 1, ω 2 = πf 2 For accurate results, use frequency points returned by freqspace to create Hd. (See the entry for freqspace for more information.) h = fsamp2(f1,f2,Hd,[m n]) produces an m-by-n FIR filter by matching the filter response at the points in the vectors f1 and f2. The frequency vectors f1 and f2 are in normalized frequency, where 1.0 corresponds to half the sampling frequency, or π radians. The resulting filter fits the desired response as closely as possible in the least squares sense. For best results, there must be at least m*n desired frequency points. fsamp2 issues a warning if you specify fewer than m*n points. Class Support The input matrix Hd can be of class double or of any integer class. All other inputs to fsamp2 must be of class double. All outputs are of class double. Example Use fsamp2 to design an approximately symmetric two-dimensional bandpass filter with passband between 0.1 and 0.5 (normalized frequency, where 1.0 corresponds to half the sampling frequency, or π radians): 1 Create a matrix Hd that contains the desired bandpass response. Use freqspace to create the frequency range vectors f1 and f2. 14-136 fsamp2 [f1,f2] = freqspace(21,'meshgrid'); Hd = ones(21); r = sqrt(f1.^2 + f2.^2); Hd((r<0.1)|(r>0.5)) = 0; colormap(jet(64)) mesh(f1,f2,Hd) 2 Design the filter that passes through this response. h = fsamp2(Hd); freqz2(h) 14-137 fsamp2 Algorithm fsamp2 computes the filter h by taking the inverse discrete Fourier transform of the desired frequency response. If the desired frequency response is real and symmetric (zero phase), the resulting filter is also zero phase. See Also conv2, filter2, freqspace, ftrans2, fwind1, fwind2 Reference [1] Lim, Jae S. Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1990. pp. 213-217. 14-138 fspecial Purpose 14fspecial Create 2-D special filters Syntax h = fspecial(type) h = fspecial(type,parameters) Description h = fspecial(type) creates a two-dimensional filter, h, of the specified type. fspecial returns h as a correlation kernel, which is the appropriate form to use with imfilter. type is a string having one of these values: • 'gaussian' for a Gaussian lowpass filter • 'sobel' for a Sobel horizontal edge-emphasizing filter • 'prewitt' for a Prewitt horizontal edge-emphasizing filter • 'laplacian' for a filter approximating the two-dimensional Laplacian operator • 'log' for a Laplacian of Gaussian filter • 'average' for an averaging filter • 'unsharp' for an unsharp contrast enhancement filter h = fspecial(type,parameters) accepts a filter type plus additional modifying parameters particular to the type of filter chosen. If you omit these arguments, fspecial uses default values for the parameters. The following list shows the syntax for each filter type. Where applicable, additional parameters are also shown. • h = fspecial('average',hsize) returns an averaging filter, h, of size hsize. The argument hsize can be a vector specifying the number of rows and columns in h, or it can be a scalar, in which case h is a square matrix. The default value for hsize is [3 3]. • h = fspecial('disk',radius) returns a circular averaging filter (pillbox) within the square matrix of side 2*radius+1. The default radius is 5. • h = fspecial('gaussian',hsize,sigma) returns a rotationally symmetric Gaussian lowpass filter of size hsize with standard deviation sigma (positive). hsize can be a vector specifying the number of rows and columns in h, or it can be a scalar, in which case h is a square matrix. The default value for hsize is [3 3]; the default value for sigma is 0.5. • h = fspecial('laplacian',alpha) returns a 3-by-3 filter approximating the shape of the two-dimensional Laplacian operator. The parameter alpha 14-139 fspecial controls the shape of the Laplacian and must be in the range 0.0 to 1.0. The default value for alpha is 0.2. • h = fspecial('log',hsize,sigma) returns a rotationally symmetric Laplacian of Gaussian filter of size hsize with standard deviation sigma (positive). hsize can be a vector specifying the number of rows and columns in h, or it can be a scalar, in which case h is a square matrix. The default value for hsize is [5 5] and 0.5 for sigma. • h = fspecial('motion',len,theta) returns a filter to approximate, once convolved with an image, the linear motion of a camera by len pixels, with an angle of theta degrees in a counter-clockwise direction. The filter becomes a vector for horizontal and vertical motions. The default len is 9, the default theta is 0, which corresponds to a horizontal motion of 9 pixels. • h = fspecial('prewitt') returns a 3-by-3 filter, h, (shown below) that emphasizes horizontal edges by approximating a vertical gradient. If you need to emphasize vertical edges, transpose the filter, h'. [1 1 1 0 0 0 -1 -1 -1 ] To find vertical edges, or for x-derivatives, use h'. • h = fspecial('sobel') returns a 3-by-3 filter, h, (shown below) that emphasizes horizontal edges using the smoothing effect by approximating a vertical gradient. If you need to emphasize vertical edges, transpose the filter, h'. [1 2 1 0 0 0 -1 -2 -1 ] • h = fspecial('unsharp',alpha) returns a 3-by-3 unsharp contrast enhancement filter. fspecial creates the unsharp filter from the negative of the Laplacian filter with parameter alpha. alpha controls the shape of the Laplacian and must be in the range 0.0 to 1.0. The default value for alpha is 0.2. Class Support Example 14-140 h is of class double. I = imread('saturn.tif'); subplot(2,2,1);imshow(I);title('Original Image'); fspecial H = fspecial('motion',50,45); MotionBlur = imfilter(I,H); subplot(2,2,2);imshow(MotionBlur);title('Motion Blurred Image'); H = fspecial('disk',10); blurred = imfilter(I,H); subplot(2,2,3);imshow(blurred);title('Blurred Image'); H = fspecial('unsharp'); sharpened = imfilter(I,H); subplot(2,2,4);imshow(sharpened);title('Sharpened Image'); Original Image Blurred Image Algorithms Motion Blurred Image Sharpened Image fspecial creates Gaussian filters using 14-141 fspecial h g ( n 1, n 2 ) = e – ( n 1 + n 2 ) ⁄ ( 2σ ) 2 2 2 h g ( n 1, n 2 ) h ( n1, n 2 ) = -------------------------hg ∑∑ n1 n2 fspecial creates Laplacian filters using 2 2 2 ∂ ∂ ∇ = --------- + --------2 2 ∂x ∂y α --4 2 4 1 – α∇ ≈ ------------------ -----------(α + 1) 4 α --4 1–α ------------4 α --4 1 – α-----------4 α --4 –1 1 – α-----------4 fspecial creates Laplacian of Gaussian (LoG) filters using h g ( n 1, n 2 ) = e – ( n 1 + n 2 ) ⁄ ( 2σ ) 2 2 2 2 2 2 ( n 1 + n 2 – 2σ )h g ( n 1, n 2 ) h ( n1, n 2 ) = ----------------------------------------------------------------------6 hg 2πσ ∑∑ n1 n2 fspecial creates averaging filters using ones(n(1),n(2))/(n(1)*n(2)) fspecial creates unsharp filters using –α 1 ----------------(α + 1) α – 1 –α See Also 14-142 α–1 α+5 α–1 –α α–1 –α conv2, edge, filter2, fsamp2, fwind1, fwind2, imfilter fspecial del2 in the MATLAB Function Reference 14-143 ftrans2 Purpose 14ftrans2 Design two-dimensional FIR filter using frequency transformation Syntax h = ftrans2(b,t) h = ftrans2(b) Description h = ftrans2(b,t) produces the two-dimensional FIR filter h that corresponds to the one-dimensional FIR filter b using the transform t. (ftrans2 returns h as a computational molecule, which is the appropriate form to use with filter2.) b must be a one-dimensional, odd-length (Type I) FIR filter such as can be returned by fir1, fir2, or remez in the Signal Processing Toolbox. The transform matrix t contains coefficients that define the frequency transformation to use. If t is m-by-n and b has length Q, then h is size ((m-1)*(Q-1)/2+1)-by-((n-1)*(Q-1)/2+1). h = ftrans2(b) uses the McClellan transform matrix t. t = [1 2 1; 2 -4 2; 1 2 1]/8; Remarks The transformation below defines the frequency response of the two-dimensional filter returned by ftrans2, H ( ω 1, ω 2 ) = B ( ω ) cos ω = T ( ω 1, ω 2 ) where B(ω) is the Fourier transform of the one-dimensional filter b, N B(ω) = ∑ b ( n )e – jωn n = –N and T(ω1,ω2) is the Fourier transform of the transformation matrix t. T ( ω 1, ω 2 ) = ∑ ∑ t ( n1, n2 )e – jω 1 n 1 – jω 2 n 2 e n2 n1 The returned filter h is the inverse Fourier transform of H(ω1,ω2). 1 h ( n1, n 2 ) = -------------2 ( 2π ) 14-144 π π ∫–π ∫–π H ( ω1, ω2 )e jω 1 n 1 jω 2 n 2 e dω 1 dω 2 ftrans2 Example Use ftrans2 to design an approximately circularly symmetric two-dimensional bandpass filter with passband between 0.1 and 0.6 (normalized frequency, where 1.0 corresponds to half the sampling frequency, or π radians): 1 Since ftrans2 transforms a one-dimensional FIR filter to create a two-dimensional filter, first design a one-dimensional FIR bandpass filter using the Signal Processing Toolbox function remez. colormap(jet(64)) b = remez(10,[0 0.05 0.15 0.55 0.65 1],[0 0 1 1 0 0]); [H,w] = freqz(b,1,128,'whole'); plot(w/pi-1,fftshift(abs(H))) 2 Use ftrans2 with the default McClellan transformation to create the desired approximately circularly symmetric filter. h = ftrans2(b); freqz2(h) 14-145 ftrans2 See Also conv2, filter2, fsamp2, fwind1, fwind2 Reference [1] Lim, Jae S. Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1990. pp. 218-237. 14-146 fwind1 Purpose 14fwind1 Design two-dimensional FIR filter using one-dimensional window method Syntax h = fwind1(Hd,win) h = fwind1(Hd,win1,win2) h = fwind1(f1,f2,Hd,...) Description fwind1 designs two-dimensional FIR filters using the window method. fwind1 uses a one-dimensional window specification to design a two-dimensional FIR filter based on the desired frequency response Hd. fwind1 works with one-dimensional windows only; use fwind2 to work with two-dimensional windows. h = fwind1(Hd,win) designs a two-dimensional FIR filter h with frequency response Hd. (fwind1 returns h as a computational molecule, which is the appropriate form to use with filter2.) fwind1 uses the one-dimensional window win to form an approximately circularly symmetric two-dimensional window using Huang’s method. You can specify win using windows from the Signal Processing Toolbox, such as boxcar, hamming, hanning, bartlett, blackman, kaiser, or chebwin. If length(win) is n, then h is n-by-n. Hd is a matrix containing the desired frequency response sampled at equally spaced points between -1.0 and 1.0 (in normalized frequency, where 1.0 corresponds to half the sampling frequency, or π radians) along the x and y frequency axes. For accurate results, use frequency points returned by freqspace to create Hd. (See the entry for freqspace for more information.) h = fwind1(Hd,win1,win2) uses the two one-dimensional windows win1 and win2 to create a separable two-dimensional window. If length(win1) is n and length(win2) is m, then h is m-by-n. h = fwind1(f1,f2,Hd,...) lets you specify the desired frequency response Hd at arbitrary frequencies (f1 and f2) along the x and y axes. The frequency vectors f1 and f2 should be in the range -1.0 to 1.0, where 1.0 corresponds to half the sampling frequency, or π radians. The length of the window(s) controls the size of the resulting filter, as above. Class Support The input matrix Hd can be of class double or of any integer class. All other inputs to fwind1 must be of class double. All outputs are of class double. 14-147 fwind1 Example Use fwind1 to design an approximately circularly symmetric two-dimensional bandpass filter with passband between 0.1 and 0.5 (normalized frequency, where 1.0 corresponds to half the sampling frequency, or π radians): 1 Create a matrix Hd that contains the desired bandpass response. Use freqspace to create the frequency range vectors f1 and f2. [f1,f2] = freqspace(21,'meshgrid'); Hd = ones(21); r = sqrt(f1.^2 + f2.^2); Hd((r<0.1)|(r>0.5)) = 0; colormap(jet(64)) mesh(f1,f2,Hd) 2 Design the filter using a one-dimensional Hamming window. h = fwind1(Hd,hamming(21)); freqz2(h) 14-148 fwind1 Algorithm fwind1 takes a one-dimensional window specification and forms an approximately circularly symmetric two-dimensional window using Huang’s method, w ( n 1, n 2 ) = w ( t ) 2 t= 2 n1 + n 2 where w(t) is the one-dimensional window and w(n1,n2) is the resulting two-dimensional window. Given two windows, fwind1 forms a separable two-dimensional window. w ( n 1, n 2 ) = w 1 ( n 1 )w 2 ( n 2 ) fwind1 calls fwind2 with Hd and the two-dimensional window. fwind2 computes h using an inverse Fourier transform and multiplication by the two-dimensional window. 1 h d ( n 1, n 2 ) = -------------2 ( 2π ) π π ∫–π ∫–π Hd ( ω 1, ω 2 )e jω 1 n 1 jω 2 n 2 e dω 1 dω 2 h ( n1, n 2 ) = h d ( n 1, n 2 )w ( n 1, n 2 ) 14-149 fwind1 See Also conv2, filter2, fsamp2, freqspace, ftrans2, fwind2 Reference [1] Lim, Jae S. Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1990. 14-150 fwind2 Purpose 14fwind2 Design two-dimensional FIR filter using two-dimensional window method Syntax h = fwind2(Hd,win) h = fwind2(f1,f2,Hd,win) Description Use fwind2 to design two-dimensional FIR filters using the window method. fwind2 uses a two-dimensional window specification to design a two-dimensional FIR filter based on the desired frequency response Hd. fwind2 works with two-dimensional windows; use fwind1 to work with one-dimensional windows. h = fwind2(Hd,win) produces the two-dimensional FIR filter h using an inverse Fourier transform of the desired frequency response Hd and multiplication by the window win. Hd is a matrix containing the desired frequency response at equally spaced points in the Cartesian plane. fwind2 returns h as a computational molecule, which is the appropriate form to use with filter2. h is the same size as win. For accurate results, use frequency points returned by freqspace to create Hd. (See the entry for freqspace for more information. h = fwind2(f1,f2,Hd,win) lets you specify the desired frequency response Hd at arbitrary frequencies (f1 and f2) along the x and y axes. The frequency vectors f1 and f2 should be in the range -1.0 to 1.0, where 1.0 corresponds to half the sampling frequency, or π radians. h is the same size as win. Class Support The input matrix Hd can be of class double or of any integer class. All other inputs to fwind2 must be of class double. All outputs are of class double. Example Use fwind2 to design an approximately circularly symmetric two-dimensional bandpass filter with passband between 0.1 and 0.5 (normalized frequency, where 1.0 corresponds to half the sampling frequency, or π radians): 1 Create a matrix Hd that contains the desired bandpass response. Use freqspace to create the frequency range vectors f1 and f2. [f1,f2] = freqspace(21,'meshgrid'); Hd = ones(21); r = sqrt(f1.^2 + f2.^2); Hd((r<0.1)|(r>0.5)) = 0; 14-151 fwind2 colormap(jet(64)) mesh(f1,f2,Hd) 2 Create a two-dimensional Gaussian window using fspecial. win = fspecial('gaussian',21,2); win = win ./ max(win(:)); % Make the maximum window value be 1. mesh(win) 14-152 fwind2 3 Design the filter using the window from step 2. h = fwind2(Hd,win); freqz2(h) 14-153 fwind2 Algorithm fwind2 computes h using an inverse Fourier transform and multiplication by the two-dimensional window win. 1 h d ( n 1, n 2 ) = -------------2 ( 2π ) π π ∫–π ∫–π Hd ( ω 1, ω 2 )e jω 1 n 1 jω 2 n 2 e dω 1 dω 2 h ( n1, n 2 ) = h d ( n 1, n 2 )w ( n 1, n 2 ) See Also conv2, filter2, fsamp2, freqspace, ftrans2, fwind1 Reference [1] Lim, Jae S. Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1990. pp. 202-213. 14-154 getheight Purpose 14getheight Get height of structuring element Syntax H = getheight(SE) Description H = getheight(SE) returns an array the same size as getnhood(SE) containing the height associated with each of the structuring element neighbors. H is all zeros for a flat structuring element. Class Support SE is a STREL object. H is of class double. Example See Also se = strel(ones(3,3),magic(3)); getheight(se) strel, getnhood 14-155 getimage Purpose 14getimage Get image data from axes Syntax A = getimage(h) [x,y,A] = getimage(h) [...,A,flag] = getimage(h) [...] = getimage Description A = getimage(h) returns the first image data contained in the Handle Graphics object h. h can be a figure, axes, image, or texture-mapped surface. A is identical to the image CData; it contains the same values and is of the same class (uint8 or double) as the image CData. If h is not an image or does not contain an image or texture-mapped surface, A is empty. [x,y,A] = getimage(h) returns the image XData in x and the YData in y. XData and YData are two-element vectors that indicate the range of the x-axis and y-axis. [...,A,flag] = getimage(h) returns an integer flag that indicates the type of image h contains. This table summarizes the possible values for flag. Flag Type of Image 0 Not an image; A is returned as an empty matrix 1 Intensity image with values in standard range ([0,1] for double arrays, [0,255] for uint8 arrays, [0,65535] for uint16 arrays) 2 Indexed image 3 Intensity data, but not in standard range 4 RGB image [...] = getimage returns information for the current axes. It is equivalent to [...] = getimage(gca). Class Support 14-156 The output array A is of the same class as the image CData. All other inputs and outputs are of class double. getimage Example This example illustrates obtaining the image data from an image displayed directly from a file. imshow rice.tif I = getimage; 14-157 getneighbors Purpose 14getneighbors Get structuring element neighbor locations and heights Syntax [offsets,heights] = getneighbors(SE) Description [offsets,heights] = getneighbors(SE) returns the relative locations and corresponding heights for each of the neighbors in the structuring element object SE. offsets is a P-by-N array where P is the number of neighbors in the structuring element and N is the dimensionality of the structuring element. Each row of offsets contains the location of the corresponding neighbor, relative to the center of the structuring element. heights is a P-element column vector containing the height of each structuring element neighbor. Class Support Example SE is a STREL object. The return values, offsets and heights, are arrays of double precision values. se = strel([1 0 1],[5 0 -5]) [offsets,heights] = getneighbors(se) se = Nonflat STREL object containing 2 neighbors. Neighborhood: 1 0 Height: 5 0 1 -5 offsets = 0 -1 0 1 heights = 5 -5 See Also 14-158 strel, getnhood, getheight getnhood Purpose Syntax Description 14getnhood Get structuring element neighborhood nhood = getnhood(SE) nhood = getnhood(SE) returns the neighborhood associated with the structuring element SE. Class Support Example See Also SE is a STREL object. nhood is a logical array. se = strel(eye(5)); nhood = getnhood(se) strel, getneighbors 14-159 getsequence Purpose 14getsequence Extract sequence of decomposed structuring elements Syntax SEQ = getsequence(SE) Description SEQ = getsequence(SE), where SE is a structuring element array, returns another structuring element array SEQ containing the individual structuring elements that form the decomposition of SE. SEQ is equivalent to SE, but the elements of SEQ have no decomposition. Class Support SE and SEQ are arrays of STREL objects. Example The strel function uses decomposition for square structuring elements larger than 3-by-3. Use getsequence to extract the decomposed structuring elements. se = strel('square',5) seq = getsequence(se) se = Flat STREL object containing 25 neighbors. Decomposition: 2 STREL objects containing a total of 10 neighbors Neighborhood: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 seq = 2x1 array of STREL objects 1 1 1 1 1 Use imdilate with the 'full' option to see that dilating sequentially with the decomposed structuring elements really does form a 5-by-5 square: imdilate(1,seq,'full') See Also 14-160 imdilate, imerode, strel gray2ind Purpose 14gray2ind Convert an intensity image to an indexed image Syntax [X,map] = gray2ind(I,n) [X,map] = gray2ind(BW,n) Description gray2ind scales, then rounds, an intensity image to produce an equivalent indexed image. [X,map] = gray2ind(I,n) converts the intensity image I to an indexed image X with colormap gray(n). If n is omitted, it defaults to 64. [X,map] = gray2ind(BW,n) converts the binary image, BW, to an indexed image, X, with colormap gray(n). If n is omitted, it defaults to 2. n must be an integer between 1 and 65536. Class Support The input image, I, must be a real, nonsparse array of class logical, uint8, uint16, or double. It can have any dimension. The class of the output image, X, is uint8 if the colormap length is less than or equal to 256; otherwise it is uint16. See Also ind2gray 14-161 grayslice Purpose 14grayslice Create indexed image from intensity image using multilevel thresholding Syntax X = grayslice(I,n) X = grayslice(I,v) Description X = grayslice(I,n) thresholds the intensity image I using cutoff values 1 2 – 1- , returning an indexed image in X. ---, ---, … ,n ------------ n n n X = grayslice(I,v), where v is a vector of values between 0 and 1, thresholds I using the values of v, returning an indexed image in X. You can view the thresholded image using imshow(X,map) with a colormap of appropriate length. Class Support The input image, I, can be of class uint8, uint16, or double. Note that the threshold values are always between 0 and 1, even if I is of class uint8 or uint16. In this case, each threshold value is multiplied by 255 or 65535 to determine the actual threshold to use. The class of the output image, X, depends on the number of threshold values, as specified by n or length(v). If the number of threshold values is less than 256, then X is of class uint8, and the values in X range from 0 to n or length(v). If the number of threshold values is 256 or greater, X is of class double, and the values in X range from 1 to n+1 or length(v)+1. Example See Also 14-162 I = imread('ngc4024m.tif'); X = grayslice(I,16); imshow(I) figure, imshow(X,jet(16)) gray2ind graythresh Purpose 14graythresh Compute global image threshold using Otsu's method Syntax level = graythresh(I) Description level = graythresh(I) computes a global threshold (level) that can be used to convert an intensity image to a binary image with im2bw. level is a normalized intensity value that lies in the range [0, 1]. The graythresh function uses Otsu's method, which chooses the threshold to minimize the intraclass variance of the black and white pixels. Multidimensional arrays are converted automatically to 2-D arrays using reshape. The graythresh function ignores any nonzero imaginary part of I. Class Support The input image, I, can be of class uint8, uint16, or double and it must be nonsparse. The return value, level, is a double scalar. See Also im2bw Reference N. Otsu, “A Threshold Selection Method from Gray-Level Histograms,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 9, no. 1, pp. 62-66, 1979. 14-163 histeq Purpose Syntax 14histeq Enhance contrast using histogram equalization J = histeq(I,hgram) J = histeq(I,n) [J,T] = histeq(I,...) newmap = histeq(X,map,hgram) newmap = histeq(X,map) [newmap,T] = histeq(X,...) Description histeq enhances the contrast of images by transforming the values in an intensity image, or the values in the colormap of an indexed image, so that the histogram of the output image approximately matches a specified histogram. J = histeq(I,hgram) transforms the intensity image I so that the histogram of the output intensity image J with length(hgram) bins approximately matches hgram. The vector hgram should contain integer counts for equally spaced bins with intensity values in the appropriate range: [0, 1] for images of class double, [0, 255] for images of class uint8, and [0, 65535] for images of class uint16. histeq automatically scales hgram so that sum(hgram) = prod(size(I)). The histogram of J will better match hgram when length(hgram) is much smaller than the number of discrete levels in I. J = histeq(I,n) transforms the intensity image I, returning in J an intensity image with n discrete gray levels. A roughly equal number of pixels is mapped to each of the n levels in J, so that the histogram of J is approximately flat. (The histogram of J is flatter when n is much smaller than the number of discrete levels in I.) The default value for n is 64. [J,T] = histeq(I,...) returns the grayscale transformation that maps gray levels in the intensity image I to gray levels in J. newmap = histeq(X,map,hgram) transforms the colormap associated with the indexed image X so that the histogram of the gray component of the indexed image (X,newmap) approximately matches hgram. The histeq function returns the transformed colormap in newmap. length(hgram) must be the same as size(map,1). 14-164 histeq newmap = histeq(X,map) transforms the values in the colormap so that the histogram of the gray component of the indexed image X is approximately flat. It returns the transformed colormap in newmap. [newmap,T] = histeq(X,...) returns the grayscale transformation T that maps the gray component of map to the gray component of newmap. Class Support For syntaxes that include an intensity image I as input, I can be of class uint8, uint16, or double, and the output image J has the same class as I. For syntaxes that include an indexed image X as input, X can be of class uint8 or double; the output colormap is always of class double. Also, the optional output T (the gray level transform) is always of class double. Example Enhance the contrast of an intensity image using histogram equalization. I = imread('tire.tif'); J = histeq(I); imshow(I) figure, imshow(J) Display the resulting histograms. imhist(I,64) figure; imhist(J,64) 14-165 histeq Algorithm When you supply a desired histogram hgram, histeq chooses the grayscale transformation T to minimize c 1 ( T ( k ) ) – c0 ( k ) where c0 is the cumulative histogram of A, c1 is the cumulative sum of hgram for all intensities k. This minimization is subject to the constraints that T must be monotonic and c1(T(a)) cannot overshoot c0(a) by more than half the distance between the histogram counts at a. histeq uses this transformation to map the gray levels in X (or the colormap) to their new values. b = T( a) If you do not specify hgram, histeq creates a flat hgram, hgram = ones(1,n)*prod(size(A))/n; and then applies the previous algorithm. See Also 14-166 brighten, imadjust, imhist hsv2rgb Purpose 14hsv2rgb Convert hue-saturation-value (HSV) values to RGB color space hsv2rgb is a function in MATLAB. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-167 idct2 Purpose Syntax Description 14idct2 Compute two-dimensional inverse discrete cosine transform B = idct2(A) B = idct2(A,m,n) B = idct2(A,[m n]) B = idct2(A) returns the two-dimensional inverse discrete cosine transform (DCT) of A. B = idct2(A,m,n) or B = idct2(A,[m n]) pads A with zeros to size m-by-n before transforming. If [m n] < size(A), idct2 crops A before transforming. For any A, idct2(dct2(A)) equals A to within roundoff error. Class Support The input matrix A can be of class double or of any numeric class. The output matrix B is of class double. Algorithm idct2 computes the two-dimensional inverse DCT using M–1 N–1 A mn = ∑ ∑ π ( 2m + 1 )p π ( 2n + 1 )q α p α q B pq cos ------------------------------- cos ----------------------------- , 2M 2N 0≤m≤M–1 0≤n≤N–1 p=0 q=0 1 ⁄ M, p = 0 αp = 2 ⁄ M, 1 ≤ p ≤ M – 1 1 ⁄ N, q = 0 αq = 2 ⁄ N, 1 ≤ q ≤ N – 1 See Also dct2, dctmtx, fft2, ifft2 References [1] Jain, Anil K. Fundamentals of Digital Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1989. pp. 150-153. [2] Pennebaker, William B., and Joan L. Mitchell. JPEG: Still Image Data Compression Standard. New York: Van Nostrand Reinhold, 1993. 14-168 ifft2 Purpose 14ifft2 Compute two-dimensional inverse fast Fourier transform ifft2 is a function in MATLAB. To get help for this function, select MATLAB Help from the Help menu and view the online function reference pages. 14-169 ifftn Purpose 14ifftn Compute N-dimensional inverse fast Fourier transform ifftn is a function in MATLAB. To get help for this function, select MATLAB Help from the Help menu and view the online function reference pages. 14-170 im2bw Purpose 14im2bw Convert an image to a binary image, based on threshold Syntax BW = im2bw(I,level) BW = im2bw(X,map,level) BW = im2bw(RGB,level) Description im2bw produces binary images from indexed, intensity, or RGB images. To do this, it converts the input image to grayscale format (if it is not already an intensity image), and then converts this grayscale image to binary by thresholding. The output binary image BW has values of 0 (black) for all pixels in the input image with luminance less than level and 1 (white) for all other pixels. (Note that you specify level in the range [0,1], regardless of the class of the input image.) BW = im2bw(I,level) converts the intensity image I to black and white. BW = im2bw(X,map,level) converts the indexed image X with colormap map to black and white. BW = im2bw(RGB,level) converts the RGB image RGB to black and white. Note The function graythresh can be used to compute the level argument automatically. Class Support Example The input image can be of class uint8, uint16, or double and it must be nonsparse. The output image, BW, is of class logical. load trees BW = im2bw(X,map,0.4); imshow(X,map) figure, imshow(BW) 14-171 im2bw See Also 14-172 graythresh, ind2gray, rgb2gray im2col Purpose 14im2col Rearrange image blocks into columns Syntax B = im2col(A,[m n],block_type) B = im2col(A,[m n]) B = im2col(A,'indexed',...) Description B = im2col(A,[m n],block_type) rearranges image blocks into columns. block_type is a string that can have one of these values: • 'distinct' for m-by-n distinct blocks • 'sliding' for m-by-n sliding blocks (default) B = im2col(A,[m n],'distinct') rearranges each distinct m-by-n block in the image A into a column of B. im2col pads A with zeros, if necessary, so its size is an integer multiple of m-by-n. If A = [A11 A12;A21 A22], where each Aij is m-by-n, then B = [A11(:) A12(:) A21(:) A22(:)]. B = im2col(A,[m n],'sliding') converts each sliding m-by-n block of A into a column of B, with no zero padding. B has m*n rows and will contain as many columns as there are m-by-n neighborhoods of A. If the size of A is [mm nn], then the size of B is (m*n)-by-((mm-m+1)*(nn-n+1)). B = im2col(A,[m n]) uses the default block_type of 'sliding'. For the sliding block case, each column of B contains the neighborhoods of A reshaped as nhood(:) where nhood is a matrix containing an m-by-n neighborhood of A. im2col orders the columns of B so that they can be reshaped to form a matrix in the normal way. For example, suppose you use a function, such as sum(B), that returns a scalar for each column of B. You can directly store the result in a matrix of size (mm-m+1)-by-(nn-n+1), using these calls. B = im2col(A,[m n],'sliding'); C = reshape(sum(B),mm-m+1,nn-n+1); B = im2col(A,'indexed',...) processes A as an indexed image, padding with zeros if the class of A is uint8, or ones if the class of A is double. Class Support The input image, A, can be numeric or logical. The output matrix, B, is of the same class as the input image. See Also blkproc, col2im, colfilt, nlfilter 14-173 im2double Purpose 14im2double Convert image array to double precision Syntax I2 = im2double(I1) RGB2 = im2double(RGB1) I = im2double(BW) X2 = im2double(X1,'indexed') Description im2double takes an image as input, and returns an image of class double. If the input image is of class double, the output image is identical to it. If the input image is of class logical, uint8, or uint16, im2double returns the equivalent image of class double, rescaling or offsetting the data as necessary. I2 = im2double(I1) converts the intensity image I1 to double precision, rescaling the data if necessary. RGB2 = im2double(RGB1) converts the truecolor image RGB1 to double precision, rescaling the data if necessary. I = im2double(BW) converts the binary image BW to an intensity image of class double. X2 = im2double(X1,'indexed') converts the indexed image X1 to double precision, offsetting the data if necessary. See Also 14-174 double, im2uint8, uint8 im2java Purpose 14im2java Convert image to Java image im2java is a MATLAB function. To get help for this function, select MATLAB Help from the Help menu and view the online function reference pages. 14-175 im2uint8 Purpose 14im2uint8 Convert image array to eight-bit unsigned integers Syntax I2 = im2uint8(I1) RGB2 = im2uint8(RGB1) I = im2uint8(BW) X2 = im2uint8(X1,'indexed') Description im2uint8 takes an image as input, and returns an image of class uint8. If the input image is of class uint8, the output image is identical to it. If the input image is of class logical, uint16, or double, im2uint8 returns the equivalent image of class uint8, rescaling or offsetting the data as necessary. I2 = im2uint8(I1) converts the intensity image I1 to uint8, rescaling the data if necessary. RGB2 = im2uint8(RGB1) converts the truecolor image RGB1 to uint8, rescaling the data if necessary. I = im2uint8(BW) converts the binary image BW to an intensity image of class uint8. X2 = im2uint8(X1,'indexed') converts the indexed image X1 to uint8, offsetting the data if necessary. Note that it is not always possible to convert an indexed image to uint8. If X1 is of class double, max(X1(:)) must be 256 or less; if X1 is of class uint16, max(X1(:)) must be 255 or less. To convert a uint16 indexed image to uint8 by reducing the number of colors, use imapprox. See Also 14-176 im2uint16, double, im2double, uint8, imapprox, uint16 im2uint16 Purpose 14im2uint16 Convert image array to 16-bit unsigned integers Syntax I2 = im2uint16(I1) RGB2 = im2uint16(RGB1) I = im2uint16(BW) X2 = im2uint16(X1,'indexed') Description im2uint16 takes an image as input, and returns an image of class uint16. If the input image is of class uint16, the output image is identical to it. If the input image is of class double or uint8, im2uint16 returns the equivalent image of class uint16, rescaling or offsetting the data as necessary. I2 = im2uint16(I1) converts the intensity image I1 to uint16, rescaling the data if necessary. RGB2 = im2uint16(RGB1) converts the truecolor image RGB1 to uint16, rescaling the data if necessary. I = im2uint16(BW) converts the binary image BW to an intensity image of class uint16. X2 = im2uint16(X1,'indexed') converts the indexed image X1 to uint16, offsetting the data if necessary. Note that it is not always possible to convert an indexed image to uint16. If X1 is of class double, max(X1(:)) must be 65536 or less. Note im2uint16 does not support binary images. See Also im2uint8, double, im2double, uint8, uint16, imapprox 14-177 imabsdiff Purpose 14imabsdiff Compute absolute difference of two images Syntax Z = imabsdiff(X,Y) Description Z = imabsdiff(X,Y) subtracts each element in array Y from the corresponding element in array X and returns the absolute difference in the corresponding element of the output array Z. X and Y are real, nonsparse numeric arrays with the same class and size. Z has the same class and size as X and Y. If X and Y are integer arrays, elements in the output that exceed the range of the integer type are truncated. If X and Y are double arrays, you can use the expression abs(X-Y) instead of this function. Examples This example calculates the absolute difference between two uint8 arrays. Note that the absolute value prevents negative values from being rounded to zero in the result, as they are with imsubtract. X = uint8([ 255 10 75; 44 225 100]); Y = uint8([ 50 50 50; 50 50 50 ]); Z = imabsdiff(X,Y) Z = 205 6 40 175 25 50 Display the absolute difference between a filtered image and the original. I = imread('cameraman.tif'); J = uint8(filter2(fspecial('gaussian'), I)); K = imabsdiff(I,J); imshow(K,[]) % [] = scale data automatically See also 14-178 imadd, imcomplement, imdivide, imlincomb, immultiply, imsubtract imadd Purpose 14imadd Add two images, or add a constant to an image Syntax Z = imadd(X,Y) Description Z = imadd(X,Y) adds each element in array X with the corresponding element in array Y and returns the sum in the corresponding element of the output array Z. X and Y are real, nonsparse numeric arrays with the same size and class, or Y is a scalar double. The array returned, Z, has the same size and class, or Y is a scalar double. Z has the same size and class as X. If X and Y are integer arrays, elements in the output that exceed the range of the integer type are truncated, and fractional values are rounded. If X and Y are double arrays, you can use the expression X+Y instead of this function. Examples Add two uint8 arrays. Note the truncation that occurs when the values exceed 255. X Y Z Z = uint8([ 255 0 75; 44 225 100]); = uint8([ 50 50 50; 50 50 50 ]); = imadd(X,Y) = 255 94 50 255 125 150 Add two images together and specify an output class. I = imread('rice.tif'); J = imread('cameraman.tif'); K = imadd(I,J,'uint16'); imshow(K,[]) Add a constant to an image. I = imread('rice.tif'); J = imadd(I,50); subplot(1,2,1), imshow(I) subplot(1,2,2), imshow(J) See also imabsdiff, imcomplement, imdivide, imlincomb, immultiply, imsubtract 14-179 imadjust Purpose 14imadjust Adjust image intensity values or colormap Syntax J = imadjust(I,[low_in high_in],[low_out high_out],gamma) newmap = imadjust(map,[low_in high_in],[low_out high_out],gamma) RGB2 = imadjust(RGB1,...) Description J = imadjust(I,[low_in high_in],[low_out high_out],gamma) maps the values in intensity image I to new values in J such that values between low_in and high_in map to values between low_out and high_out. Values below low_in and above high_in are clipped; that is, values below low_in map to low_out, and those above high_in map to high_out. You can use an empty matrix ([]) for [low_in high_in] or for [low_out high_out] to specify the default of [0 1]. gamma specifies the shape of the curve describing the relationship between the values in I and J. If gamma is less than 1, the mapping is weighted toward higher (brighter) output values. If gamma is greater than 1, the mapping is weighted toward lower (darker) output values. If you omit the argument, gamma defaults to 1 (linear mapping). newmap = imadjust(map,[low_in; high_in],[low_out;high_out],gamma) transforms the colormap associated with an indexed image. If low_in, high_in, low_out, high_out, and gamma are scalars, then the same mapping applies to red, green and blue components. Unique mappings for each color component are possible when low_in and high_in are both 1-by-3 vectors low_out and high_out are both 1-by-3 vectors, or gamma is a 1-by-3 vector. The rescaled colormap, newmap, is the same size as map. RGB2 = imadjust(RGB1,...) performs the adjustment on each image plane (red, green, and blue) of the RGB image RGB1. As with the colormap adjustment, you can apply unique mappings to each plane. Note If high_out < low_out, the output image is reversed, as in a photographic negative. 14-180 imadjust The function stretchlim can be used with imadjust to apply an automatically computed contrast stretch. Class Support Example For syntax variations that include an input image (rather than a colormap), the input image can be of class uint8, uint16, or double. The output image has the same class as the input image. For syntax variations that include a colormap, the input and output colormaps are of class double. I = imread('pout.tif'); J = imadjust(I,[0.3 0.7],[]); imshow(I), figure, imshow(J) RGB1 = imread('flowers.tif'); RGB2 = imadjust(RGB1,[.2 .3 0; .6 .7 1],[]); imshow(RGB1), figure, imshow(RGB2) 14-181 imadjust See Also 14-182 brighten, histeq, stretchlim imapprox Purpose 14imapprox Approximate indexed image by one with fewer colors Syntax [Y,newmap] = imapprox(X,map,n) [Y,newmap] = imapprox(X,map,tol) Y = imapprox(X,map,newmap) [...] = imapprox(...,dither_option) Description [Y,newmap] = imapprox(X,map,n) approximates the colors in the indexed image X and associated colormap map by using minimum variance quantization. imapprox returns indexed image Y with colormap newmap, which has at most n colors. [Y,newmap] = imapprox(X,map,tol) approximates the colors in X and map through uniform quantization. newmap contains at most (floor(1/tol)+1)^3 colors. tol must be between 0 and 1.0. Y = imapprox(X,map,newmap) approximates the colors in map by using colormap mapping to find the colors in newmap that best match the colors in map. Y = imapprox(...,dither_option) enables or disables dithering. dither_option is a string that can have one of these values: • 'dither' dithers, if necessary, to achieve better color resolution at the expense of spatial resolution (default). • 'nodither' maps each color in the original image to the closest color in the new map. No dithering is performed. Class Support The input image, X, can be of class uint8, uint16, or double. The output image Y is of class uint8 if the length of newmap is less than or equal to 256. If the length of newmap is greater than 256, X is of class double. Algorithm imapprox uses rgb2ind to create a new colormap that uses fewer colors. See Also cmunique, dither, rgb2ind 14-183 imbothat Purpose 14imbothat Perform bottom-hat filtering Syntax IM2 = imbothat(IM,SE) IM2 = imbothat(IM,NHOOD) Description IM2 = imbothat(IM,SE) performs morphological bottom-hat filtering on the grayscale or binary input image, IM, returning the filtered image, IM2. The argument SE is a structuring element returned by the strel function. SE must be a single structuring element object, not an array containing multiple structuring element objects. IM2 = imbothat(IM,NHOOD) performs morphological bottom hat filtering where NHOOD is an array of 0’s and 1’s that specifies the size and shape of the structuring element. This is equivalent to imbothat(IM,strel(NHOOD)). Class Support IM can be numeric or logical and must be nonsparse. The output image has the same class as the input image. If the input is binary (logical), then the structuring element must be flat. Example Top-hat and bottom-hat filtering can be used together to enhance contrast in an image. 1 Read the image into the MATLAB workspace. I = imread('pout.tif'); imshow(I), title('Original') 14-184 imbothat 2 Add the original image to the top-hat filtered image, and then subtract the bottom-hat filtered image. se = strel('disk',3); J = imsubtract(imadd(I,imtophat(I,se)), imbothat(I,se)); figure, imshow(J), title('Contrast filtered') See Also imtophat, strel 14-185 imclearborder Purpose 14imclearborder Suppress light structures connected to image border Syntax IM2 = imclearborder(IM) IM2 = imclearborder(IM,CONN) Description IM2 = imclearborder(IM) suppresses structures that are lighter than their surroundings and that are connected to the image border. IM can be an intensity or binary image. The output image, IM2, is intensity or binary, respectively. The default connectivity is 8 for two dimensions, 26 for three dimensions, and conndef(ndims(BW),'maximal') for higher dimensions. Note For intensity images, imclearborder tends to reduce the overall intensity level in addition to suppressing border structures. IM2 = imclearborder(IM,CONN) specifies the desired connectivity. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may also be defined in a more general way for any dimension by using for CONN a 3-by-3-by- ... -by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. 14-186 imclearborder Note A pixel on the edge of the input image might not be considered to be a “border” pixel if a nondefault connectivity is specified. For example, if conn = [0 0 0; 1 1 1; 0 0 0], elements on the first and last row are not considered to be border pixels because, according to that connectivity definition, they are not connected to the region outside of the image. Class Support IM can be a numeric or logical array of any dimension, and it must be nonsparse and real. IM2 has the same class as IM. Example The following examples use this simple binary image to illustrate the effect of of imclearborder when you specify different connectivities. BW = 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Using a 4-connected neighborhood, the pixel at (5,2) is not considered connected to the border pixel (4,1), so it is not cleared. BWc1 = imclearborder(BW,4) BWc1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14-187 imclearborder Using an 8-connected neighborhood, pixel (5,2) is considered connected to pixel (4,1) so both are cleared. BWc2 = imclearborder(BW,8) BWc2 = 0 0 0 0 0 0 0 0 0 Algorithm 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 imclearborder uses morphological reconstruction where: • the mask image is the input image • the marker image is zero everywhere except along the border, where it equals the mask image See Also conndef Reference [1] P. Soille, Morphological Image Analysis: Principles and Applications, Springer, 1999, pp. 164-165. 14-188 imclose Purpose 14imclose Close an image Syntax IM2 = imclose(IM,SE) IM2 = imclose(IM,NHOOD) Description IM2 = imclose(IM,SE) performs morphological closing on the grayscale or binary image IM, returning the closed image, IM2. The structuring element, SE, must be a single structuring element object, as opposed to an array of objects. IM2 = imclose(IM,NHOOD) performs closing with the structuring element strel(NHOOD), where NHOOD is an array of 0’s and 1’s that specifies the structuring element neighborhood. Class Support IM can be any numeric or logical class and any dimension, and must be nonsparse. If IM is logical, then SE must be flat. IM2 has the same class as IM. Example This examples closes an image to merge together the small features in the image that are close together. The example then opens the image to remove the isolated white pixels. 1 Read the image into the MATLAB workspace and threshold it. I = imread('pearlite.tif'); bw = ~im2bw(I,graythresh(I)); figure, imshow(bw), title('Step 1: threshold') 14-189 imclose 2 Close the image with a a disk-shaped structuring element. se = strel('disk',6); bw2 = imclose(bw,se); figure, imshow(bw2), title('Step 2: closing') 14-190 imclose 3 Open the image with the same structuring element. bw3 = imopen(bw2,se); figure, imshow(bw3), title('Step 3: opening') 14-191 imclose See Also 14-192 imopen, imdilate, imerode, strel imcomplement Purpose 14imcomplement Complement image Syntax IM2 = imcomplement(IM) Description IM2 = imcomplement(IM) computes the complement of the image IM. IM can be a binary, intensity, or RGB image. IM2 has the same class and size as IM. In the complement of a binary image, zeros becomes ones and ones become zeros; black and white are reversed. In the complement of an intensity or RGB image, each pixel value is subtracted from the maximum pixel value supported by the class (or 1.0 for double-precision images) and the difference is used as the pixel value in the output image. In the output image, dark areas become lighter and light areas become darker. Note If IM is an intensity or RGB image of class double, you can use the expression 1-IM instead of this function. If IM is a binary image, you can use the expression ~IM instead of this function. Examples Create the complement of a uint8 array. X = uint8([ 255 10 75; 44 225 100]); X2 = imcomplement(X) X2 = 0 245 180 211 30 155 Reverse black and white in a binary image. bw = imread('text.tif'); bw2 = imcomplement(bw); subplot(1,2,1),imshow(bw) subplot(1,2,2),imshow(bw2) Create the complement of an intensity image. I = imread('bonemarr.tif'); J = imcomp(I); imshow(I), figure, imshow(J) 14-193 imcomplement See Also 14-194 imabsdiff, imadd, imdivide, imlincomb, immultiply, imsubtract imcontour Purpose 14imcontour Create a contour plot of image data Syntax imcontour(I,n) imcontour(I,v) imcontour(x,y,...) imcontour(...,LineSpec) [C,h] = imcontour(...) Description imcontour(I,n) draws a contour plot of the intensity image I, automatically setting up the axes so their orientation and aspect ratio match the image. n is the number of equally spaced contour levels in the plot; if you omit the argument, the number of levels and the values of the levels are chosen automatically. imcontour(I,v) draws a contour plot of I with contour lines at the data values specified in vector v. The number of contour levels is equal to length(v). imcontour(x,y,...) uses the vectors x and y to specify the x- and y-axis limits. imcontour(...,LineSpec) draws the contours using the line type and color specified by LineSpec. Marker symbols are ignored. [C,h] = imcontour(...) returns the contour matrix C and a vector of handles to the objects in the plot. (The objects are actually patches, and the lines are the edges of the patches.) You can use the clabel function with the contour matrix C to add contour labels to the plot. Class Support Example The input image can be of class uint8, uint16, double, or logical. I = imread('ic.tif'); imcontour(I,3) 14-195 imcontour See Also 14-196 clabel, contour, LineSpec in the MATLAB Function Reference imcrop Purpose Syntax 14imcrop Crop an image I2 = imcrop(I) X2 = imcrop(X,map) RGB2 = imcrop(RGB) I2 = imcrop(I,rect) X2 = imcrop(X,map,rect) RGB2 = imcrop(RGB,rect) [...] = imcrop(x,y,...) [A,rect] = imcrop(...) [x,y,A,rect] = imcrop(...) Description imcrop crops an image to a specified rectangle. In the syntaxes below, imcrop displays the input image and waits for you to specify the crop rectangle with the mouse. I2 = imcrop(I) X2 = imcrop(X,map) RGB2 = imcrop(RGB) If you omit the input arguments, imcrop operates on the image in the current axes. To specify the rectangle: • For a single-button mouse, press the mouse button and drag to define the crop rectangle. Finish by releasing the mouse button. • For a 2- or 3-button mouse, press the left mouse button and drag to define the crop rectangle. Finish by releasing the mouse button. If you hold down the Shift key while dragging, or if you press the right mouse button on a 2- or 3-button mouse, imcrop constrains the bounding rectangle to be a square. When you release the mouse button, imcrop returns the cropped image in the supplied output argument. If you do not supply an output argument, imcrop displays the output image in a new figure. 14-197 imcrop You can also specify the cropping rectangle noninteractively, using these syntaxes I2 = imcrop(I,rect) X2 = imcrop(X,map,rect) RGB2 = imcrop(RGB,rect) rect is a four-element vector with the form [xmin ymin width height]; these values are specified in spatial coordinates. To specify a nondefault spatial coordinate system for the input image, precede the other input arguments with two, two-element vectors specifying the XData and YData. For example, [...] = imcrop(x,y,...) If you supply additional output arguments, imcrop returns information about the selected rectangle and the coordinate system of the input image. For example, [A,rect] = imcrop(...) [x,y,A,rect] = imcrop(...) A is the output image. x and y are the XData and YData of the input image. Class Support The input image A can be of class logical, uint8, uint16, or double. The output image B is of the same class as A. rect is always of class double. Remarks Because rect is specified in terms of spatial coordinates, the width and height elements of rect do not always correspond exactly with the size of the output image. For example, suppose rect is [20 20 40 30], using the default spatial coordinate system. The upper-left corner of the specified rectangle is the center of the pixel (20,20) and the lower-right corner is the center of the pixel (50,60). The resulting output image is 31-by-41, not 30-by-40, because the output image includes all pixels in the input image that are completely or partially enclosed by the rectangle. Example 14-198 I = imread('ic.tif'); I2 = imcrop(I,[60 40 100 90]); imshow(I) figure, imshow(I2) imcrop See Also zoom 14-199 imdilate Purpose 14imdilate Dilate image Syntax IM2 IM2 IM2 IM2 Description IM2 = imdilate(IM,SE) dilates the grayscale, binary, or packed binary image IM, returning the dilated image, IM2. The argument SE is a structuring element object, or array of structuring element objects, returned by the strel function. = = = = imdilate(IM,SE) imdilate(IM,NHOOD) imdilate(IM,SE,PACKOPT) imdilate(...,PADOPT) If IM is logical and the structuring element is flat, imdilate performs binary dilation; otherwise, it performs grayscale dilation. If SE is an array of structuring element objects, imdilate performs multiple dilations of the input image, using each structuring element in SE in succession. IM2 = imdilate(IM,NHOOD) dilates the image IM, where NHOOD is a matrix of 0’s and 1’s that specifies the structuring element neighborhood. This is equivalent to the syntax imdilate(IM,strel(NHOOD)). The imdilate function determines the center element of the neighborhood by floor((size(NHOOD)+1)/2). IM2 = imdilate(IM,SE,PACKOPT) or imdilate(IM,NHOOD,PACKOPT) specifies whether IM is a packed binary image. PACKOPT can have either of the following values. 14-200 'ispacked' IM is treated as a packed binary image as produced by bwpack. IM must be a 2-D uint32 array and SE must be a flat 2-D structuring element. If the value of PACKOPT is 'ispacked', PADOPT must be 'same'. 'notpacked' IM is treated as a normal array. This is the default value. imdilate IM2 = imdilate(...,PADOPT) specifies the size of the output image. PADOPT can have either of the following values. 'same' Make the output image the same size as the input image. This is the default value. If the value of PACKOPT is 'ispacked', PADOPT must be 'same'. 'full' Compute the full dilation. PADOPT is analogous to the optional SHAPE argument to the conv2 and filter2 functions. Class Support IM can be logical or numeric and must be real and nonsparse. It can have any dimension. If IM is logical, SE must be flat. The output has the same class as the input. If the input is packed binary, then the output is also packed binary. Examples This example dilates a binary image with a vertical line structuring element. bw = imread('text.tif'); se = strel('line',11,90); bw2 = imdilate(bw,se); imshow(bw), title('Original') figure, imshow(bw2), title('Dilated') This example dilates a grayscale image with a rolling ball structuring element. I = imread('cameraman.tif'); se = strel('ball',5,5); 14-201 imdilate I2 = imdilate(I,se); imshow(I), title('Original') figure, imshow(I2), title('Dilated') To determine the domain of the composition of two flat structuring elements, dilate the scalar value 1 with both structuring elements in sequence, using the 'full' option. se1 = strel('line',3,0) se1 = Flat STREL object containing 3 neighbors. Neighborhood: 1 1 1 se2 = strel('line',3,90) se2 = Flat STREL object containing 3 neighbors. Neighborhood: 1 1 1 composition = imdilate(1,[se1 se2],'full') composition = 1 1 1 1 1 1 14-202 imdilate 1 Algorithm 1 1 imdilate automatically takes advantage of the decomposition of a structuring element object (if it exists). Also, when performing binary dilation with a structuring element object that has a decomposition, imdilate automatically uses binary image packing to speed up the dilation. Dilation using bit packing is described in [2]. See Also bwpack, bwunpack, conv2, filter2, imclose, imerode, imopen, strel References [1] Robert M. Haralick and Linda G. Shapiro, Computer and Robot Vision, vol. I, Addison-Wesley, 1992, pp. 158-205. [2] van den Boomgaard and van Balen, “Image Transforms Using Bitmapped Binary Images,” Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing, vol. 54, no. 3, May, 1992, pp. 254-258. 14-203 imdivide Purpose 14imdivide Divide one image into another, or divide an image by a constant Syntax Z = imdivide(X,Y) Description Z = imdivide(X,Y) divides each element in the array X by the corresponding element in array Y and returns the result in the corresponding element of the output array Z. X and Y are real, nonsparse numeric arrays with the same size and class, or Y can be a scalar double. Z has the same size and class as X and Y. If X is an integer array, elements in the output that exceed the range of integer type are truncated, and fractional values are rounded. If X and Y are double arrays, you can use the expression X./Y instead of this function. Example Divide two uint8 arrays. Note that fractional values greater than or equal to 0.5 are rounded up to the nearest integer. X Y Z Z = uint8([ 255 10 75; 44 225 100]); = uint8([ 50 20 50; 50 50 50 ]); = imdivide(X,Y) = 5 1 2 1 5 2 Estimate and divide out the background of the rice image. I = imread('rice.tif'); blocks = blkproc(I,[32 32],'min(x(:))'); background = imresize(blocks,[256 256],'bilinear'); Ip = imdivide(I,background); imshow(Ip,[]) % [] = let imshow scale data automatically Divide an image by a constant factor. I = imread('rice.tif'); J = imdivide(I,2); subplot(1,2,1), imshow(I) subplot(1,2,2), imshow(J) See Also 14-204 imabsdiff, imadd, imcomplement, imlincomb, immultiply, imsubtract imerode Purpose 14imerode Erode image Syntax IM2 IM2 IM2 IM2 Description IM2 = imerode(IM,SE) erodes the grayscale, binary, or packed binary image IM, returning the eroded image, IM2. The argument SE is a structuring element object, or array of structuring element objects, returned by the strel function. = = = = imerode(IM,SE) imerode(IM,NHOOD) imerode(IM,SE,PACKOPT,M) imerode(...,PADOPT) If IM is logical and the structuring element is flat, imerode performs binary dilation; otherwise it performs grayscale erosion. If SE is an array of structuring element objects, imerode performs multiple erosions of the input image, using each structuring element in SE in succession. IM2 = imerode(IM,NHOOD) erodes the image IM, where NHOOD is an array of 0’s and 1’s that specifies the structuring element neighborhood. This is equivalent to the syntax imerode(IM,strel(NHOOD)). The imerode function determines the center element of the neighborhood by floor((size(NHOOD)+1)/2) IM2 = imerode(IM,SE,PACKOPT,M) or imerode(IM,NHOOD,PACKOPT,M) specifies whether IM is a packed binary image and, if it is, provides the row dimension, M, of the original unpacked image. PACKOPT can have either of the following values. 'ispacked' IM is treated as a packed binary image as produced by bwpack. IM must be a 2-D uint32 array and SE must be a flat 2-D structuring element. 'notpacked' IM is treated as a normal array. This is the default value. If PACKOPT is 'ispacked', you must specify a value for M. 14-205 imerode IM2 = imerode(...,PADOPT) specifies the size of the output image. PADOPT can have either of the following values. 'same' Make the output image the same size as the input image. This is the default value. If the value of PACKOPT is 'ispacked', PADOPT must be 'same'. 'full' Compute the full erosion. PADOPT is analogous to the SHAPE input to the CONV2 and FILTER2 functions. Class Support IM can be numeric or logical and it can be of any dimension. If IM is logical, SE must be flat. The output has the same class as the input. If the input is packed binary, then the output is also packed binary. Examples This example erodes a binary image with a vertical line. bw = imread('text.tif'); se = strel('line',11,90); bw2 = imerode(bw,se); imshow(bw), title('Original') figure, imshow(bw2), title('Eroded') This example erodes a grayscale image with a rolling ball. I = imread('cameraman.tif'); se = strel('ball',5,5); I2 = imerode(I,se); 14-206 imerode imshow(I), title('Original') figure, imshow(I2), title('Eroded') Algorithm Notes imerode automatically takes advantage of the decomposition of a structuring element object (if a decomposition exists). Also, when performing binary dilation with a structuring element object that has a decomposition, imerode automatically uses binary image packing to speed up the dilation. Erosion using bit packing is described in [2]. See Also bwpack, bwunpack, conv2, filter2, imclose, imdilate, imopen, strel References [1] Robert M. Haralick and Linda G. Shapiro, Computer and Robot Vision, vol. I, Addison-Wesley, 1992, pp. 158-205. [2] van den Boomgaard and van Balen, “Image Transforms Using Bitmapped Binary Images,” Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing, vol. 54, no. 3, May, 1992, pp. 254-258. 14-207 imextendedmax Purpose 14imextendedmax Extended-maxima transform Syntax BW = imextendedmax(I,H) Description BW = imextendedmax(I,H) computes the extended-maxima transform, which is the regional maxima of the H-maxima transform. H is a nonnegative scalar. Regional maxima are connected components of pixels with the same intensity value, t, whose external boundary pixels all have a value less than t. By default, imextendedmax uses 8-connected neighborhoods for 2-D images and 26-connected neighborhoods for 3-D images. For higher dimensions, imextendedmax uses conndef(ndims(I),'maximal'). BW = imextendedmax(I,H,CONN) computes the extended-maxima transform, where CONN specifies the connectivity. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may be defined in a more general way for any dimension by using for CONN a 3-by-3-by- ...-by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. Class Support 14-208 I can be of any nonsparse numeric class and any dimension. BW has the same size as I and is always logical. imextendedmax Example I = imread('bonemarr.tif'); BW = imextendedmax(I,40); imshow(I), figure, imshow(BW) See Also conndef, imextendedmin, imreconstruct Reference [1] Pierre Soille, Morphological Image Analysis: Principles and Applications, Springer-Verlag, 1999, pp. 170-171. 14-209 imextendedmin Purpose 14imextendedmin Extended-minima transform Syntax BW = imextendedmin(I,h) Description BW = imextendedmin(I,h) computes the extended-minima transform, which is the regional minima of the H-minima transform. h is a nonnegative scalar. Regional minima are connected components of pixels with the same intensity value, t, whose external boundary pixels all have a value greater than t. By default, imextendedmin uses 8-connected neighborhoods for 2-D images, and 26-connected neighborhoods for 3-D images. For higher dimensions, imextendedmin uses conndef(ndims(I),'maximal'). BW = imextendedmin(I,h,CONN) computes the extended-minima transform, where CONN specifies the connectivity. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may be defined in a more general way for any dimension by using for CONN a 3-by-3-by-...-by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. Class Support 14-210 I can be of any nonsparse numeric class and any dimension. BW has the same size as I and is always logical. imextendedmin Example I = imread('bonemarr.tif'); BW = imextendedmin(I,10); imshow(I), figure, imshow(BW) See Also conndef, imextendedmax, imreconstruct Reference [1] Pierre Soille, Morphological Image Analysis: Principles and Applications, Springer-Verlag, 1999, pp. 170-171. 14-211 imfeature Purpose 14imfeature Compute feature measurements for image regions Note This function is obsolete and may be removed in future versions. Use regionprops instead. Syntax stats = imfeature(L,measurements) stats = imfeature(L,measurements,n) Description stats = imfeature(L,measurements) computes a set of measurements for each labeled region in the label matrix L. Positive integer elements of L correspond to different regions. For example, the set of elements of L equal to 1 corresponds to region 1; the set of elements of L equal to 2 corresponds to region 2; and so on. stats is a structure array of length max(L(:)). The fields of the structure array denote different measurements for each region, as specified by measurements. measurements can be a comma-separated list of strings, a cell array containing strings, the single string 'all', or the single string 'basic'. The set of valid measurement strings includes the following. 'Area' 'Image' 'EulerNumber' 'Centroid' 'FilledImage' 'Extrema' 'BoundingBox' 'FilledArea' 'EquivDiameter' 'MajorAxisLength' 'ConvexHull' 'Solidity' 'MinorAxisLength' 'ConvexImage' 'Extent' 'Eccentricity' 'ConvexArea' 'PixelList' 'Orientation' Measurement strings are case insensitive and can be abbreviated. If measurements is the string 'all', then all of the above measurements are computed. If measurements is not specified or if it is the string 'basic', then these measurements are computed: 'Area', 'Centroid', and 'BoundingBox'. 14-212 imfeature stats = imfeature(L,measurements,n) specifies the type of connectivity used in computing the 'FilledImage', 'FilledArea', and 'EulerNumber' measurements. n can have a value of either 4 or 8, where 4 specifies 4-connected objects and 8 specifies 8-connected objects; if the argument is omitted, it defaults to 8. Definitions 'Area' – Scalar; the actual number of pixels in the region. (This value may differ slightly from the value returned by bwarea, which weights different patterns of pixels differently.) 'Centroid' – 1-by-2 vector; the x- and y-coordinates of the center of mass of the region. 'BoundingBox' – 1-by-4 vector; the smallest rectangle that can contain the region. The format of the vector is [x y width height], where x and y are the x- and y-coordinates of the upper-left corner of the rectangle, and width and height are the width and height of the rectangle. Note that x and y are always noninteger values, because they are the spatial coordinates for the upper-left corner of a pixel in the image; for example, if this pixel is the third pixel in the fifth row of the image, then x = 2.5 and y = 4.5. This figure illustrates the centroid and bounding box. The region consists of the white pixels; the green box is the bounding box, and the red dot is the centroid. 'MajorAxisLength' – Scalar; the length (in pixels) of the major axis of the ellipse that has the same second-moments as the region. 'MinorAxisLength' – Scalar; the length (in pixels) of the minor axis of the ellipse that has the same second-moments as the region. 'Eccentricity' – Scalar; the eccentricity of the ellipse that has the same second-moments as the region. The eccentricity is the ratio of the distance between the foci of the ellipse and its major axis length. The value is between 0 and 1. (0 and 1 are degenerate cases; an ellipse whose eccentricity is 0 is actually a circle, while an ellipse whose eccentricity is 1 is a line segment.) 14-213 imfeature 'Orientation' – Scalar; the angle (in degrees) between the x-axis and the major axis of the ellipse that has the same second-moments as the region. This figure illustrates the axes and orientation of the ellipse. The left side of the figure shows an image region and its corresponding ellipse. The right side shows the same ellipse, with features indicated graphically; the solid blue lines are the axes, the red dots are the foci, and the orientation is the angle between the horizontal dotted line and the major axis. 'Image' – Binary image of the same size as the bounding box of the region; the on pixels correspond to the region, and all other pixels are off. 'FilledImage' – Binary image of the same size as the bounding box of the region; the on pixels correspond to the region, with all holes filled in. 'FilledArea' – Scalar; the number of on pixels in FilledImage. This figure illustrates Image' and FilledImage'. Original image, containing a single region 14-214 'Image' 'FilledImage' imfeature 'ConvexHull' – p-by-2 matrix; the smallest convex polygon that can contain the region. Each row of the matrix contains the x- and y-coordinates of one vertex of the polygon. 'ConvexImage' – Binary image (uint8); the convex hull, with all pixels within the hull filled in (i.e., set to on). (For pixels that the boundary of the hull passes through, imfeature uses the same logic as roipoly to determine whether the pixel is inside or outside the hull.) The image is the size of the bounding box of the region. 'ConvexArea' – Scalar; the number of pixels in 'ConvexImage'. 'EulerNumber' – Scalar; equal to the number of objects in the region minus the number of holes in those objects. 'Extrema' – 8-by-2 matrix; the extremal points in the region. Each row of the matrix contains the x- and y-coordinates of one of the points; the format of the vector is [top-left top-right right-top right-bottom bottom-right bottom-left left-bottom left-top]. This figure illustrates the extrema of two different regions. In the region on the left, each extremal point is distinct; in the region on the right, certain extremal points (e.g., top-left and left-top) are identical. top-left top-right top-left top-right left-top right-top left-top right-top left-bottom right-bottom left-bottom right-bottom bottom-left bottom-right bottom-lef t bottom-right 'EquivDiameter' – Scalar; the diameter of a circle with the same area as the region. Computed as sqrt(4*Area/pi). 'Solidity' – Scalar; the proportion of the pixels in the convex hull that are also in the region. Computed as Area/ConvexArea. 'Extent' – Scalar; the proportion of the pixels in the bounding box that are also in the region. Computed as the Area divided by area of the bounding box. 14-215 imfeature 'PixelList' – p-by-2 matrix; the actual pixels in the region. Each row of the matrix contains the x- and y-coordinates of one pixel in the region. Class Support The input label matrix L can be of class double or of any integer class. Remarks The comma-separated list syntax for structure arrays is very useful when working with the output of imfeature. For example, for a field that contains a scalar, you can use a this syntax to create a vector containing the value of this field for each region in the image. For instance, if stats is a structure array with field Area, then these two expressions are equivalent stats(1).Area, stats(2).Area, ..., stats(end).Area and stats.Area Therefore, you can use these calls to create a vector containing the area of each region in the image. stats = imfeature(L,'Area'); allArea = [stats.Area]; allArea is a vector of the same length as the structure array stats. The function ismember is useful in conjunction with imfeature for selecting regions based on certain criteria. For example, these commands create a binary image containing only the regions in text.tif whose area is greater than 80. idx = find([stats.Area] > 80); BW2 = ismember(L,idx); Most of the measurements take very little time to compute. The exceptions are these, which may take significantly longer, depending on the number of regions in L: • 'ConvexHull' • 'ConvexImage' • 'ConvexArea' • 'FilledImage' 14-216 imfeature Note that computing certain groups of measurements takes about the same amount of time as computing just one of them, because imfeature takes advantage of intermediate computations used in both computations. Therefore, it is fastest to compute all of the desired measurements in a single call to imfeature. Example BW = imread('text.tif'); L = bwlabel(BW); stats = imfeature(L,'all'); stats(23) ans = Area: Centroid: BoundingBox: MajorAxisLength: MinorAxisLength: Eccentricity: Orientation: ConvexHull: ConvexImage: ConvexArea: Image: FilledImage: FilledArea: EulerNumber: Extrema: EquivDiameter: Solidity: Extent: PixelList: See Also 89 [95.6742 192.9775] [87.5000 184.5000 16 15] 19.9127 14.2953 0.6961 9.0845 [13x2 double] [15x16 logical ] 205 [15x16 logical ] [15x16 logical ] 122 0 [ 8x2 double] 10.6451 0.4341 0.3708 [89x2 double] bwlabel ismember in the MATLAB Function Reference 14-217 imfill Purpose 14imfill Fill image regions Syntax BW2 = imfill(BW) [BW2,LOCATIONS] = imfill(BW) BW2 = imfill(BW,LOCATIONS) BW2 = imfill(BW,LOCATIONS,CONN) BW2 = imfill(BW,'holes') BW2 = imfill(BW,CONN,'holes') I2 = imfill(I,'holes') I2 = imfill(I,CONN,'holes') Description BW2 = imfill(BW,LOCATIONS) performs a flood-fill operation on background pixels of the input binary image BW, starting from the points specified in LOCATIONS. LOCATIONS can be a P-by-1 vector, in which case it contains the linear indices of the starting locations. LOCATIONS can also be a P-by-ndims(BW) matrix, in which case each row contains the array indices of one of the starting locations. BW2 = imfill(BW,'holes') fills holes in the input binary image. A hole is a set of background pixels that cannot be reached by filling in the background from the edge of the image. I2 = imfill(I,'holes') fills holes in an input intensity image, I. In this case, a hole is an area of dark pixels surrounded by lighter pixels. Interactive Use BW2 = imfill(BW) displays the binary image, BW, on the screen and lets you select the starting locations using the mouse. Click the mouse button to add points. Press <BackSpace> or <Delete> to remove the previously selected point. A shift-click, right-click, or double-click selects a final point and then starts the fill operation; pressing <Return> finishes the selection without adding a point. Interactive use is supported only for 2-D images. The syntax [BW2,LOCATIONS] = imfill(BW) can be used to get the starting points selected using the mouse. The output LOCATIONS is a vector of linear indices into the input image. Specifying Connectivity 14-218 By default, imfill uses 4-connected background neighbors for 2-D inputs and 6-connected background neighbors for 3-D inputs. For higher dimensions the imfill default background connectivity is determined by using conndef(NUM_DIMS,'minimal'). You can override the default connectivity with these syntaxes: BW2 = imfill(BW,LOCATIONS,CONN) BW2 = imfill(BW,CONN,'holes') I2 = imfill(I,CONN,'holes') To override the default connectivity and interactively specify the starting locations, use this syntax: BW2 = imfill(BW,0,CONN) CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may be defined in a more general way for any dimension by using for CONN a 3-by-3-by- ... -by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. Class Support The input image can be numeric or logical, and it must be real and nonsparse. It can have any dimension. The output image has the same class as the input image. Examples Fill in the background of a binary image from a specified starting location: 14-219 imfill BW1 = [1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0] BW2 = imfill(BW1,[3 3],8) Fill in the holes of a binary image: BW4 = ~im2bw(imread('blood1.tif')); BW5 = imfill(BW4,'holes'); imshow(BW4), figure, imshow(BW5) Fill in the holes of an intensity image: I = imread('enamel.tif'); I2 = imcomplement(imfill(imcomplement(I),'holes')); imshow(I), figure, imshow(I2) Algorithm imfill uses an algorithm based on morphological reconstruction [1]. See Also bwselect, imreconstruct, roifill Reference [1] Pierre Soille, Morphological Image Analysis: Principles and Applications, Springer-Verlag, 1999, pp. 173-174. 14-220 imfilter Purpose 14imfilter Multidimensional image filtering Syntax B = imfilter(A,H) B = imfilter(A,H,option1,option2,...) Description B = imfilter(A,H) filters the multidimensional array A with the multidimensional filter H. The array, A, can be a nonsparse numeric array of any class and dimension. The result, B, has the same size and class as A. Each element of the output, B, is computed using double-precision floating point. If A is an integer array, then output elements that exceed the range of the integer type are truncated, and fractional values are rounded. B = imfilter(A,H,option1,option2,...) performs multidimensional filtering according to the specified options. Option arguments can have the following values. Boundary Options Option Description X Input array values outside the bounds of the array are implicitly assumed to have the value X. When no boundary option is specified, imfilter uses X = 0. 'symmetric' Input array values outside the bounds of the array are computed by mirror-reflecting the array across the array border. 'replicate' Input array values outside the bounds of the array are assumed to equal the nearest array border value. 'circular' Input array values outside the bounds of the array are computed by implicitly assuming the input array is periodic. 14-221 imfilter Output Size Options Option Description 'same' The output array is the same size as the input array. This is the default behavior when no output size options are specified. 'full' The output array is the full filtered result, and so is larger than the input array. Correlation and Convolution Options Option Description 'corr' imfilter performs multidimensional filtering using correlation, which is the same way that filter2 performs filtering. When no correlation or convolution option is specified, imfilter uses correlation. 'conv' imfilter performs multidimensional filtering using convolution. N-D convolution is related to N-D correlation by a reflection of the filter matrix. Examples This example starts by reading an image of a bouquet of flowers into a three-dimensional array of uint8 values, called rgb. The imshow function displays the original image, entitled 'Original'. rgb = imread('flowers.tif'); imshow(rgb), title('Original') Create a filter, h, that can be used to approximate linear camera motion. Use imfilter on the three-dimensional RGB image, rgb, to create a new image, rgb2. h = fspecial('motion', 50, 45); rgb2 = imfilter(rgb, h); figure, imshow(rgb2), title('Filtered') 14-222 imfilter Note that imfilter is more memory efficient than some other filtering operations in that it outputs an array of the same data type as the input image array. In this example, the output is an array of uint8. whos rgb2 Name rgb2 Size 362x500x3 Bytes 543000 Class uint8 array Filter the image once more, this time specifying the replicate boundary option and observe that the picture now extends to the border. rgb3 = imfilter(rgb, h, 'replicate'); figure, imshow(rgb3), title('Filtered with boundary replication') See Also conv2, convn, filter2 14-223 imfinfo Purpose 14imfinfo Information about graphics file imfinfo is a MATLAB function. To get help for this function, select MATLAB Help from the Help menu and view the online function reference pages. 14-224 imhist Purpose 14imhist Display a histogram of image data Syntax imhist(I,n) imhist(X,map) [counts,x] = imhist(...) Description imhist(I,n) displays a histogram with n bins for the intensity image I above a grayscale colorbar of length n. If you omit the argument, imhist uses a default value of n = 256 if I is a grayscale image, or n = 2 if I is a binary image. imhist(X,map) displays a histogram for the indexed image X. This histogram shows the distribution of pixel values above a colorbar of the colormap map. The colormap must be at least as long as the largest index in X. The histogram has one bin for each entry in the colormap. [counts,x] = imhist(...) returns the histogram counts in counts and the bin locations in x so that stem(x,counts) shows the histogram. For indexed images, it returns the histogram counts for each colormap entry; the length of counts is the same as the length of the colormap. Note For intensity images, the n bins of the histogram are each half-open intervals of width A ⁄ ( n – 1 ) . In particular, the p th bin is the half-open interval A ( p – 1.5 ) ⁄ ( n – 1 ) ≤ x < A ( p – 0.5 ) ⁄ ( n – 1 ) The scale factor A depends on the image class. A is 1 if the intensity image is double; A is 255 if the intensity image is uint8; and A is 65535 if the intensity image is uint16. Class Support Example The input image can be of class logical, uint8, uint16, or double. I = imread('pout.tif'); imhist(I) 14-225 imhist See Also histeq hist is in the MATLAB Function Reference 14-226 imhmax Purpose 14imhmax H-maxima transform Syntax I2 = imhmax(I,h) Description I2 = imhmax(I,h) suppresses all maxima in the intensity image, I, whose height is less than h, where h is a scalar. Regional maxima are connected components of pixels with the same intensity value, t, whose external boundary pixels all have a value less than t. By default, imhmax uses 8-connected neighborhoods for 2-D images, and 26-connected neighborhoods for 3-D images. For higher dimensions, imhmax uses conndef(ndims(I),'maximal'). I2 = imhmax(I,h,CONN) computes the H-maxima transform, where CONN specifies the connectivity. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may be defined in a more general way for any dimension by using for CONN a 3-by-3-by- ...-by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. Class Support I can be of any nonsparse numeric class and any dimension. I2 has the same size and class as I. 14-227 imhmax Example a = zeros(10,10); a(2:4,2:4) = 3; % maxima 3 higher than surround a(6:8,6:8) = 8; % maxima 8 higher than surround b = imhmax(a,4); % only the maxima higher than 4 survive. See Also conndef, imhmin, imreconstruct Reference [1] Pierre Soille, Morphological Image Analysis: Principles and Applications, Springer-Verlag, 1999, pp. 170-171. 14-228 imhmin Purpose 14imhmin H-minima transform Syntax I2 = imhmin(I,h) Description I2 = imhmin(I,h) suppresses all minima in the intensity image, I, whose depth is less than h, where h is a scalar. Regional minima are connected components of pixels with the same intensity value, t, whose external boundary pixels all have a value greater than t. By default, imhmin uses 8-connected neighborhoods for 2-D images, and 26-connected neighborhoods for 3-D images. For higher dimensions, imhmin uses conndef(ndims(I),'maximal'). I2 = imhmin(I,h,CONN) computes the H-minima transform, where CONN specifies the connectivity. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may be defined in a more general way for any dimension by using for CONN a 3-by-3-by- ...-by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. Class Support I can be of any nonsparse numeric class and any dimension. I2 has the same size and class as I. 14-229 imhmin Example a = 10*ones(10,10); a(2:4,2:4) = 7; % maxima 3 lower than surround a(6:8,6:8) = 2; % maxima 8 lower than surround b = imhmin(a,4); % only the minima lower than 4 survive. See Also conndef, imhmax, imreconstruct Reference [1] Pierre Soille, Morphological Image Analysis: Principles and Applications, Springer-Verlag, 1999, pp. 170-171. 14-230 imimposemin Purpose 14imimposemin Impose minima Syntax I2 = imimposemin(I,BW) Description I2 = imimposemin(I,BW) modifies the intensity image I using morphological reconstruction so it only has regional minima wherever BW is nonzero. BW is a binary image the same size as I. By default, imimposemin uses 8-connected neighborhoods for 2-D images and 26-connected neighborhoods for 3-D images. For higher dimensions, imimposemin uses conndef(ndims(I),'mimimum'). I2 = imimposemin(I,H,CONN) specifies the connectivity, where CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may also be defined in a more general way for any dimension by using for CONN a 3-by-3-by-...-by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. Class Support I can be of any nonsparse numeric class and any dimension. BW must be a nonsparse numeric array with the same size as I. I2 has the same size and class as I. 14-231 imimposemin Example Modify the image in bonemarr.tif so that is only has regional minima at one location. 1 First read in and display the image to be processed, called the mask image. I = imread('bonemarr.tif'); imshow(I) 2 Create the marker image that will be used to process the mask image. The example creates a binary image the same size as the mask image, filled with zeros. It then sets a small area of the binary image to 1; these pixels define the location of the minima in the mask image. bw = zeros(size(I)); bw(98:102,101:105) = 1; To show where these pixels of interest fall on the original image, this code superimposes the marker over the mask. The small white square marks the spot. This code is not essential to the impose minima operation. J = I; J(bw ~= 0) = 255; figure, imshow(J) 14-232 imimposemin 3 Impose the minina on the input image by performing morphological reconstruction of the mask image with the marker image. Note how all the dark areas of the original image, except those marked, are lighter. K = imimposemin(I,bw); figure, imshow(K) To illustrate how this operation removes all minima in the original image except the imposed minima, the example uses imregionalmin to find all regional minima in the image. bw2 = imregionalmin(K); figure, imshow(bw2) 14-233 imimposemin Algorithm imimposemin uses a technique based on morphological reconstruction. See Also conndef, imreconstruct, imregionalmin 14-234 imlincomb Purpose 14imlincomb Compute linear combination of images Syntax Z = imlincomb(K1,A1,K2,A2,...,Kn,An) Z = imlincomb(K1,A1,K2,A2,...,Kn,An,K) Z = imlincomb(..., output_class) Description Z = imlincomb(K1,A1,K2,A2,...,Kn,An) computes: K1*A1 + K2*A2 + ... + Kn*An where K1, K2, through Kn are real, double scalars and A1, A2, through An are real, nonsparse, numeric arrays with the same class and size. Z has the same class and size as A1. Z = imlincomb(K1,A1,K2,A2,...,Kn,An,K) computes: K1*A1 + K2*A2 + ... + Kn*An + K where imlincomb adds K, a real, double scalar, to the sum of the products of K1 through Kn and A1 through An. Z = imlincomb(...,output_class) lets you specify the class of Z. output_class is a string containing the name of a numeric class. Note When performing a series of arithmetic operations on a pair of images, you can achieve more accurate results if you use imlincomb to combine the operations, rather than nesting calls to the individual arithmetic functions, such as imadd. When you nest calls to the arithmetic functions, and the input arrays are of an integer class, each function truncates and rounds the result before passing it to the next function, thus losing accuracy in the final result. imlincomb computes each element of the output, Z, individually in double-precision, floating point. If Z is an integer array, imlincomb truncates elements of Z that exceed the range of the integer type, and rounds off fractional values. Example Scale an image by a factor of two. I = imread('cameraman.tif'); J = imlincomb(2,I); 14-235 imlincomb imshow(J) Form a difference image with the zero value shifted to 128. I = imread('cameraman.tif'); J = uint8(filter2(fspecial('gaussian'), I)); K = imlincomb(1,I,-1,J,128); % K(r,c) = I(r,c) - J(r,c) + 128 imshow(K) To illustrate how imlincomb performs all the arithmetic operations before truncating the result, compare the results of calculating the average of two arrays, X and Y, using nested arithmetic functions and then using imlincomb. Consider the values in row 1, column 1: 255 in the X and 50 in Y. In the version that uses nested arithmetic functions, imadd adds 255 and 50 and truncates the result to 255 before passing it to imdivide. The average returned in Z(1,1) is 128. X Y Z Z = uint8([ 255 10 75; 44 225 100]); = uint8([ 50 20 50; 50 50 50 ]); = imdivide(imadd(X,Y),2) = 128 15 63 47 128 75 imlincomb performs the addition and division in double precision and only truncates the final result. The average returned in Z2(1,1) is 153. Z2 = imlincomb(.5,X,.5,Y) Z2 = 153 15 63 47 138 75 Form a difference image with the zero value shifted to 128. I = imread('cameraman.tif'); J = uint8(filter2(fspecial('gaussian'), I)); K = imlincomb(1,I,-1,J,128); % K(r,c) = I(r,c) - J(r,c) + 128 imshow(K) See Also 14-236 imadd, imcomplement, imdivide, immultiply, imsubtract immovie Purpose 14immovie Make a movie of a multiframe indexed image Syntax mov = immovie(X,map) mov = immovie(RGB) Description mov = immovie(X,map) returns the movie structure array, mov, from the images in the multiframe indexed image X with the colormap map. As it creates the movie array, it displays the movie frames on the screen. You can play the movie using the MATLAB movie function. For details about the movie structure array, see the reference page for getframe. X comprises multiple indexed images, all having the same size and all using the colormap map. X is an m-by-n-by-1-by-k array, where k is the number of images. mov = immovie(RGB) returns the movie structure array mov from the images in the multiframe, truecolor image RGB. RGB comprises multiple truecolor images, all having the same size. RGB is an m-by-n-by-3-by-k array, where k is the number of images. Remarks You can also use the MATLAB function avifile to make movies from images. The avifile function creates AVI files. In addition, you can convert an existing MATLAB movie into an AVI file by using the movie2avi function. Class Support An indexed image can be uint8, uint16, double, or logical. A truecolor image can be uint8, uint16, or double. mov is a MATLAB movie structure. Example See Also load mri mov = immovie(D,map); movie(mov,3) avifile, getframe, montage, movie, movie2avi 14-237 immultiply Purpose 14immultiply Multiply two images, or multiply an image by a constant Syntax Z = immultiply(X,Y) Description Z = immultiply(X,Y) multiplies each element in array X by the corresponding element in array Y and returns the product in the corresponding element of the output array Z. If X and Y are real numeric arrays with the same size and class, then Z has the same size and class as X. If X is a numeric array and Y is a scalar double, then Z has the same size and class as X. If X is logical and Y is numeric, then Z has the same size and class as Y. If X is numeric and Y is logical, then Z has the same size and class as X. immultiply computes each element of Z individually in double-precision floating point. If X is an integer array, then elements of Z exceeding the range of the integer type are truncated, and fractional values are rounded. If X and Y are double arrays, you can use the expression X.*Y instead of this function. Example Multiply an image by itself. Note how the example converts the class of the image from uint8 to uint16 before performing the multiplication to avoid truncating the results. I = imread('moon.tif'); I16 = uint16(I); J = immultiply(I16,I16); imshow(I), figure, imshow(J) Scale an image by a constant factor: I = imread('moon.tif'); J = immultiply(I,0.5); subplot(1,2,1), imshow(I) subplot(1,2,2), imshow(J) See also 14-238 imabsdiff, imadd, imcomplement, imdivide, imlincomb, imsubtract imnoise Purpose 14imnoise Add noise to an image Syntax J = imnoise(I,type) J = imnoise(I,type,parameters) Description J = imnoise(I,type) adds noise of given type to the intensity image I. type is a string that can have one of these values: • 'gaussian' for Gaussian white noise • 'localvar' for zero-mean Gaussian white noise with an intensity-dependent variance • 'poisson' for Poisson noise • 'salt & pepper' for “on and off” pixels • 'speckle' for multiplicative noise J = imnoise(I,type,parameters) accepts an algorithm type plus additional modifying parameters particular to the type of algorithm chosen. If you omit these arguments, imnoise uses default values for the parameters. Here are examples of the different noise types and their parameters: • J = imnoise(I,'gaussian',m,v) adds Gaussian white noise of mean m and variance v to the image I. The default is zero mean noise with 0.01 variance. • J = imnoise(I,'localvar',V) adds zero-mean, Gaussian white noise of local variance, V, to the image I. V is an array of the same size as I. • J = imnoise(I,'localvar',image_intensity,var) adds zero-mean, Gaussian noise to an image I, where the local variance of the noise, var, is a function of the image intensity values in I. The image_intensity and var arguments are vectors of the same size, and plot(image_intensity,var) plots the functional releationship between noise variance and image intensity. The image_intensity vector must contain normalized intensity values ranging from 0 to 1. • J = imnoise(I,'poisson') generates Poisson noise from the data instead of adding artificial noise to the data. In order to respect Poisson statistics, the intensities of unit8 and uint16 images must correspond to the number of photons (or any other quanta of information). Double-precision images are used when the number of photons per pixel can be much larger than 65535 (but less than 10^12); the intensities values vary between 0 and 1 and correspond to the number of photons divided by 10^12. 14-239 imnoise • J = imnoise(I,'salt & pepper',d) adds salt and pepper noise to the image I, where d is the noise density. This affects approximately d*prod(size(I)) pixels. The default is 0.05 noise density. • J = imnoise(I,'speckle',v) adds multiplicative noise to the image I, using the equation J = I+n*I, where n is uniformly distributed random noise with mean 0 and variance v. The default for v is 0.04. Class Support I can be of class uint8, uint16, or double. The output image J is of the same class as I. If I has more than two dimensions it is treated as a multidimensional intensity image and not as an RGB image. Example I = imread('eight.tif'); J = imnoise(I,'salt & pepper',0.02); imshow(I) figure, imshow(J) See Also rand, randn in the MATLAB Function Reference 14-240 imopen Purpose 14imopen Open an image Syntax IM2 = imopen(IM,SE) IM2 = imopen(IM,NHOOD) Description IM2 = imopen(IM,SE) performs morphological opening on the grayscale or binary image IM with the structuring element SE. The argument SE must be a single structuring element object, as opposed to an array of objects. IM2 = imopen(IM,NHOOD) performs opening with the structuring element strel(NHOOD), where NHOOD is an array of 0’s and 1’s that specifies the structuring element neighborhood. Class Support IM can be any numeric or logical class and any dimension, and must be nonsparse. If IM is logical, then SE must be flat. IM2 has the same class as IM. Example This example uses imopen to filter out the smaller objects in an image. 1 Read the image into the MATLAB workspace and threshold it. I = imread('nodules1.tif'); bw = ~im2bw(I,graythresh(I)); imshow(bw), title('Thresholded Image') 14-241 imopen 2 Create a disk-shaped structuring element and open the image. se = strel('disk',5); bw2 = imopen(bw,se); figure, imshow(bw2), title('After opening') See Also 14-242 imclose, imdilate, imerode, strel impixel Purpose Syntax 14impixel Determine pixel color values P = impixel(I) P = impixel(X,map) P = impixel(RGB) P = impixel(I,c,r) P = impixel(X,map,c,r) P = impixel(RGB,c,r) [c,r,P] = impixel(...) P = impixel(x,y,I,xi,yi) P = impixel(x,y,X,map,xi,yi) P = impixel(x,y,RGB,xi,yi) [xi,yi,P] = impixel(x,y,...) Description impixel returns the red, green, and blue color values of specified image pixels. In the syntaxes below, impixel displays the input image and waits for you to specify the pixels with the mouse. P = impixel(I) P = impixel(X,map) P = impixel(RGB) If you omit the input arguments, impixel operates on the image in the current axes. Use normal button clicks to select pixels. Press Backspace or Delete to remove the previously selected pixel. A shift-click, right-click, or double-click adds a final pixel and ends the selection; pressing Return finishes the selection without adding a pixel. When you finish selecting pixels, impixel returns an m-by-3 matrix of RGB values in the supplied output argument. If you do not supply an output argument, impixel returns the matrix in ans. You can also specify the pixels noninteractively, using these syntaxes. P = impixel(I,c,r) P = impixel(X,map,c,r) P = impixel(RGB,c,r) 14-243 impixel r and c are equal-length vectors specifying the coordinates of the pixels whose RGB values are returned in P. The kth row of P contains the RGB values for the pixel (r(k),c(k)). If you supply three output arguments, impixel returns the coordinates of the selected pixels. For example, [c,r,P] = impixel(...) To specify a nondefault spatial coordinate system for the input image, use these syntaxes. P = impixel(x,y,I,xi,yi) P = impixel(x,y,X,map,xi,yi) P = impixel(x,y,RGB,xi,yi) x and y are two-element vectors specifying the image XData and YData. xi and yi are equal-length vectors specifying the spatial coordinates of the pixels whose RGB values are returned in P. If you supply three output arguments, impixel returns the coordinates of the selected pixels. [xi,yi,P] = impixel(x,y,...) Class Support The input image can be of class uint8, uint16, double, or logical. All other inputs and outputs are of class double. Remarks impixel works with indexed, intensity, and RGB images. impixel always returns pixel values as RGB triplets, regardless of the image type: • For an RGB image, impixel returns the actual data for the pixel. The values are either uint8 integers or double floating-point numbers, depending on the class of the image array. • For an indexed image, impixel returns the RGB triplet stored in the row of the colormap that the pixel value points to. The values are double floating-point numbers. • For an intensity image, impixel returns the intensity value as an RGB triplet, where R=G=B. The values are either uint8 integers or double floating-point numbers, depending on the class of the image array. Example 14-244 RGB = imread('flowers.tif'); c = [12 146 410]; impixel r = [104 156 129]; pixels = impixel(RGB,c,r) pixels = 61 253 237 See Also 59 240 37 101 0 44 improfile, pixval 14-245 improfile Purpose Syntax 14improfile Compute pixel-value cross-sections along line segments c = improfile c = improfile(n) c = improfile(I,xi,yi) c = improfile(I,xi,yi,n) [cx,cy,c] = improfile(...) [cx,cy,c,xi,yi] = improfile(...) [...] = improfile(x,y,I,xi,yi) [...] = improfile(x,y,I,xi,yi,n) [...] = improfile(...,method) Description improfile computes the intensity values along a line or a multiline path in an image. improfile selects equally spaced points along the path you specify, and then uses interpolation to find the intensity value for each point. improfile works with grayscale intensity images and RGB images. If you call improfile with one of these syntaxes, it operates interactively on the image in the current axes. c = improfile c = improfile(n) n specifies the number of points to compute the intensity value for. If you do not provide this argument, improfile chooses a value for n, roughly equal to the number of pixels the path traverses. You specify the line or path using the mouse, by clicking on points in the image. Press Backspace or Delete to remove the previously selected point. A shift-click, right-click, or double-click adds a final point and ends the selection; pressing Return finishes the selection without adding a point. When you finish selecting points, improfile returns the interpolated data values in c. c is an n-by-1 vector if the input is a grayscale intensity image, or an n-by-1-by-3 array if the input is an RGB image. 14-246 improfile If you omit the output argument, improfile displays a plot of the computed intensity values. If the specified path consists of a single line segment, improfile creates a two-dimensional plot of intensity values versus the distance along the line segment; if the path consists of two or more line segments, improfile creates a three-dimensional plot of the intensity values versus their x- and y-coordinates. You can also specify the path noninteractively, using these syntaxes. c = improfile(I,xi,yi) c = improfile(I,xi,yi,n) xi and yi are equal-length vectors specifying the spatial coordinates of the endpoints of the line segments. You can use these syntaxes to return additional information. [cx,cy,c] = improfile(...) [cx,cy,c,xi,yi] = improfile(...) cx and cy are vectors of length n, containing the spatial coordinates of the points at which the intensity values are computed. To specify a nondefault spatial coordinate system for the input image, use these syntaxes. [...] = improfile(x,y,I,xi,yi) [...] = improfile(x,y,I,xi,yi,n) x and y are two-element vectors specifying the image XData and YData. [...] = improfile(...,method) uses the specified interpolation method. method is a string that can have one of these values: • 'nearest' (default) uses nearest neighbor interpolation. • 'bilinear' uses bilinear interpolation. • 'bicubic' uses bicubic interpolation. If you omit the method argument, improfile uses the default method of 'nearest'. Class Support The input image can be uint8, uint16, double, or logical. All other inputs and outputs must be double. 14-247 improfile Example See Also I = imread('alumgrns.tif'); x = [35 338 346 103]; y = [253 250 17 148]; improfile(I,x,y), grid on impixel, pixval interp2 in the MATLAB Function Reference 14-248 imread Purpose 14imread Read image from graphics file imread is a MATLAB function. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-249 imreconstruct Purpose 14imreconstruct Morphological reconstruction Syntax IM = imreconstruct(MARKER,MASK) IM = imreconstruct(MARKER,MASK,CONN) Description IM = imreconstruct(MARKER,MASK) performs morphological reconstruction of the image MARKER under the image MASK. MARKER and MASK can be two intensity images or two binary images with the same size. The returned image, IM is an intensity or binary image, respectively. MARKER must be the same size as MASK, and its elements must be less than or equal to the corresponding elements of MASK. By default, imreconstruct uses 8-connected neighborhoods for 2-D images and 26-connected neighborhoods for 3-D images. For higher dimensions, imreconstruct uses conndef(ndims(I),'maximal'). IM = imreconstruct(MARKER,MASK,CONN) performs morphological reconstruction with the specified connectivity. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may be defined in a more general way for any dimension by using for CONN a 3-by-3-by- ... -by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. 14-250 imreconstruct Morphological reconstruction is the algorithmic basis for several other Image Processing Toolbox functions, including imclearborder, imextendedmax, imextendedmin, imfill, imhmax, imhmin, and imimposemin. Class Support MARKER and MASK must be nonsparse numeric or logical arrays with the same class and any dimension. IM is of the same class as MARKER and MASK. Algorithm imreconstruct uses the “fast hybrid grayscale reconstruction” algorithm described in [1]. See Also imclearborder, imextendedmax, imextendedmin, imfill, imhmax, imhmin, imimposemin Reference [1] Luc Vincent, “Morphological Grayscale Reconstruction in Image Analysis: Applications and Efficient Algorithms,” IEEE Transactions on Image Processing, vol. 2, no. 2, April 1993, pp. 176-201. 14-251 imregionalmax Purpose 14imregionalmax Find regional maxima Syntax BW = imregionalmax(I) BW = imregionalmax(I,CONN) Description BW = imregionalmax(I) finds the regional maxima of I. imregionalmax returns a binary image, BW, the same size as I, that identifies the locations of the regional maxima in I. In BW, pixels that are set to 1 identify regional maxima; all other pixels are set to 0. Regional maxima are connected components of pixels with the same intensity value, t, whose external boundary pixels all have a value less than t. By default, imregionalmax uses 8-connected neighborhoods for 2-D images and 26-connected neighborhoods for 3-D images. For higher dimensions, imregionalmax uses conndef(ndims(I),'maximal'). BW = imregionalmax(I,CONN) computes the regional maxima of I using the specified connectivity. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may be defined in a more general way for any dimension by using for CONN a 3-by-3-by- ...-by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. 14-252 imregionalmax Class Support Example I can be any nonsparse, numeric class and any dimension. BW is logical. A = 10*ones(10,10); A(2:4,2:4) = 22; % maxima 12 higher than surround A(6:8,6:8) = 33; % maxima 23 higher than surround A(2,7) = 44; A(3,8) = 45; A(4,9) = 44; A = 10 10 10 10 10 10 10 10 10 10 22 22 22 10 10 44 10 10 10 22 22 22 10 10 10 45 10 10 22 22 22 10 10 10 10 44 10 10 10 10 10 10 10 10 10 10 10 10 10 10 33 33 33 10 10 10 10 10 10 33 33 33 10 10 10 10 10 10 33 33 33 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 regmax = imregionalmax(A) regmax = 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 See Also 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 10 10 10 10 10 10 10 10 10 10 0 0 0 0 0 0 0 0 0 0 conndef, imreconstruct, imregionalmin 14-253 imregionalmin Purpose 14imregionalmin Find regional minima Syntax BW = imregionalmin(I) BW = imregionalmin(I,CONN) Description BW = imregionalmin(I) computes the regional minima of I. The output binary image BW has value 1 corresponding to the pixels of I that belong to regional minima and 0 otherwise. BW is the same size as I. Regional minima are connected components of pixels with the same intensity value, t, whose external boundary pixels all have a value greater than t. By default, imregionalmin uses 8-connected neighborhoods for 2-D images and 26-connected neighborhoods for 3-D images. For higher dimensions, imregionalmin uses conndef(ndims(I),'maximal'). BW = imregionalmin(I,CONN) specifies the desired connectivity. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may be defined in a more general way for any dimension by using for CONN a 3-by-3-by- ...-by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. Class Support 14-254 I can be any nonsparse, numeric class and any dimension. BW is logical. imregionalmin Example A = 10*ones(10,10); A(2:4,2:4) = 3; % minima A(6:8,6:8) = 8; % minima A = 10 10 10 10 10 7 7 7 10 7 7 7 10 7 7 7 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 B = imregionalmin(A) B = 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 See Also 0 1 1 1 0 0 0 0 0 0 3 lower than surround 8 lower than surround 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 2 2 2 10 10 0 0 0 0 0 0 0 0 0 0 10 10 10 10 10 2 2 2 10 10 0 0 0 0 0 1 1 1 0 0 10 10 10 10 10 2 2 2 10 10 0 0 0 0 0 1 1 1 0 0 10 10 10 10 10 10 10 10 10 10 0 0 0 0 0 1 1 1 0 0 10 10 10 10 10 10 10 10 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 conndef, imreconstruct, imregionalmax 14-255 imresize Purpose Syntax 14imresize Resize an image B = imresize(A,m,method) B = imresize(A,[mrows ncols],method) B = imresize(...,method,n) B = imresize(...,method,h) Description imresize resizes an image of any type using the specified interpolation method. method is a string that can have one of these values: • 'nearest' (default) uses nearest neighbor interpolation. • 'bilinear' uses bilinear interpolation. • 'bicubic' uses bicubic interpolation. If you omit the method argument, imresize uses the default method of 'nearest'. B = imresize(A,m,method) returns an image that is m times the size of A. If m is between 0 and 1.0, B is smaller than A. If m is greater than 1.0, B is larger than A. B = imresize(A,[mrows ncols],method) returns an image of size [mrows ncols]. If the specified size does not produce the same aspect ratio as the input image has, the output image is distorted. When the specified output size is smaller than the size of the input image, and method is 'bilinear' or 'bicubic', imresize applies a lowpass filter before interpolation to reduce aliasing. The default filter size is 11-by-11. You can specify a different order for the default filter using [...] = imresize(...,method,n) n is an integer scalar specifying the size of the filter, which is n-by-n. If n is 0 (zero), imresize omits the filtering step. You can also specify your own filter h using [...] = imresize(...,method,h) 14-256 imresize h is any two-dimensional FIR filter (such as those returned by ftrans2, fwind1, fwind2, or fsamp2). Class Support The input image, A, can be numeric or logical and it must be nonsparse. The output image, B, is of the same class as the input image. See Also imrotate, imtransform, tformarray; interp2 in the MATLAB Function Reference 14-257 imrotate Purpose 14imrotate Rotate an image Syntax B = imrotate(A,angle,method) B = imrotate(A,angle,method,'crop') Description B = imrotate(A,angle,method) rotates the image A by angle degrees in a counter-clockwise direction, using the specified interpolation method. method is a string that can have one of these values: • 'nearest' (default) uses nearest neighbor interpolation. • 'bilinear' uses bilinear interpolation. • 'bicubic' uses bicubic interpolation. If you omit the method argument, imrotate uses the default method of 'nearest'. The returned image matrix B is, in general, larger than A to include the whole rotated image. imrotate sets invalid values on the periphery of B to 0. B = imrotate(A,angle,method,'crop') rotates the image A through angle degrees and returns the central portion which is the same size as A. Class Support The input image, A, can be numeric or logical and it must be nonsparse. The output image, B, is of the same class as the input image. Remarks To rotate the image clockwise, specify a negative angle. Example 14-258 I = imread('ic.tif'); J = imrotate(I,-4,'bilinear','crop'); imshow(I) figure, imshow(J) imrotate See Also imcrop, imresize, imtransform, tformarray 14-259 imshow Purpose Syntax 14imshow Display an image imshow(I,n) imshow(I,[low high]) imshow(BW) imshow(X,map) imshow(RGB) imshow(...,display_option) imshow(x,y,A,...) imshow filename h = imshow(...) Description imshow(I,n) displays the intensity image I with n discrete levels of gray. If you omit n, imshow uses 256 gray levels on 24-bit displays, or 64 gray levels on other systems. imshow(I,[low high]) displays I as a grayscale intensity image, specifying the data range for I. The value low (and any value less than low) displays as black, the value high (and any value greater than high) displays as white, and values in between display as intermediate shades of gray. imshow uses the default number of gray levels. If you use an empty matrix ([]) for [low high], imshow uses [min(I(:)) max(I(:))]; the minimum value in I displays as black, and the maximum value displays as white. imshow(BW) displays the binary image BW. Values of 0 display as black, and values of 1 display as white. imshow(X,map) displays the indexed image X with the colormap map. imshow(RGB) displays the truecolor image RGB. imshow(...,display_option) displays the image, calling truesize if display_option is 'truesize', or suppressing the call to truesize if display_option is 'notruesize'. Either option string can be abbreviated. If you do not supply this argument, imshow determines whether to call truesize based on the setting of the 'ImshowTruesize' preference. imshow(x,y,A,...) uses the two-element vectors x and y to establish a nondefault spatial coordinate system, by specifying the image XData and YData. 14-260 imshow imshow filename displays the image stored in the graphics file filename. imshow calls imread to read the image from the file, but the image data is not stored in the MATLAB workspace. The file must be in the current directory or on the MATLAB path. h = imshow(...) returns the handle to the image object created by imshow. Class Support The input image can be of class logical, uint8, uint16, or double, and it must be nonsparse. Remarks You can use the iptsetpref function to set several toolbox preferences that modify the behavior of imshow. For example: • 'ImshowBorder' controls whether imshow displays the image with a border around it. • 'ImshowAxesVisible' controls whether imshow displays the image with the axes box and tick labels. • 'ImshowTruesize' controls whether imshow calls the truesize function. Note that the display_option argument to imshow enables you to override the 'ImshowTruesize' preference. For more information about these preferences, see the reference entry for iptsetpref. See Also getimage, imread, iptgetpref, iptsetpref, subimage, truesize, warp image, imagesc in the MATLAB Function Reference 14-261 imsubtract Purpose 14imsubtract Subtract one image from another, or subtract a constant from an image Syntax Z = imsubtract(X,Y) Description Z = imsubtract(X,Y) subtracts each element in array Y from the corresponding element in array X and returns the difference in the corresponding element of the output array Z. X and Y are real, nonsparse numeric arrays of the same size and class, or Y is a double scalar. The array returned, Z, has the same size and class as X. If X is an integer array, then elements of the output that exceed the range of the integer type are truncated, and fractional values are rounded. If X and Y are double arrays, then you can use the expression X-Y instead of this function. Examples Subtract two uint8 arrays. Note that negative results are rounded to 0. X Y Z Z = uint8([ 255 10 75; 44 225 100]); = uint8([ 50 50 50; 50 50 50 ]); = imadd(X,Y) = 205 0 0 175 25 50 Estimate and subtract the background of the rice image: I = imread('rice.tif'); blocks = blkproc(I,[32 32],'min(x(:))'); background = imresize(blocks,[256 256],'bilinear'); Ip = imsubtract(I,background); imshow(Ip,[]) Subtract a constant value from the rice image: I = imread('rice.tif'); Iq = imsubtract(I,50); subplot(1,2,1), imshow(I) subplot(1,2,2), imshow(Iq) See also 14-262 imabsdiff, imadd, imcomplement, imdivide, imlincomb, immultiply imtophat Purpose 14imtophat Perform top-hat filtering Syntax IM2 = imtophat(IM,SE) IM2 = imtophat(IM,NHOOD) Description IM2 = imtophat(IM,SE) performs morphological top-hat filtering on the grayscale or binary input image IM using the structuring element SE, where SE is returned by strel. SE must be a single structuring element object, not an array containing multiple structuring element objects. IM2 = imtophat(IM,NHOOD), where NHOOD is an array of 0’s and 1’s that specifies the size and shape of the structuring element, is the same as imptophat(IM,strel(NHOOD)). Class Support IM can be numeric or logical and must be nonsparse. The output image, IM2, has the same class as the input image. If the input is binary (logical), then the structuring element must be flat. Example You can use top-hat filtering to correct uneven illumination when the background is dark. This example uses top-hat filtering with a disk-shaped structuring element to remove the uneven background illumination from the image rice.tif. 1 Read the image into the MATLAB workspace. I = imread('rice.tif'); imshow(I), title('Original') 14-263 imtophat 2 Create the structuring element and perform top-hat filtering of the image. se = strel('disk',12); J = imtophat(I,se); figure, imshow(J), title('Step 1: Top-hat filtering') 3 Use imadjust and stretchlim to make the result more easily visible. K = imadjust(J,stretchlim(J)); figure, imshow(K), title('Step 2: Contrast adjustment') See Also 14-264 imbothat, strel imtransform Purpose 14imtransform Apply 2-D spatial transformation to image Syntax B = imtransform(A,TFORM) B = imtransform(A,TFORM,INTERP) [B,XDATA,YDATA] = imtransform(...) [B,XDATA,YDATA] = imtransform(...,PARAM1,VAL1,PARAM2,VAL2,...) Description B = imtransform(A,TFORM) transforms the image A according to the 2-D spatial transformation defined by TFORM, which is a spatial transformation structure (TFORM) as returned by maketform or cp2tform. If ndims(A) > 2, such as for an RGB image, then the same 2-D transformation is automatically applied to all 2-D planes along the higher dimensions. When you use this syntax, imtransform automatically shifts the origin of your output image to make as much of the transformed image visible as possible. If you are using imtransform to do image registration, this syntax is not likely to give you the results you expect; you may want to set 'XData' and 'YData' explicitly. For more information, see Parameters, as well as Example 3. B = imtransform(A,TFORM,INTERP) specifies the form of interpolation to use. INTERP can be one of the strings 'nearest', 'bilinear', or 'bicubic'. Alternatively, INTERP can be a RESAMPLER structure as returned by makeresampler. This option allows more control over how resampling is performed. The default value for INTERP is 'bilinear'. [B,XDATA,YDATA] = imtransform(...) returns the location of the output image B in the output X-Y space. XDATA and YDATA are two-element vectors. The elements of XDATA specify the x-coordinates of the first and last columns of B. The elements of YDATA specify the y-coordinates of the first and last rows of B. Normally, imtransform computes XDATA and YDATA automatically so that B contains the entire transformed image A. However, you can override this automatic computation; see below. [B,XDATA,YDATA] = imtransform(...,PARAM1,VAL1,PARAM2,VAL2,...) specifies parameters that control various aspects of the spatial transformation. Parameter names can be abbreviated, and case does not matter. 14-265 imtransform Parameters This table lists all the parameters you can specify. Note that parameter names can be abbreviated and are not case-sensitive. Parameter Description 'UData' 'VData' Both of these parameters are two-element real vectors. 'UData' and 'VData' specify the spatial location of the image A in the 2-D input space, U-V. The two elements of 'UData' give the u-coordinates (horizontal) of the first and last columns of A, respectively. The two elements of 'VData' give the v-coordinates (vertical) of the first and last rows of A, respectively. The default values for 'UData' and 'VData' are [1 size(A,2)] and [1 size(A,1)], respectively. 'XData' 'YData' Both of these parameters are two-element real vectors. 'XData' and 'YData' specify the spatial location of the output image B in the 2-D output space X-Y. The two elements of 'XData' give the x-coordinates (horizontal) of the first and last columns of B, respectively. The two elements of 'YData' give the y-coordinates (vertical) of the first and last rows of B, respectively. If 'XData' and 'YData' are not specified, imtransform estimates values for them that will completely contain the entire transformed output image. 14-266 imtransform Parameter Description 'XYScale' A one- or two-element real vector. The first element of 'XYScale' specifies the width of each output pixel in X-Y space. The second element (if present) specifies the height of each output pixel. If 'XYScale' has only one element, then the same value is used for both width and height. If 'XYScale' is not specified but 'Size' is, then 'XYScale' is computed from 'Size', 'XData', and 'YData'. If neither 'XYScale' nor 'Size' is provided, then the scale of the input pixels is used for 'XYScale'. 'Size' A two-element vector of nonnegative integers. 'Size' specifies the number of rows and columns of the output image B. For higher dimensions, the size of B is taken directly from the size of A. In other words, size(B,k) equals size(A,k) for k > 2. If 'Size' is not specified, then it is computed from 'XData', 'YData', and 'XYScale'. 14-267 imtransform Parameter Description 'FillValues' An array containing one or several fill values. Fill values are used for output pixels when the corresponding transformed location in the input image is completely outside the input image boundaries. If A is 2-D, 'FillValues' must be a scalar. However, if A's dimension is greater than two, then 'FillValues' can be an array whose size satisfies the following constraint: size(fill_values,k) must either equal size(A,k+2) or 1. For example, if A is a uint8 RGB image that is 200-by-200-by-3, then possibilities for 'FillValues' include: 0 - fill with black [0;0;0] - also fill with black 255 - fill with white [255;255;255] - also fill with white [0;0;255] - fill with blue [255;255;0] - fill with yellow If A is 4-D with size 200-by-200-by-3-by-10, then 'FillValues' can be a scalar, 1-by-10, 3-by-1, or 3-by-10. Notes • When you do not specify the output-space location for B using 'XData' and 'YData', imtransform estimates them automatically using the function findbounds. For some commonly-used transformations, such as affine or projective, for which a forward-mapping is easily computable, findbounds is fast. For transformations that do not have a forward mapping, such as the polynomial ones computed by cp2tform, findbounds can take significantly longer. If you can specify 'XData' and 'YData' directly for such transformations, imtransform may run noticeably faster. • The automatic estimate of 'XData' and 'YData' using findbounds is not guaranteed in all cases to completely contain all the pixels of the transformed input image. 14-268 imtransform • The output values XDATA and YDATA may not exactly equal the input 'XData and 'YData' parameters. This can happen either because of the need for an integer number or rows and columns, or if you specify values for 'XData', 'YData', 'XYScale', and 'Size' that are not entirely consistent. In either case, the first element of XDATA and YDATA always equals the first element of 'XData' and 'YData', respectively. Only the second elements of XDATA and YDATA might be different. • imtransform assumes spatial-coordinate conventions for the transformation TFORM. Specifically, the first dimension of the transformation is the horizontal or x-coordinate, and the second dimension is the vertical or y-coordinate. Note that this is the reverse of the array subscripting convention in MATLAB. • TFORM must be a 2-D transformation to be used with imtransform. For arbitrary-dimensional array transformations, see tformarray. Class Support The input image, A, can be of any nonsparse numeric class, real or complex, or it can be of class logical. The class of B is the same as the class of A. Example Example 1 Apply a horizontal shear to an intensity image. I = imread('cameraman.tif'); tform = maketform('affine',[1 0 0; .5 1 0; 0 0 1]); J = imtransform(I,tform); imshow(I), figure, imshow(J) Example 2 A projective transformation can map a square to a quadrilateral. In this example, set up an input coordinate system so that the input image fills the unit square and then transform the image into the quadrilateral with vertices (0 0), (1 0), (1 1), (0 1) to the quadrilateral with vertices (-4 2), (-8 3), (-3 -5), and (6 3). Fill with gray and use bicubic interpolation. Make the output size the same as the input size. I = imread('cameraman.tif'); udata = [0 1]; vdata = [0 1]; % input coordinate system tform = maketform('projective',[ 0 0; 1 0; 1 1; 0 1],... [-4 2; -8 -3; -3 -5; 6 3]); 14-269 imtransform [B,xdata,ydata] = imtransform(I,tform,'bicubic','udata',udata,... 'vdata',vdata,... 'size',size(I),... 'fill',128); subplot(1,2,1), imshow(udata,vdata,I), axis on subplot(1,2,2), imshow(xdata,ydata,B), axis on Example 3 Register an aerial photo to an orthophoto. unregistered = imread('westconcordaerial.png'); figure, imshow(unregistered) figure, imshow('westconcordorthophoto.png') load westconcordpoints % load some points that were already picked t_concord = cp2tform(input_points,base_points,'projective'); info = imfinfo('westconcordorthophoto.png'); registered = imtransform(unregistered,t_concord,... 'XData',[1 info.Width], 'YData',[1 info.Height]); figure, imshow(registered) See Also 14-270 cp2tform, imresize, imrotate, maketform, makeresampler, tformarray imwrite Purpose 14imwrite Write image to graphics file imwrite is a function in MATLAB. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-271 ind2gray Purpose 14ind2gray Convert an indexed image to an intensity image Syntax I = ind2gray(X,map) Description I = ind2gray(X,map) converts the image X with colormap map to an intensity image I. ind2gray removes the hue and saturation information from the input image while retaining the luminance. Class Support X can be of class uint8, uint16, or double. I is of class double. Example load trees I = ind2gray(X,map); imshow(X,map) figure,imshow(I) Algorithm ind2gray converts the colormap to NTSC coordinates using rgb2ntsc, and sets the hue and saturation components (I and Q) to zero, creating a gray colormap. ind2gray then replaces the indices in the image X with the corresponding grayscale intensity values in the gray colormap. See Also gray2ind, imshow, rgb2ntsc 14-272 ind2rgb Purpose 14ind2rgb Convert an indexed image to an RGB image Syntax RGB = ind2rgb(X,map) Description RGB = ind2rgb(X,map) converts the matrix X and corresponding colormap map to RGB (truecolor) format. Class Support X can be of class uint8, uint16, or double. RGB is an m-by-n-by-3 array of class double. See Also ind2gray, rgb2ind 14-273 iptgetpref Purpose 14iptgetpref Get Image Processing Toolbox preference Syntax value = iptgetpref(prefname) Description value = iptgetpref(prefname) returns the value of the Image Processing Toolbox preference specified by the string prefname. For Preferences for a complete list of valid preference names. Preference names are not case-sensitive and can be abbreviated. iptgetpref without an input argument displays the current setting of all Image Processing Toolbox preferences. Example value = iptgetpref('ImshowAxesVisible') value = off See Also 14-274 imshow, iptsetpref iptsetpref Purpose 14iptsetpref Set Image Processing Toolbox preferences or display valid values Syntax iptsetpref(prefname,value) Description iptsetpref(prefname) displays the valid values for prefname. See Preferences for a complete list of available preferences. iptsetpref(prefname,value) sets the Image Processing Toolbox preference specified by the string prefname to value. The setting persists until the end of the current MATLAB session, or until you change the setting. (To make the value persist between sessions, put the command in your startup.m file.) Preferences This table describes the available preferences. Note that the preference names are case insensitive and can be abbreviated. Preference Name Values Description 'ImshowBorder' 'loose' (default) or 'tight' If 'ImshowBorder' is 'loose', imshow displays the image with a border between the image and the edges of the figure window, thus leaving room for axes labels, titles, etc. If 'ImshowBorder' is 'tight', imshow adjusts the figure size so that the image entirely fills the figure. Note: There may still be a border if the image is very small, or if there are other objects besides the image and its axes in the figure. 'ImshowAxesVisible' 'on' or 'off' (default) If 'ImshowAxesVisible' is 'on', imshow displays the image with the axes box and tick labels. If 'ImshowAxesVisible' is 'off', imshow displays the image without the axes box and tick labels. 14-275 iptsetpref Preference Name Values Description 'ImshowTruesize' 'auto' (default) or 'manual' If 'ImshowTruesize' is 'manual', imshow does not call truesize. If 'ImshowTruesize' is 'auto', imshow automatically decides whether to call truesize. (imshow calls truesize if there will be no other objects in the resulting figure besides the image and its axes.) You can override this setting for an individual display by specifying the display_option argument to imshow, or you can call truesize manually after displaying the image. 'TruesizeWarning' 'on' (default) or 'off' If 'TruesizeWarning' is 'on', the truesize function displays a warning if the image is too large to fit on the screen. (The entire image is still displayed, but at less than true size.) If 'TruesizeWarning' is 'off', truesize does not display the warning. Note: This preference applies even when you call truesize indirectly, such as through imshow. Example See Also iptsetpref('ImshowBorder','tight') imshow, iptgetpref, truesize axis in the MATLAB Function Reference 14-276 iradon Purpose 14iradon Compute inverse Radon transform Syntax I = iradon(P,theta) I = iradon(P,theta,interp,filter,d,n) [I,h] = iradon(...) Description I = iradon(P,theta) reconstructs the image I from projection data in the two-dimensional array P. The columns of P are parallel beam projection data. iradon assumes that the center of rotation is the center point of the projections, which is defined as ceil(size(P,1)/2). theta describes the angles (in degrees) at which the projections were taken. It can be either a vector containing the angles or a scalar specifying D_theta, the incremental angle between projections. If theta is a vector, it must contain angles with equal spacing between them. If theta is a scalar specifying D_theta, the projections are taken at angles theta = m*D_theta, where m = 0,1,2,...,size(P,2) 1. If the input is the empty matrix ([]), D_theta defaults to 180/size(P,2). iradon uses the filtered back-projection algorithm to perform the inverse Radon transform. The filter is designed directly in the frequency domain and then multiplied by the FFT of the projections. The projections are zero-padded to a power of 2 before filtering to prevent spatial domain aliasing and to speed up the FFT. I = iradon(P,theta,interp,filter,d,n) specifies parameters to use in the inverse Radon transform. You can specify any combination of the last four arguments. iradon uses default values for any of these arguments that you omit. interp specifies the type of interpolation to use in the backprojection. The available options are listed in order of increasing accuracy and computational complexity: • 'nearest' – nearest neighbor interpolation • 'linear' – linear interpolation (default) • 'spline' – spline interpolation filter specifies the filter to use for frequency domain filtering. filter is a string that specifies any of the following standard filters: 14-277 iradon • 'Ram-Lak' – The cropped Ram-Lak or ramp filter (default). The frequency response of this filter is | f |. Because this filter is sensitive to noise in the projections, one of the filters listed below may be preferable. These filters multiply the Ram-Lak filter by a window that de-emphasizes high frequencies. • 'Shepp-Logan' – The Shepp-Logan filter multiplies the Ram-Lak filter by a sinc function. • 'Cosine' – The cosine filter multiplies the Ram-Lak filter by a cosine function. • 'Hamming' – The Hamming filter multiplies the Ram-Lak filter by a Hamming window. • 'Hann' – The Hann filter multiplies the Ram-Lak filter by a Hann window. d is a scalar in the range (0,1] that modifies the filter by rescaling its frequency axis. The default is 1. If d is less than 1, the filter is compressed to fit into the frequency range [0,d], in normalized frequencies; all frequencies above d are set to 0. n is a scalar that specifies the number of rows and columns in the reconstructed image. If n is not specified, the size is determined from the length of the projections. n = 2*floor(size(P,1)/(2*sqrt(2))) If you specify n, iradon reconstructs a smaller or larger portion of the image, but does not change the scaling of the data. If the projections were calculated with the radon function, the reconstructed image may not be the same size as the original image. [I,h] = iradon(...) returns the frequency response of the filter in the vector h. Class Support Example 14-278 All input arguments and output arguments must be of class double. P = phantom(128); R = radon(P,0:179); I = iradon(R,0:179,'nearest','Hann'); imshow(P) figure, imshow(I) iradon Algorithm iradon uses the filtered backprojection algorithm to perform the inverse Radon transform. The filter is designed directly in the frequency domain and then multiplied by the FFT of the projections. The projections are zero-padded to a power of 2 before filtering to prevent spatial domain aliasing and to speed up the FFT. See Also radon, phantom References [1] Kak, Avinash C., and Malcolm Slaney, Principles of Computerized Tomographic Imaging. New York: IEEE Press. 14-279 isbw Purpose 14isbw Return true for a binary image Syntax flag = isbw(A) Description flag = isbw(A) returns 1 if A is a binary image and 0 otherwise. The input image, A, is considered to be a binary image if it is a nonsparse logical array. Class Support The input image, A, can be any MATLAB array. See Also isind, isgray, isrgb 14-280 isflat Purpose 14isflat Return true for flat structuring element Syntax TF = isflat(SE) Description TF = isflat(SE) returns true (1) if the structuring element SE is flat; otherwise it returns false (0). If SE is an array of STREL objects, then TF is the same size as SE. Class Support SE is a STREL object. TF is a double precision value. See Also strel 14-281 isgray Purpose 14isgray Return true for intensity image Syntax flag = isgray(A) Description flag = isgray(A) returns 1 if A is a grayscale intensity image and 0 otherwise. isgray uses these criteria to decide if A is an intensity image: • If A is of class double, all values must be in the range [0,1], and the number of dimensions of A must be 2. • If A is of class uint16 or uint8, the number of dimensions of A must be 2. Note A four-dimensional array that contains multiple intensity images returns 0, not 1. Class Support The input image, A, can be of class logical, uint8, uint16, or double. See Also isbw, isind, isrgb 14-282 isind Purpose 14isind Return true for an indexed image Syntax flag = isind(A) Description flag = isind(A) returns 1 if A is an indexed image and 0 otherwise. isind uses these criteria to determine if A is an indexed image: • If A is of class double, all values in A must be integers greater than or equal to 1, and the number of dimensions of A must be 2. • If A is of class uint8 or uint16, the number of dimensions of A must be 2. Note A four-dimensional array that contains multiple indexed images returns 0, not 1. Class Support A can be of class logical, uint8, uint16, or double. See Also isbw, isgray, isrgb 14-283 isrgb Purpose 14isrgb Return true for an RGB image Syntax flag = isrgb(A) Description flag = isrgb(A) returns 1 if A is an RGB truecolor image and 0 otherwise. isrgb uses these criteria to determine if A is an RGB image: • If A is of class double, all values must be in the range [0,1], and A must be m-by-n-by-3. • If A is of class uint16 or uint8, A must be m-by-n-by-3. Note A four-dimensional array that contains multiple RGB images returns 0, not 1. Class Support A can be of class logical, uint8, uint16, or double. See Also isbw, isgray, isind 14-284 label2rgb Purpose 14label2rgb Convert a label matrix into an RGB image Syntax RGB RGB RGB RGB Description RGB = label2rgb(L) converts a label matrix L, such as those returned by bwlabel or watershed, into an RGB color image for the purpose of visualizing the labeled regions. The label2rgb function determines the color to assign to = = = = label2rgb(L) label2rgb(L,map) label2rgb(L,map,zerocolor) label2rgb(L,map,zerocolor,order) each object based on the number of objects in the label matrix and range of colors in the colormap. The label2rgb function picks colors from the entire range. RGB = label2rgb(L,map) defines the colormap to be used in the RGB image. The colormap, map, can have any of the following values: • n-by-3 colormap matrix • String containing the name of a MATLAB colormap function, such as 'jet' or 'gray' (See colormap for a list of supported colormaps.) • Function handle of a colormap function, such as @jet or @gray If you do not specify map, the default value is 'colorcube'. RGB = label2rgb(L,map,zerocolor) defines the RGB color of the elements labeled zero (i.e., the background) in the input label matrix L. As the value of zerocolor, specify an RGB triple or one of the strings listed in this table. Value Color 'b' blue 'c' cyan 'g' green 'k' black 'm' magenta 'r' red 14-285 label2rgb Value Color 'w' white 'y' yellow If you do not specify zerocolor, the default value for zero-labeled elements is [1 1 1] (white). RGB = label2rgb(L,map,zerocolor,order) controls how label2rgb assigns colormap colors to regions in the label matrix. If order is 'noshuffle' (the default), label2rgb assigns colormap colors to label matrix regions in numerical order. If order is 'shuffle', label2rgb assigns colormap colors pseudo-randomly. Class Support Example See Also 14-286 The input label matrix L can have any nonsparse, numeric class. It must contain finite, nonnegative integers. The output of label2rgb is of class uint8. I = imread('rice.tif'); figure, imshow(I), title('original image') BW = im2bw(I, graythresh(I)); L = bwlabel(BW); RGB = label2rgb(L); RGB2 = label2rgb(L, 'spring', 'c', 'shuffle'); imshow(RGB), figure, imshow(RGB2) bwlabel, bwlabeln, ismember, watershed makelut Purpose 14makelut Construct a lookup table for use with applylut Syntax lut = makelut(fun,n) lut = makelut(fun,n,P1,P2,...) Description lut = makelut(fun,n) returns a lookup table for use with applylut. fun is either a string containing the name of a function or an inline function object. The function should take a 2-by-2 or 3-by-3 matrix of 1’s and 0’s as input and return a scalar. n is either 2 or 3, indicating the size of the input to fun. makelut creates lut by passing all possible 2-by-2 or 3-by-3 neighborhoods to fun, one at a time, and constructing either a 16-element vector (for 2-by-2 neighborhoods) or a 512-element vector (for 3-by-3 neighborhoods). The vector consists of the output from fun for each possible neighborhood. lut = makelut(fun,n,P1,P2,...) passes the additional parameters P1,P2,..., to fun. Class Support lut is returned as a vector of class double. Example In this example, the function returns 1 (true) if the number of 1’s in the neighborhood is 2 or greater, and returns 0 (false) otherwise. makelut then uses the function to construct a lookup table for 2-by-2 neighborhoods. f = inline('sum(x(:)) >= 2'); lut = makelut(f,2) lut = 0 0 0 1 0 1 1 1 0 1 1 1 14-287 makelut 1 1 1 1 See Also 14-288 applylut makeresampler Purpose 14makeresampler Create resampling structure Syntax R = makeresampler(interpolant,padmethod) Description R = makeresampler(interpolant,padmethod) creates a separable resampler structure for use with tformarray and imtransform. The interpolant argument specifies the interpolating kernel that the separable resampler uses. In its simplest form, interpolant can have any of the following strings as a value. Interpolant Description 'cubic' Cubic interpolation 'linear' Linear interpolation 'nearest' Nearest neighbor interpolation If you are using a custom interpolating kernel, you can specify interpolant as a cell array in either of these forms {half_width, positive_half} half_width is a positive scalar designating the half width of a symmetric interpolating kernel. positive_half is a vector of values regularly sampling the kernel on the closed interval [0 positive_half] {half_width, interp_fcn} interp_fcn is a function handle that returns interpolating kernel values, given an array of input values in the interval [0 positive_half] To specify the interpolation method independently along each dimension, you can combine both types of interpolant specifications. The number of elements in the cell array must equal the number of transform dimensions. For example, if you specify this value for interpolant {'nearest', 'linear', {2 KERNEL_TABLE}} 14-289 makeresampler the resampler uses nearest-neighbor interpolation along the first transform dimension, linear interpolation along the second dimension, and a custom table-based interpolation along the third. The padmethod argument controls how the resampler interpolates or assigns values to output elements that map close to or outside the edge of input array. The following table lists all the possible values of padmethod. Pad Method Description 'bound' Assigns values from the fill value array to points that map outside the array and repeats border elements of array for points that map inside the array. (Same as 'replicate'.) When interpolant is 'nearest', this pad method produces the same results as 'fill'. 'bound' is like 'fill', but avoids mixing fill values and input image values. 'circular' Pads array with circular repetition of elements within the dimension. Same as padarray. 'fill' Generates an output array with smooth-looking edges (except when using nearest neighbor interpolation). For output points that map near the edge of the input array (either inside or outside), it combines input image and fill values. When interpolant is 'nearest', this pad method produces the same results as 'bound'. 'replicate' Pads array by repeating border elements of array. Same as padarray. 'symmetric' Pads array with mirror reflections of itself. Same as padarray. In the case of 'fill', 'replicate', 'circular', or 'symmetric', the resampling performed by tformarray or imtransform occurs in two logical steps: 1 Pad the array A infinitely to fill the entire input transform space 14-290 makeresampler 2 Evaluate the convolution of the padded A with the resampling kernel at the output points specified by the geometric map Each nontransform dimension is handled separately. The padding is virtual, (accomplished by remapping array subscripts) for performance and memory efficiency. If you implement a custom resampler, you can implement these behaviors. Custom Resamplers The syntaxes described above construct a resampler structure that uses the separable resampler function that ships with the Image Processing Toolbox. It is also possible to create a resampler structure that uses a user-written resampler by using this syntax R = makeresampler(PropertyName,PropertyValue,...) The makeresampler function supports the following properties. Property Description 'Type' Can have the values 'separable' or 'custom' and must always be supplied. If 'Type' is 'separable', the only other properties that can be specified are 'Interpolant' and 'PadMethod', and the result is equivalent to using the makeresampler(interpolant,padmethod) syntax. If 'Type' is 'custom', you must specify the'NDims' and 'ResampleFcn' properties and, optionally, the 'CustomData' property. 'PadMethod' See the padmethod argument for more information. 'Interpolant' See the interpolant argument for more information. 'NDims' 'NDims' is a positive integer and indicates what dimensionality the custom resampler can handle. Use a value of Inf to indicate that the custom resampler can handle any dimension. If 'Type' is 'custom', NDims is required. 14-291 makeresampler Property Description 'ResampleFcn' This property is a handle to a function that performs the resampling. The function is called with the following interface. B = resample_fcn(A,M,TDIMS_A,TDIMS_B,FSIZE_A,FSIZE_B,F,R) See the help for tformarray for information about the inputs A, TDIMS_A, TDIMS_B, and F. The argument M is an array that maps the transform subscript space of B to the transform subscript space of A. If A has N transform dimensions (N=length(TDIMS_A)) and B has P transform dimensions (P=length(TDIMS_B)), then ndims(M)=P+1, if N>1 and P if N==1, and size(M,P+1)=N. The first P dimensions of M correspond to the output transform space, permuted according to the order in which the output transform dimensions are listed in TDIMS_B. (In general TDIMS_A and TDIMS_B need not be sorted in ascending order, although such a limitation may be imposed by specific resamplers.) Thus, the first P elements of size(M) determine the sizes of the transform dimensions of B. The input transform coordinates to which each point is mapped are arrayed across the final dimension of M, following the order given in TDIMS_A. M must be double. FSIZE_A and FSIZE_B are the full sizes of A and B, padded with 1’s as necessary to be consistent with TDIMS_A, TDIMS_B, and size(A). 'CustomData' Example User-defined. Stretch an image in the y-direction using separable resampler that applies in cubic interpolation in the y-direction and nearest neighbor interpolation in the x-direction. (This is equivalent to, but faster than, applying bicubic interpolation.) A = imread('moon.tif'); resamp = makeresampler({'nearest','cubic'},'fill'); stretch = maketform('affine',[1 0; 0 1.3; 0 0]); B = imtransform(A,stretch,resamp); See Also 14-292 imtransform, tformarray maketform Purpose 14maketform Create geometric transformation structure Syntax T = maketform(transformtype,...) Description T = maketform(transformtype,...) creates a multidimensional spatial transformation structure (called a TFORM struct) that can be used with the tformfwd, tforminv, fliptform, imtransform, or tformarray functions. transformtype can be any of the following spatial transformation types. maketform supports a special syntax for each transformation type. See the following sections for information about these syntaxes. Transform Types Transform Type Description 'affine' Affine transformation in 2-D or N-D 'projective' Projective transformation in 2-D or N-D 'custom' User-defined transformation, may be N-D to M-D 'box' Independent affine transformation (scale and shift) in each dimension 'composite' Composition of an arbitrary number of more basic transformations Affine T = maketform('affine',A) builds a TFORM struct, T, for an N-dimensional affine transformation. A is a nonsingular real (N+1)-by-(N+1) or (N+1)-by-N matrix. If A is (N+1)-by-(N+1), the last column of A must be [zeros(N,1);1]. Otherwise, A is augmented automatically, such that its last column is [zeros(N,1);1]. The matrix A defines a forward transformation such that tformfwd(U,T), where U is a 1-by-N vector, returns a 1-by-N vector X, such that X=U*A(1:N,1:N)+A(N+1,1:N). T has both forward and inverse transformations. T = maketform('affine',U,X) builds a TFORM struct, T, for a two-dimensional affine transformation that maps each row of U to the corresponding row of X. The U and X arguments are each 3-by-2 and define the corners of input and output triangles. The corners may not be collinear. 14-293 maketform Projective T = maketform('projective',A) builds a TFORM struct for an N-dimensional projective transformation. A is a nonsingular real (N+1)-by-(N+1) matrix. A(N+1,N+1) cannot be 0. The matrix A defines a forward transformation such that tformfwd(U,T), where U is a 1-by-N vector, returns a 1-by-N vector X, such that X = W(1:N)/W(N+1), where W=[U 1]*A. The transformation structure, T, has both forward and inverse transformations. T = maketform('projective',U,X) builds a TFORM struct, T, for a two-dimensional projective transformation that maps each row of U to the corresponding row of X. The U and X arguments are each 4-by-2 and define the corners of input and output quadrilaterals. No three corners may be collinear. Custom T = maketform('custom',NDIMS_IN,NDIMS_OUT,... FORWARD_FCN,INVERSE_FCN,TDATA) builds a custom TFORM struct, T, based on user-provided function handles and parameters. NDIMS_IN and NDIMS_OUT are the numbers of input and output dimensions. FORWARD_FCN and INVERSE_FCN are function handles to forward and inverse functions. Those functions must support the syntaxes: Forward function: X = FORWARD_FCN(U,T) Inverse function U = INVERSE_FCN(X,T) where U is a P-by-NDIMS_IN matrix whose rows are points in the transformation's input space, and X is a P-by-NDIMS_OUT matrix whose rows are points in the transformation's output space. The TDATA argument can be any MATLAB array and is typically used to store parameters of the custom transformation. It is accessible to FORWARD_FCN and INVERSE_FCN via the “tdata” field of T. Either FORWARD_FCN or INVERSE_FCN can be empty, although at least INVERSE_FCN must be defined to use T with tformarray or imtransform. Box T = maketform('box',tsize,LOW,HIGH) or T = maketform('box',INBOUNDS, OUTBOUNDS) builds an N-dimensional affine TFORM struct, T. The tsize argument is an N-element vector of positive integers. LOW and HIGH are also N-element vectors. The transformation maps 14-294 maketform an input box defined by the opposite corners ones(1,N) and tsize or, alternatively, by corners INBOUNDS(1,:) and INBOUND(2,:) to an output box defined by the opposite corners LOW and HIGH or OUTBOUNDS(1,:) and OUTBOUNDS(2,:). LOW(K) and HIGH(K) must be different unless tsize(K) is 1, in which case the affine scale factor along the K-th dimension is assumed to be 1.0. Similarly, INBOUNDS(1,K) and INBOUNDS(2,K) must be different unless OUTBOUNDS(1,K) and OUTBOUNDS(1,K) are the same, and vice versa. The 'box' TFORM is typically used to register the row and column subscripts of an image or array to some “world” coordinate system. Composite T = maketform('composite',T1,T2,...,TL) or T = maketform('composite', [T1 T2 ... TL]) builds a TFORM struct, T, whose forward and inverse functions are the functional compositions of the forward and inverse functions of the T1, T2, ..., TL. For example, if L = 3, then tformfwd(U,T) is the same as tformfwd(tformfwd(tformfwd(U,T3),T2),T1). The components T1 through TL must be compatible in terms of the numbers of input and output dimensions. T has a defined forward transform function only if all of the component transforms have defined forward transform functions. T has a defined inverse transform function only if all of the component functions have defined inverse transform functions. Example Make and apply an affine transformation. T = maketform('affine',[.5 0 0; .5 2 0; 0 0 1]); tformfwd([10 20],T) I = imread('cameraman.tif'); I2 = imtransform(I,T); imshow(I2) See Also tformfwd, tforminv, fliptform, imtransform, tformarray 14-295 mat2gray Purpose 14mat2gray Convert a matrix to a grayscale intensity image Syntax I = mat2gray(A,[amin amax]) I = mat2gray(A) Description I = mat2gray(A,[amin amax]) converts the matrix A to the intensity image I. The returned matrix I contains values in the range 0 (black) to 1.0 (full intensity or white). amin and amax are the values in A that correspond to 0 and 1.0 in I. I = mat2gray(A) sets the values of amin and amax to the minimum and maximum values in A. Class Support Example See Also 14-296 The input array, A, and the output image, I, are of class double. I = imread('rice.tif'); J = filter2(fspecial('sobel'),I); K = mat2gray(J); imshow(I) figure, imshow(K) gray2ind mean2 Purpose 14mean2 Compute the mean of the elements of a matrix Syntax B = mean2(A) Description B = mean2(A) computes the mean of the values in A. Class Support The input image, A, can be numeric or logical. The output image, B, is a scalar of class double. Algorithm mean2 computes the mean of an array A using mean(A(:)). See Also std2 mean, std in the MATLAB Function Reference 14-297 medfilt2 Purpose 14medfilt2 Perform two-dimensional median filtering Syntax B = medfilt2(A,[m n]) B = medfilt2(A) B = medfilt2(A,'indexed',...) Description Median filtering is a nonlinear operation often used in image processing to reduce “salt and pepper” noise. Median filtering is more effective than convolution when the goal is to simultaneously reduce noise and preserve edges. B = medfilt2(A,[m n]) performs median filtering of the matrix A in two dimensions. Each output pixel contains the median value in the m-by-n neighborhood around the corresponding pixel in the input image. medfilt2 pads the image with zeros on the edges, so the median values for the points within [m n]/2 of the edges may appear distorted. B = medfilt2(A) performs median filtering of the matrix A using the default 3-by-3 neighborhood. B = medfilt2(A,'indexed',...) processes A as an indexed image, padding with zeros if the class of A is uint8, or ones if the class of A is double. Class Support The input image, A, can be of class logical, uint8, uint16, or double (unless the 'indexed' syntax is used, in which case A cannot be of class uint16). The output image, B, is of the same class as A. Remarks If the input image A is of an integer class, all of the output values are returned as integers. If the number of pixels in the neighborhood (i.e., m*n) is even, some of the median values may not be integers. In these cases, the fractional parts are discarded. Logical input is treated similarly. For example, suppose you call medfilt2 using 2-by-2 neighborhoods, and the input image is a uint8 array that includes this neighborhood. 1 5 4 8 medfilt2 returns an output value of 4 for this neighborhood, although the true median is 4.5. 14-298 medfilt2 Example This example adds salt and pepper noise to an image, then restores the image using medfilt2. I = imread('eight.tif'); J = imnoise(I,'salt & pepper',0.02); K = medfilt2(J); imshow(J) figure, imshow(K) Algorithm medfilt2 uses ordfilt2 to perform the filtering. See Also filter2, ordfilt2, wiener2 Reference [1] Lim, Jae S. Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1990. pp. 469-476. 14-299 montage Purpose 14montage Display multiple image frames as a rectangular montage Syntax montage(I) montage(BW) montage(X,map) montage(RGB) h = montage(...) Description montage displays all of the frames of a multiframe image array in a single image object, arranging the frames so that they roughly form a square. montage(I) displays the k frames of the intensity image array I. I is m-by-n-by-1-by-k. montage(BW) displays the k frames of the binary image array BW. BW is m-by-n-by-1-by-k. montage(X,map) displays the k frames of the indexed image array X, using the colormap map for all frames. X is m-by-n-by-1-by-k. montage(RGB) displays the k frames of the truecolor image array RGB. RGB is m-by-n-by-3-by-k. h = montage(...) returns the handle to the image object. Class Support Example 14-300 The input image can be of class logical, uint8, uint16, or double. load mri montage(D,map) montage See Also immovie 14-301 nlfilter Purpose 14nlfilter Perform general sliding-neighborhood operations Syntax B = nlfilter(A,[m n],fun) B = nlfilter(A,[m n],fun,P1,P2,...) B = nlfilter(A,'indexed',...) Description B = nlfilter(A,[m n],fun) applies the function fun to each m-by-n sliding block of A. fun is a function that accepts an m-by-n matrix as input, and returns a scalar result. c = fun(x) c is the output value for the center pixel in the m-by-n block x. nlfilter calls fun for each pixel in A. nlfilter zero pads the m-by-n block at the edges, if necessary. B = nlfilter(A,[m n],fun,P1,P2,...) passes the additional parameters P1,P2,..., to fun. B = nlfilter(A,'indexed',...) processes A as an indexed image, padding with ones if A is of class double and zeros if A is of class uint8. Class Support The input image, A, can be of any class supported by fun. The class of B depends on the class of the output from fun. Remarks nlfilter can take a long time to process large images. In some cases, the colfilt function can perform the same operation much faster. Example fun can be a function_handle, created using @. This example produces the same result as calling medfilt2 with a 3-by-3 neighborhood. B = nlfilter(A,[3 3],@myfun); where myfun is an M-file containing function scalar = myfun(x) scalar = median(x(:)); fun can also be an inline object. The example above can be written as fun = inline('median(x(:))'); See Also 14-302 blkproc, colfilt normxcorr2 Purpose 14normxcorr2 Normalized two-dimensional cross-correlation Syntax C = normxcorr2(TEMPLATE,A) Description C = normxcorr2(TEMPLATE,A) computes the normalized cross-correlation of the matrices TEMPLATE and A. The matrix A must be larger than the matrix TEMPLATE for the normalization to be meaningful. The values of TEMPLATE cannot all be the same. The resulting matrix, C, contains the correlation coefficients, which may range in value from -1.0 to 1.0. Class Support The input matrices can be of class uint8, uint16, or double. Algorithm normxcorr2 uses the following general procedure: 1 Calculate cross-correlation in the spatial or the frequency domain, depending on size of images. 2 Calculate local sums by precomputing running sums. [1] 3 Use local sums to normalize the cross-correlation to get correlation coefficients. [2] Example T = .2*ones(11); % make light gray plus on dark gray background T(6,3:9) = .6; T(3:9,6) = .6; BW = T>0.5; % make white plus on black background imshow(BW), title('Binary') figure, imshow(T), title('Template') % make new image that offsets template T T_offset = .2*ones(21); offset = [3 5]; % shift by 3 rows, 5 columns T_offset( (1:size(T,1))+offset(1), (1:size(T,2))+offset(2) ) = T; figure, imshow(T_offset), title('Offset Template') % cross-correlate BW and T_offset to recover offset cc = normxcorr2(BW,T_offset); [max_cc, imax] = max(abs(cc(:))); [ypeak, xpeak] = ind2sub(size(cc),imax(1)); corr_offset = [ (ypeak-size(T,1)) (xpeak-size(T,2)) ]; isequal(corr_offset,offset) % 1 means offset was recovered 14-303 normxcorr2 See Also corrcoef References [1] J. P. Lewis, “Fast Normalized Cross-Correlation", Industrial Light & Magic. <http://www.idiom.com/~zilla/Papers/nvisionInterface/nip.html> [2] Robert M. Haralick and Linda G. Shapiro, Computer and Robot Vision, Volume II, Addison-Wesley, 1992, pp. 316-317. 14-304 ntsc2rgb Purpose 14ntsc2rgb Convert NTSC values to RGB color space Syntax rgbmap = ntsc2rgb(yiqmap) RGB = ntsc2rgb(YIQ) Description rgbmap = ntsc2rgb(yiqmap) converts the m-by-3 NTSC (television) color values in yiqmap to RGB color space. If yiqmap is m-by-3 and contains the NTSC luminance (Y) and chrominance (I and Q) color components as columns, then rgbmap is an m-by-3 matrix that contains the red, green, and blue values equivalent to those colors. Both rgbmap and yiqmap contain intensities in the range 0 to 1.0. The intensity 0 corresponds to the absence of the component, while the intensity 1.0 corresponds to full saturation of the component. RGB = ntsc2rgb(YIQ) converts the NTSC image YIQ to the equivalent truecolor image RGB. ntsc2rgb computes the RGB values from the NTSC components using 1.000 R G = 1.000 1.000 B 0.956 – 0.272 – 1.106 0.621 Y – 0.647 I 1.703 Q Class Support The input image or colormap must be of class double. The output is of class double. See Also rgb2ntsc, rgb2ind, ind2rgb, ind2gray 14-305 ordfilt2 Purpose 14ordfilt2 Perform two-dimensional order-statistic filtering Syntax B = ordfilt2(A,order,domain) B = ordfilt2(A,order,domain,S) B = ordfilt2(...,padopt) Description B = ordfilt2(A,order,domain) replaces each element in A by the order-th element in the sorted set of neighbors specified by the nonzero elements in domain. B = ordfilt2(A,order,domain,S), where S is the same size as domain, uses the values of S corresponding to the nonzero values of domain as additive offsets. B = ordfilt2(...,padopt) controls how the matrix boundaries are padded. Set padopt to 'zeros' (the default), or 'symmetric'. If padopt is 'zeros', A is padded with zeros at the boundaries. If padopt is 'symmetric', A is symmetrically extended at the boundaries. Class Support The class of A can be logical, uint8, uint16, or double. The class of B is the same as the class of A, unless the additive offset form of ordfilt2 is used, in which case the class of B is double. Remarks domain is equivalent to the structuring element used for binary image operations. It is a matrix containing only 1’s and 0’s; the 1’s define the neighborhood for the filtering operation. For example, B = ordfilt2(A,5,ones(3,3)) implements a 3-by-3 median filter; B = ordfilt2(A,1,ones(3,3)) implements a 3-by-3 minimum filter; and B = ordfilt2(A,9,ones(3,3)) implements a 3-by-3 maximum filter. B = ordfilt2(A,1,[0 1 0; 1 0 1; 0 1 0]) replaces each element in A by the minimum of its north, east, south, and west neighbors. The syntax that includes S (the matrix of additive offsets) can be used to implement grayscale morphological operations, including grayscale dilation and erosion. See Also 14-306 medfilt2 ordfilt2 Reference [1] Haralick, Robert M., and Linda G. Shapiro. Computer and Robot Vision, Volume I. Addison-Wesley, 1992. 14-307 otf2psf Purpose 14otf2psf Convert optical transfer function to point-spread function Syntax PSF = otf2psf(OTF) PSF = otf2psf(OTF,OUTSIZE) Description PSF = otf2psf(OTF) computes the inverse Fast Fourier Transform (IFFT) of the optical transfer function (OTF) array and creates a point-spread function (PSF), centered at the origin. By default, the PSF is the same size as the OTF. PSF = otf2psf(OTF,OUTSIZE) converts the OTF array into a PSF array, where OUTSIZE specifies the size of the output point-spread function. The size of the output array must not exceed the size of the OTF array in any dimension. To center the PSF at the origin, otf2psf circularly shifts the values of the output array down (or to the right) until the (1,1) element reaches the central position, then it crops the result to match dimensions specified by OUTSIZE. Note that this function is used in image convolution/deconvolution when the operations involve the FFT. Class Support Example See Also 14-308 OTF can be any nonsparse, numeric array. PSF is of class double. PSF = fspecial('gaussian',13,1); OTF = psf2otf(PSF,[31 31]); % PSF --> OTF PSF2 = otf2psf(OTF,size(PSF)); % OTF --> PSF2 subplot(1,2,1); surf(abs(OTF)); title('|OTF|'); axis square; axis tight subplot(1,2,2); surf(PSF2); title('corresponding PSF'); axis square; axis tight psf2otf, circshift, padarray padarray Purpose 14padarray Pad an array Syntax B = padarray(A,padsize,method,direction) Description B = padarray(A,padsize,method,direction) pads array A with padsize number of elements along each dimension, using specified method and direction. The N-th element of the padsize vector specifies the padding size for the N-th dimension of the array, A. method can be a numeric scalar, in which case it specifies the value for all padding elements, or one of the following strings which specify other methods used to determine pad values. By default, METHOD sets the pad value to the 0. Value Meaning 'circular' Pad with circular repetition of elements within the dimension. 'replicate' Pad by repeating border elements of array. 'symmetric' Pad array with mirror reflections of itself. direction specifies where to pad the array along each dimension. By default, direction is 'post'. Class Support Value Meaning 'pre' Pad before the first element of each dimension. 'post' Pad after the last element of each dimension. This is the default. 'both' Pads before the first element and after the last element of each dimension. When padding with a constant value, A can be numeric or logical. When padding using the 'circular', 'replicate', or 'symmetric' methods, A can be of any class. B is of the same class as A. 14-309 padarray Example Add three elements of padding to the beginning of a vector. The padding elements contain mirror copies of the array. b = padarray([1 2 3 4],3,'symmetric','pre') b = 3 2 1 1 2 3 4 Add three elements of padding to the end of the first dimension of the array and two elements of padding to the end of the second dimension. Use the value of the last array element as the padding value. B = padarray([1 2; 3 4],[3 2],'replicate','post') B = 1 3 3 3 3 2 4 4 4 4 2 4 4 4 4 2 4 4 4 4 Add three elements of padding to each dimension of a three-dimensional array. Each pad element contains the value 0. A = [ 1 2; 3 4]; B = [ 5 6; 7 8]; C = cat(3,A,B) C(:,:,1) = 1 3 2 4 C(:,:,2) = 5 7 6 8 D = padarray(C,[3 3],0,'both') 14-310 padarray See Also D(:,:,1) = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D(:,:,2) = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 7 0 0 0 0 0 0 6 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 circshift, imfilter 14-311 phantom Purpose 14phantom Generate a head phantom image Syntax P = phantom(def,n) P = phantom(E,n) [P,E] = phantom(...) Description P = phantom(def,n) generates an image of a head phantom that can be used to test the numerical accuracy of radon and iradon or other two-dimensional reconstruction algorithms. P is a grayscale intensity image that consists of one large ellipse (representing the brain) containing several smaller ellipses (representing features in the brain). def is a string that specifies the type of head phantom to generate. Valid values are: • 'Shepp-Logan' – a test image used widely by researchers in tomography. • 'Modified Shepp-Logan' (default) – a variant of the Shepp-Logan phantom in which the contrast is improved for better visual perception. n is a scalar that specifies the number of rows and columns in P. If you omit the argument, n defaults to 256. P = phantom(E,n) generates a user-defined phantom, where each row of the matrix E specifies an ellipse in the image. E has six columns, with each column containing a different parameter for the ellipses. This table describes the columns of the matrix. 14-312 Column Parameter Meaning Column 1 A Additive intensity value of the ellipse Column 2 a Length of the horizontal semi-axis of the ellipse Column 3 b Length of the vertical semi-axis of the ellipse Column 4 x0 x-coordinate of the center of the ellipse phantom Column Parameter Meaning Column 5 y0 y-coordinate of the center of the ellipse Column 6 phi Angle (in degrees) between the horizontal semi-axis of the ellipse and the x-axis of the image For purposes of generating the phantom, the domains for the x- and y-axes span [–1,1]. Columns 2 through 5 must be specified in terms of this range. [P,E] = phantom(...) returns the matrix E used to generate the phantom. Class Support All inputs and all outputs must be of class double. Remarks For any given pixel in the output image, the pixel’s value is equal to the sum of the additive intensity values of all ellipses that the pixel is a part of. If a pixel is not part of any ellipse, its value is 0. The additive intensity value A for an ellipse can be positive or negative; if it is negative, the ellipse will be darker than the surrounding pixels. Note that, depending on the values of A, some pixels may have values outside the range [0,1]. Example P = phantom('Modified Shepp-Logan',200); imshow(P) 14-313 phantom Reference [1] Jain, Anil K. Fundamentals of Digital Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1989. p. 439. See Also radon, iradon 14-314 pixval Purpose 14pixval Display information about image pixels Syntax pixval on pixval off pixval pixval(fig,option) Purpose pixval on turns on interactive display of information about image pixels in the current figure. pixval installs a black bar at the bottom of the figure, which displays the (x,y) coordinates for whatever pixel the cursor is currently over, and the color information for that pixel. If the image is binary or intensity, the color information is a single intensity value. If the image is indexed or RGB, the color information is an RGB triplet. The values displayed are the actual data values, regardless of the class of the image array, or whether the data is in normal image range. If you click on the image and hold down the mouse button while you move the cursor, pixval also displays the Euclidean distance between the point you clicked on and the current cursor location. pixval draws a line between these points to indicate the distance being measured. When you release the mouse button, the line and the distance display disappear. You can move the display bar by clicking on it and dragging it to another place in the figure. pixval off turns interactive display off in the current figure. You can also turn off the display by clicking the button on the right side of the display bar. pixval toggles interactive display on or off in the current figure. pixval(fig,option) applies the pixval command to the figure specified by fig. option is string containing 'on' or 'off'. See Also impixel, improfile 14-315 psf2otf Purpose 14psf2otf Convert point-spread function to optical transfer function. Syntax OTF = psf2otf(PSF) OTF = psf2otf(PSF,OUTSIZE) Description OTF = psf2otf(PSF) computes the Fast Fourier Transform (FFT) of the point-spread function (PSF) array and creates the optical transfer function array, OTF, that is not influenced by the PSF off-centering. By default, the OTF array is the same size as the PSF array. OTF = psf2otf(PSF,OUTSIZE) converts the PSF array into an OTF array, where OUTSIZE specifies the size of the OTF array. The OUTSIZE cannot be smaller than the PSF array size in any dimension. To ensure that the OTF is not altered due to PSF off-centering, psf2otf post-pads the PSF array (down or to the right) with zeros to match dimensions specified in OUTSIZE, then circularly shifts the values of the PSF array up (or to the left) until the central pixel reaches (1,1) position. Note that this function is used in image convolution/deconvolution when the operations involve the FFT. Class Support Example See Also 14-316 PSF can be any nonsparse, numeric array. OTF is of class double. PSF = fspecial('gaussian',13,1); OTF = psf2otf(PSF,[31 31]); % PSF --> OTF subplot(1,2,1); surf(PSF); title('PSF'); axis square; axis tight subplot(1,2,2); surf(abs(OTF)); title('corresponding |OTF|'); axis square; axis tight otf2psf, circshift, padarray qtdecomp Purpose Syntax 14qtdecomp Perform quadtree decomposition S S S S = = = = qtdecomp(I) qtdecomp(I,threshold) qtdecomp(I,threshold,mindim) qtdecomp(I,threshold,[mindim maxdim]) S = qtdecomp(I,fun) S = qtdecomp(I,fun,P1,P2,...) Description qtdecomp divides a square image into four equal-sized square blocks, and then tests each block to see if it meets some criterion of homogeneity. If a block meets the criterion, it is not divided any further. If it does not meet the criterion, it is subdivided again into four blocks, and the test criterion is applied to those blocks. This process is repeated iteratively until each block meets the criterion. The result may have blocks of several different sizes. S = qtdecomp(I) performs a quadtree decomposition on the intensity image I, and returns the quadtree structure in the sparse matrix S. If S(k,m) is nonzero, then (k,m) is the upper-left corner of a block in the decomposition, and the size of the block is given by S(k,m). By default, qtdecomp splits a block unless all elements in the block are equal. S = qtdecomp(I,threshold) splits a block if the maximum value of the block elements minus the minimum value of the block elements is greater than threshold. threshold is specified as a value between 0 and 1, even if I is of class uint8 or uint16. If I is uint8, the threshold value you supply is multiplied by 255 to determine the actual threshold to use; if I is uint16, the threshold value you supply is multiplied by 65535. S = qtdecomp(I,threshold,mindim) will not produce blocks smaller than mindim, even if the resulting blocks do not meet the threshold condition. S = qtdecomp(I,threshold,[mindim maxdim]) will not produce blocks smaller than mindim or larger than maxdim. Blocks larger than maxdim are split even if they meet the threshold condition. maxdim/mindim must be a power of 2. S = qtdecomp(I,fun) uses the function fun to determine whether to split a block. qtdecomp calls fun with all the current blocks of size m-by-m stacked into an m-by-m-by-k array, where k is the number of m-by-m blocks. fun should return 14-317 qtdecomp a k-element vector, containing only 1’s and 0’s, where 1 indicates that the corresponding block should be split, and 0 indicates it should not be split. (For example, if k(3) is 0, the third m-by-m block should not be split.) fun can be a function_handle, created using @, or an inline object. S = qtdecomp(I,fun,P1,P2,...) passes P1,P2,..., as additional arguments to fun. Class Support For the syntaxes that do not include a function, the input image can be of class logical, uint8, uint16, or double. For the syntaxes that include a function, the input image can be of any class supported by the function. The output matrix is always of class sparse. Remarks qtdecomp is appropriate primarily for square images whose dimensions are a power of 2, such as 128-by-128 or 512-by-512. These images can be divided until the blocks are as small as 1-by-1. If you use qtdecomp with an image whose dimensions are not a power of 2, at some point the blocks cannot be divided further. For example, if an image is 96-by-96, it can be divided into blocks of size 48-by-48, then 24-by-24, 12-by-12, 6-by-6, and finally 3-by-3. No further division beyond 3-by-3 is possible. To process this image, you must set mindim to 3 (or to 3 times a power of 2); if you are using the syntax that includes a function, the function must return 0 at the point when the block cannot be divided further. Example I = [1 1 1 1 20 20 20 22 1 1 1 1 22 22 22 22 1 2 1 1 20 22 20 20 S = qtdecomp(I,5); full(S) 14-318 1 1 1 1 22 20 20 20 2 4 10 20 1 5 9 13 3 5 15 25 2 6 10 14 6 6 7 7 3 7 11 15 6 8 7 7 4 8 12 16]; qtdecomp ans = 4 0 0 0 4 0 0 0 See Also 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 1 2 0 2 0 0 0 1 1 0 0 0 0 2 0 2 0 2 0 2 0 0 0 0 0 0 0 0 0 qtgetblk, qtsetblk 14-319 qtgetblk Purpose 14qtgetblk Get block values in quadtree decomposition Syntax [vals,r,c] = qtgetblk(I,S,dim) [vals,idx] = qtgetblk(I,S,dim) Description [vals,r,c] = qtgetblk(I,S,dim) returns in vals an array containing the dim-by-dim blocks in the quadtree decomposition of I. S is the sparse matrix returned by qtdecomp; it contains the quadtree structure. vals is a dim-by-dim-by-k array, where k is the number of dim-by-dim blocks in the quadtree decomposition; if there are no blocks of the specified size, all outputs are returned as empty matrices. r and c are vectors containing the row and column coordinates of the upper-left corners of the blocks. [vals,idx] = qtgetblk(I,S,dim) returns in idx a vector containing the linear indices of the upper-left corners of the blocks. Class Support I can be of class logical, uint8, uint16, or double. S is of class sparse. Remarks The ordering of the blocks in vals matches the columnwise order of the blocks in I. For example, if vals is 4-by-4-by-2, vals(:,:,1) contains the values from the first 4-by-4 block in I, and vals(:,:,2) contains the values from the second 4-by-4 block. Example This example continues the qtdecomp example. [vals,r,c] = qtgetblk(I,S,4) vals(:,:,1) = 1 1 1 1 14-320 1 1 1 1 1 2 1 1 1 1 1 1 qtgetblk vals(:,:,2) = 20 20 20 22 22 22 22 22 20 22 20 20 22 20 20 20 r = 1 5 c = 1 1 See Also qtdecomp, qtsetblk 14-321 qtsetblk Purpose 14qtsetblk Set block values in quadtree decomposition Syntax J = qtsetblk(I,S,dim,vals) Description J = qtsetblk(I,S,dim,vals) replaces each dim-by-dim block in the quadtree decomposition of I with the corresponding dim-by-dim block in vals. S is the sparse matrix returned by qtdecomp; it contains the quadtree structure. vals is a dim-by-dim-by-k array, where k is the number of dim-by-dim blocks in the quadtree decomposition. Class Support I can be of class logical, uint8, uint16, or double. S is of class sparse. Remarks The ordering of the blocks in vals must match the columnwise order of the blocks in I. For example, if vals is 4-by-4-by-2, vals(:,:,1) contains the values used to replace the first 4-by-4 block in I, and vals(:,:,2) contains the values for the second 4-by-4 block. Example This example continues the qtgetblock example. newvals = cat(3,zeros(4),ones(4)); J = qtsetblk(I,S,4,newvals) J = 0 0 0 0 1 1 1 1 See Also 14-322 0 0 0 0 1 1 1 1 qtdecomp, qtgetblk 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 2 4 10 20 1 5 9 13 3 5 15 25 2 6 10 14 6 6 7 7 3 7 11 15 6 8 7 7 4 8 12 16 radon Purpose 14radon Radon transform Syntax R = radon(I,theta) [R,xp] = radon(...) Description The radon function computes the Radon transform, which is the projection of the image intensity along a radial line oriented at a specified angle. R = radon(I,theta) returns the Radon transform of the intensity image I for the angle theta degrees. If theta is a scalar, the result R is a column vector containing the Radon transform for theta degrees. If theta is a vector, then R is a matrix in which each column is the Radon transform for one of the angles in theta. If you omit theta, it defaults to 0:179. [R,xp] = radon(...) returns a vector xp containing the radial coordinates corresponding to each row of R. Class Support I can be of class double, logical, or of any integer class. All other inputs and outputs are of class double. Remarks The radial coordinates returned in xp are the values along the x'-axis, which is oriented at theta degrees counterclockwise from the x-axis. The origin of both axes is the center pixel of the image, which is defined as floor((size(I)+1)/2) For example, in a 20-by-30 image, the center pixel is (10,15). Example iptsetpref('ImshowAxesVisible','on') I = zeros(100,100); I(25:75,25:75) = 1; theta = 0:180; [R,xp] = radon(I,theta); imshow(theta,xp,R,[],'notruesize'), colormap(hot), colorbar 14-323 radon See Also iradon, phantom References Bracewell, Ronald N. Two-Dimensional Imaging. Englewood Cliffs, NJ: Prentice Hall, 1995. pp. 505-537. Lim, Jae S. Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1990. pp. 42-45. 14-324 reflect Purpose 14reflect Reflect structuring element Syntax SE2 = reflect(SE) Description SE2 = reflect(SE) reflects a structuring element through its center. The effect is the same as if you rotated the structuring element's domain 180 degrees around its center (for a 2-D structuring element). If SE is an array of structuring element objects, then reflect(SE) reflects each element of SE, and SE2 has the same size as SE. Class Support SE and SE2 are STREL objects. Example se = strel([0 0 1; 0 0 0; 0 0 0]) se2 = reflect(se) se = Flat STREL object containing 1 neighbor. Neighborhood: 0 0 0 0 0 0 1 0 0 se2 = Flat STREL object containing 1 neighbor. Neighborhood: 0 0 0 0 1 0 See Also 0 0 0 strel 14-325 regionprops Purpose 14regionprops Measure properties of image regions Syntax STATS = regionprops(L,properties) Description STATS = regionprops(L,properties) measures a set of properties for each labeled region in the label matrix L. Positive integer elements of L correspond to different regions. For example, the set of elements of L equal to 1 corresponds to region 1; the set of elements of L equal to 2 corresponds to region 2; and so on. The return value, STATS, is a structure array of length max(L(:)). The fields of the structure array denote different measurements for each region, as specified by properties. properties can be a comma-separated list of strings, a cell array containing strings, the single string 'all', or the string 'basic'. This table lists the set of valid property strings. Property strings are case insensitive and can be abbreviated. 'Area' 'EquivDiameter' 'MajorAxisLength' 'BoundingBox' 'EulerNumber' 'MinorAxisLength' 'Centroid' 'Extent' 'Orientation' 'ConvexArea' 'Extrema' 'PixelIdxList' 'ConvexHull' 'FilledArea' 'PixelList' 'ConvexImage' 'FilledImage' 'Solidity' 'Eccentricity' 'Image' 'SubarrayIdx' If properties is the string 'all', then all of the above measurements are computed. If properties is not specified or if it is the string 'basic', then these measurements are computed: 'Area', 'Centroid', and 'BoundingBox'. Definitions 'Area' – Scalar; the actual number of pixels in the region. (This value may differ slightly from the value returned by bwarea, which weights different patterns of pixels differently.) 'Centroid' – 1-by-ndims(L) vector; the center of mass of the region. Note that the first element of Centroid is the horizontal coordinate (or x-coordinate) of 14-326 regionprops the center of mass, and the second element is the vertical coordinate (or y-coordinate). All other elements of Centroid are in order of dimension. 'BoundingBox' – 1-by-ndims(L)*2 vector; the smallest rectangle containing the region. BoundingBox is [ul_corner width], where ul_corner is in the form [x y z ...] and specifies the upper-left corner of the bounding box width is in the form [x_width y_width ...] and specifies the width of the bounding box along each dimension This figure illustrates the centroid and bounding box. The region consists of the white pixels; the green box is the bounding box, and the red dot is the centroid. 'MajorAxisLength' – Scalar; the length (in pixels) of the major axis of the ellipse that has the same second-moments as the region. This property is supported only for 2-D input label matrices. 'MinorAxisLength' – Scalar; the length (in pixels) of the minor axis of the ellipse that has the same second-moments as the region. This property is supported only for 2-D input label matrices. 'Eccentricity' – Scalar; the eccentricity of the ellipse that has the same second-moments as the region. The eccentricity is the ratio of the distance between the foci of the ellipse and its major axis length. The value is between 0 and 1. (0 and 1 are degenerate cases; an ellipse whose eccentricity is 0 is actually a circle, while an ellipse whose eccentricity is 1 is a line segment.) This property is supported only for 2-D input label matrices. 'Orientation' – Scalar; the angle (in degrees) between the x-axis and the major axis of the ellipse that has the same second-moments as the region. This property is supported only for 2-D input label matrices. 14-327 regionprops This figure illustrates the axes and orientation of the ellipse. The left side of the figure shows an image region and its corresponding ellipse. The right side shows the same ellipse, with features indicated graphically; the solid blue lines are the axes, the red dots are the foci, and the orientation is the angle between the horizontal dotted line and the major axis. 'Image' – Binary image (logical) of the same size as the bounding box of the region; the on pixels correspond to the region, and all other pixels are off. 'FilledImage' – Binary image (logical) of the same size as the bounding box of the region. The on pixels correspond to the region, with all holes filled in. 'FilledArea' – Scalar; the number of on pixels in FilledImage. This figure illustrates Image' and FilledImage'. Original image, containing a single region 'Image' 'FilledImage' 'ConvexHull' – p-by-2 matrix; the smallest convex polygon that can contain the region. Each row of the matrix contains the x- and y-coordinates of one vertex of the polygon. This property is supported only for 2-D input label matrices. 14-328 regionprops 'ConvexImage' – Binary image (logical); the convex hull, with all pixels within the hull filled in (i.e., set to on). (For pixels that the boundary of the hull passes through, regionprops uses the same logic as roipoly to determine whether the pixel is inside or outside the hull.) The image is the size of the bounding box of the region. This property is supported only for 2-D input label matrices. 'ConvexArea' – Scalar; the number of pixels in 'ConvexImage'. This property is supported only for 2-D input label matrices. 'EulerNumber' – Scalar; equal to the number of objects in the region minus the number of holes in those objects. This property is supported only for 2-D input label matrices. 'Extrema' – 8-by-2 matrix; the extremal points in the region. Each row of the matrix contains the x- and y-coordinates of one of the points. The format of the vector is [top-left top-right right-top right-bottom bottom-right bottom-left left-bottom left-top]. This property is supported only for 2-D input label matrices. This figure illustrates the extrema of two different regions. In the region on the left, each extremal point is distinct; in the region on the right, certain extremal points (e.g., top-left and left-top) are identical. top-left top-right top-left top-right left-top right-top left-top right-top left-bottom right-bottom left-bottom right-bottom bottom-left bottom-right bottom-lef t bottom-right 'EquivDiameter' – Scalar; the diameter of a circle with the same area as the region. Computed as sqrt(4*Area/pi). This property is supported only for 2-D input label matrices. 'Solidity' – Scalar; the proportion of the pixels in the convex hull that are also in the region. Computed as Area/ConvexArea. This property is supported only for 2-D input label matrices. 14-329 regionprops 'Extent' – Scalar; the proportion of the pixels in the bounding box that are also in the region. Computed as the Area divided by area of the bounding box. This property is supported only for 2-D input label matrices. 'PixelList' – p-by-ndims(L) matrix; the actual pixels in the region. Each row of the matrix has the form [x y z ...] and specifies the coordinates of one pixel in the region. Class Support The input label matrix L can have any numeric class. Remarks The comma-separated list syntax for structure arrays is very useful when working with the output of regionprops. For example, for a field that contains a scalar, you can use a this syntax to create a vector containing the value of this field for each region in the image. For instance, if stats is a structure array with field Area, then the following two expressions are equivalent stats(1).Area, stats(2).Area, ..., stats(end).Area and stats.Area Therefore, you can use these calls to create a vector containing the area of each region in the image. stats = regionprops(L,'Area'); allArea = [stats.Area]; allArea is a vector of the same length as the structure array stats. The function ismember is useful in conjunction with regionprops for selecting regions based on certain criteria. For example, these commands create a binary image containing only the regions in text.tif whose area is greater than 80. idx = find([stats.Area] > 80); BW2 = ismember(L,idx); Most of the measurements take very little time to compute. The exceptions are these, which may take significantly longer, depending on the number of regions in L: • 'ConvexHull' 14-330 regionprops • 'ConvexImage' • 'ConvexArea' • 'FilledImage' Note that computing certain groups of measurements takes about the same amount of time as computing just one of them because regionprops takes advantage of intermediate computations used in both computations. Therefore, it is fastest to compute all of the desired measurements in a single call to regionprops. Example BW = imread('text.tif'); L = bwlabel(BW); stats = regionprops(L,'all'); stats(23) ans = Area: Centroid: BoundingBox: MajorAxisLength: MinorAxisLength: Eccentricity: Orientation: ConvexHull: ConvexImage: ConvexArea: Image: FilledImage: FilledArea: EulerNumber: Extrema: EquivDiameter: Solidity: Extent: PixelList: See Also 89 [95.6742 192.9775] [87.5000 184.5000 16 15] 19.9127 14.2953 0.6961 9.0845 [28x2 double] [15x16 uint8 ] 205 [15x16 uint8 ] [15x16 uint8 ] 122 0 [ 8x2 double] 10.6451 0.4341 0.3708 [89x2 double] bwlabel, bwlabeln, ismember, watershed ismember is a MATLAB function 14-331 rgb2gray Purpose 14rgb2gray Convert an RGB image or colormap to grayscale Syntax I = rgb2gray(RGB) newmap = rgb2gray(map) Description rgb2gray converts RGB images to grayscale by eliminating the hue and saturation information while retaining the luminance. I = rgb2gray(RGB) converts the truecolor image RGB to the grayscale intensity image I. newmap = rgb2gray(map) returns a grayscale colormap equivalent to map. Class Support If the input is an RGB image, it can be of class uint8, uint16, or double. The output image, I, is of the same class as the input image. If the input is a colormap, the input and output colormaps are both of class double. Algorithm rgb2gray converts the RGB values to NTSC coordinates, sets the hue and saturation components to zero, and then converts back to RGB color space. See Also ind2gray, ntsc2rgb, rgb2ind, rgb2ntsc 14-332 rgb2hsv Purpose 14rgb2hsv Convert RGB values to hue-saturation-value (HSV) color space rgb2hsv is a function in MATLAB. To get help for this function, select MATLAB Help from the Help menu and view the online function reference pages. 14-333 rgb2ind Purpose 14rgb2ind Convert an RGB image to an indexed image Syntax [X,map] = rgb2ind(RGB,tol) [X,map] = rgb2ind(RGB,n) X = rgb2ind(RGB,map) [...] = rgb2ind(...,dither_option) Description rgb2ind converts RGB images to indexed images using one of three different methods: uniform quantization, minimum variance quantization, and colormap mapping. For all of these methods, rgb2ind also dithers the image unless you specify 'nodither' for dither_option. [X,map] = rgb2ind(RGB,tol) converts the RGB image to an indexed image X using uniform quantization. map contains at most (floor(1/tol)+1)^3 colors. tol must be between 0 and 1.0. [X,map] = rgb2ind(RGB,n) converts the RGB image to an indexed image X using minimum variance quantization. map contains at most n colors. n must be less than or equal to 65536. X = rgb2ind(RGB,map) converts the RGB image to an indexed image X with colormap map by matching colors in RGB with the nearest color in the colormap map. size(map,1) must be less than or equal to 65536. [...] = rgb2ind(...,dither_option) enables or disables dithering. dither_option is a string that can have one of these values: • 'dither' (default) dithers, if necessary, to achieve better color resolution at the expense of spatial resolution. • 'nodither' maps each color in the original image to the closest color in the new map. No dithering is performed. Class Support The input image can be of class uint8, uint16, or double. If the length of map is less than or equal to 256, the output image is of class uint8. Otherwise, the output image is of class uint16. Remarks If you specify tol, rgb2ind uses uniform quantization to convert the image. This method involves cutting the RGB color cube into smaller cubes of length tol. For example, if you specify a tol of 0.1, the edges of the cubes are one-tenth the length of the RGB cube. The total number of small cubes is 14-334 rgb2ind n = (floor(1/tol)+1)^3 Each cube represents a single color in the output image. Therefore, the maximum length of the colormap is n. rgb2ind removes any colors that don’t appear in the input image, so the actual colormap may be much smaller than n. If you specify n, rgb2ind uses minimum variance quantization. This method involves cutting the RGB color cube into smaller boxes (not necessarily cubes) of different sizes, depending on how the colors are distributed in the image. If the input image actually uses fewer colors than the number you specify, the output colormap is also smaller. If you specify map, rgb2ind uses colormap mapping, which involves finding the colors in map that best match the colors in the RGB image. Example See Also RGB = imread('flowers.tif'); [X,map] = rgb2ind(RGB,128); imshow(X,map) cmunique, dither, imapprox, ind2rgb, rgb2gray 14-335 rgb2ntsc Purpose 14rgb2ntsc Convert RGB values to NTSC color space Syntax yiqmap = rgb2ntsc(rgbmap) YIQ = rgb2ntsc(RGB) Description yiqmap = rgb2ntsc(rgbmap) converts the m-by-3 RGB values in rbgmap to NTSC color space. yiqmap is an m-by-3 matrix that contains the NTSC luminance (Y) and chrominance (I and Q) color components as columns that are equivalent to the colors in the RGB colormap. YIQ = rgb2ntsc(RGB) converts the truecolor image RGB to the equivalent NTSC image YIQ. Remarks In the NTSC color space, the luminance is the grayscale signal used to display pictures on monochrome (black and white) televisions. The other components carry the hue and saturation information. rgb2ntsc defines the NTSC components using Y 0.299 = I 0.596 Q 0.211 0.587 – 0.274 – 0.523 0.114 R – 0.322 G 0.312 B Class Support If the input is an RGB image, it can be of class uint8, uint16, or double; the output image is of class double. If the input is a colormap, the input and output colormaps are both of class double. See Also ntsc2rgb, rgb2ind, ind2rgb, ind2gray 14-336 rgb2ycbcr Purpose 14rgb2ycbcr Convert RGB values to YCbCr color space Syntax ycbcrmap = rgb2ycbcr(rgbmap) YCBCR = rgb2ycbcr(RGB) Description ycbcrmap = rgb2ycbcr(rgbmap) converts the RGB values in rbgmap to the YCbCr color space. ycbcrmap is an m-by-3 matrix that contains the YCbCr luminance (Y) and chrominance (Cb and Cr) color components as columns. Each row represents the equivalent color to the corresponding row in the RGB colormap. YCBCR = rgb2ycbcr(RGB) converts the truecolor image RGB to the equivalent image in the YCbCr color space. Class Support If the input is an RGB image, it can be of class uint8, uint16, or double; the output image is of the same class as the input image. If the input is a colormap, the input and output colormaps are both of class double. See Also ntsc2rgb, rgb2ntsc, ycbcr2rgb 14-337 rgbplot Purpose 14rgbplot Plot colormap rgbplot is a MATLAB function. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-338 roicolor Purpose 14roicolor Select region of interest, based on color Syntax BW = roicolor(A,low,high) BW = roicolor(A,v) Description roicolor selects a region of interest within an indexed or intensity image and returns a binary image. (You can use the returned image as a mask for masked filtering using roifilt2.) BW = roicolor(A,low,high) returns a region of interest selected as those pixels that lie within the colormap range [low high]. BW = (A >= low) & (A <= high) BW is a binary image with 0’s outside the region of interest and 1’s inside. BW = roicolor(A,v) returns a region of interest selected as those pixels in A that match the values in vector v. BW is a binary image with 1’s where the values of A match the values of v. Class Support Example See Also The input image, A, must be numeric. The output image, BW, is of class logical. I = imread('rice.tif'); BW = roicolor(I,128,255); imshow(I); figure, imshow(BW) roifilt2, roipoly 14-339 roifill Purpose Syntax 14roifill Smoothly interpolate within an arbitrary image region J = roifill(I,c,r) J = roifill(I) J = roifill(I,BW) [J,BW] = roifill(...) J = roifill(x,y,I,xi,yi) [x,y,J,BW,xi,yi] = roifill(...) Description roifill fills in a specified polygon in an intensity image. It smoothly interpolates inward from the pixel values on the boundary of the polygon by solving Laplace’s equation. roifill can be used, for example, to “erase” small objects in an image. J = roifill(I,c,r) fills in the polygon specified by c and r, which are equal-length vectors containing the row-column coordinates of the pixels on vertices of the polygon. The k-th vertex is the pixel (r(k),c(k)). J = roifill(I) displays the image I on the screen and lets you specify the polygon using the mouse. If you omit I, roifill operates on the image in the current axes. Use normal button clicks to add vertices to the polygon. Pressing Backspace or Delete removes the previously selected vertex. A shift-click, right-click, or double-click adds a final vertex to the selection and then starts the fill; pressing Return finishes the selection without adding a vertex. J = roifill(I,BW) uses BW (a binary image the same size as I) as a mask. roifill fills in the regions in I corresponding to the nonzero pixels in BW. If there are multiple regions, roifill performs the interpolation on each region independently. [J,BW] = roifill(...) returns the binary mask used to determine which pixels in I get filled. BW is a binary image the same size as I with 1’s for pixels corresponding to the interpolated region of I and 0’s elsewhere. J = roifill(x,y,I,xi,yi) uses the vectors x and y to establish a nondefault spatial coordinate system. xi and yi are equal-length vectors that specify polygon vertices as locations in this coordinate system. 14-340 roifill [x,y,J,BW,xi,yi] = roifill(...) returns the XData and YData in x and y; the output image in J; the mask image in BW; and the polygon coordinates in xi and yi. xi and yi are empty if the roifill(I,BW) form is used. If roifill is called with no output arguments, the resulting image is displayed in a new figure. Class Support Example See Also The input image I can of class uint8, uint16, or double. The input binary mask, BW, can be any numeric class or logical. The output binary mask, BW, is always logical.The output image J is of the same class as I. All other inputs and outputs are of class double. I = imread('eight.tif'); c = [222 272 300 270 221 194]; r = [21 21 75 121 121 75]; J = roifill(I,c,r); imshow(I) figure, imshow(J) roifilt2, roipoly 14-341 roifilt2 Purpose 14roifilt2 Filter a region of interest Syntax J = roifilt2(h,I,BW) J = roifilt2(I,BW,fun) J = roifilt2(I,BW,fun,P1,P2,...) Description J = roifilt2(h,I,BW) filters the data in I with the two-dimensional linear filter h. BW is a binary image the same size as I that is used as a mask for filtering. roifilt2 returns an image that consists of filtered values for pixels in locations where BW contains 1’s, and unfiltered values for pixels in locations where BW contains 0’s. For this syntax, roifilt2 calls filter2 to implement the filter. J = roifilt2(I,BW,fun) processes the data in I using the function fun. The result J contains computed values for pixels in locations where BW contains 1’s, and the actual values in I for pixels in locations where BW contains 0’s. fun can be a function_handle, created using @, or an inline object. fun should take a matrix as a single argument and return a matrix of the same size. y = fun(x) J = roifilt2(I,BW,fun,P1,P2,...) passes the additional parameters P1,P2,..., to fun. Class Support For the syntax that includes a filter, h, the input image, I, can be of class uint8, uint16, or double, and the output array, J, has the same class as the input image. For the syntax that includes a function, I can be of any class supported by fun, and the class of J depends on the class of the output from fun. Example This example continues the roipoly example. I = imread('eight.tif'); c = [222 272 300 270 221 194]; r = [21 21 75 121 121 75]; BW = roipoly(I,c,r); h = fspecial('unsharp'); J = roifilt2(h,I,BW); imshow(J), figure, imshow(J) 14-342 roifilt2 See Also filter2, roipoly 14-343 roipoly Purpose Syntax 14roipoly Select a polygonal region of interest BW = roipoly(I,c,r) BW = roipoly(I) BW = roipoly(x,y,I,xi,yi) [BW,xi,yi] = roipoly(...) [x,y,BW,xi,yi] = roipoly(...) Description Use roipoly to select a polygonal region of interest within an image. roipoly returns a binary image that you can use as a mask for masked filtering. BW = roipoly(I,c,r) returns the region of interest selected by the polygon described by vectors c and r. BW is a binary image the same size as I with 0’s outside the region of interest and 1’s inside. BW = roipoly(I) displays the image I on the screen and lets you specify the polygon using the mouse. If you omit I, roipoly operates on the image in the current axes. Use normal button clicks to add vertices to the polygon. Pressing Backspace or Delete removes the previously selected vertex. A shift-click, right-click, or double-click adds a final vertex to the selection and then starts the fill; pressing Return finishes the selection without adding a vertex. BW = roipoly(x,y,I,xi,yi) uses the vectors x and y to establish a nondefault spatial coordinate system. xi and yi are equal-length vectors that specify polygon vertices as locations in this coordinate system. [BW,xi,yi] = roipoly(...) returns the polygon coordinates in xi and yi. Note that roipoly always produces a closed polygon. If the points specified describe a closed polygon (i.e., if the last pair of coordinates is identical to the first pair), the length of xi and yi is equal to the number of points specified. If the points specified do not describe a closed polygon, roipoly adds a final point having the same coordinates as the first point. (In this case the length of xi and yi is one greater than the number of points specified.) [x,y,BW,xi,yi] = roipoly(...) returns the XData and YData in x and y; the mask image in BW; and the polygon coordinates in xi and yi. If roipoly is called with no output arguments, the resulting image is displayed in a new figure. 14-344 roipoly Class Support The input image I can be of class uint8, uint16, or double. The output image BW is of class logical. All other inputs and outputs are of class double. Remarks For any of the roipoly syntaxes, you can replace the input image I with two arguments, m and n, that specify the row and column dimensions of an arbitrary image. For example, these commands create a 100-by-200 binary mask. c = [112 112 79 79]; r = [37 66 66 37]; BW = roipoly(100,200,c,r); If you specify m and n with an interactive form of roipoly, an m-by-n black image is displayed, and you use the mouse to specify a polygon within this image. Example See Also I = imread('eight.tif'); c = [222 272 300 270 221 194]; r = [21 21 75 121 121 75]; BW = roipoly(I,c,r); imshow(I) figure, imshow(BW) roifilt2, roicolor, roifill 14-345 std2 Purpose 14std2 Compute the standard deviation of the elements of a matrix Syntax b = std2(A) Description b = std2(A) computes the standard deviation of the values in A. Class Support A can be numeric or logical. B is a scalar of class double. Algorithm std2 computes the standard deviation of the array A using std(A(:)). See Also corr2, mean2 std, mean in the MATLAB Function Reference 14-346 strel Purpose 14strel Create morphological structuring element Syntax SE = strel(shape,parameters) Description SE = strel(shape,parameters) creates a structuring element, SE, of the type specified by shape. This table lists all the supported shapes. Depending on shape, strel may take additional parameters. See the syntax descriptions that follow for details about creating each type of structuring element. Flat Structuring Elements 'arbitrary' 'pair' 'diamond' 'periodicline' 'disk' 'rectangle' 'line' 'square' 'octagon' Nonflat Structuring Elements 'arbitrary' 'ball' SE = strel('arbitrary',NHOOD) creates a flat structuring element, where NHOOD specifies the neighborhood. NHOOD is a matrix containing 1's and 0's; the location of the 1's defines the neighborhood for the morphological operation. The center (or origin) of NHOOD is it's center element, given by floor((size(NHOOD)+1)/2). You can omit the 'arbitrary' string and just use strel(NHOOD). Origin SE = 1 0 0 1 0 0 1 0 1 NHOOD = [ 1 0 0; 1 0 0; 1 0 1]; SE = strel('arbitrary',NHOOD,HEIGHT) creates a nonflat structuring element, where NHOOD specifies the neighborhood. HEIGHT is a matrix the same size as NHOOD containing the height values associated with each nonzero 14-347 strel element of NHOOD. The HEIGHT matrix must be real and finite valued. You can omit the 'arbitrary' string and just use strel(NHOOD,HEIGHT). SE = strel('ball',R,H,N) creates a nonflat, “ball-shaped” (actually an ellipsoid) structuring element whose radius in the X-Y plane is R and whose height is H. Note that R must be a nonnegative integer, H must be a real scalar, and N must be an even nonnegative integer. When N is greater than 0, the ball-shaped structuring element is approximated by a sequence of N nonflat, line-shaped structuring elements. When N equals 0, no approximation is used, and the structuring element members comprise all pixels whose centers are no greater than R away from the origin. The corresponding height values are determined from the formula of the ellipsoid specified by R and H. If N is not specified, the default value is 8. Note Morphological operations run much faster when the structuring element uses approximations (N>0) than when it does not (N=0). SE = strel('diamond',R) creates a flat, diamond-shaped structuring element, where R specifies the distance from the structuring element origin to the points of the diamond. R must be a nonnegative integer scalar. Origin SE = 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 R=3 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 SE = strel('disk',R,N) creates a flat, disk-shaped structuring element, where R specifies the radius. R must be a nonnegative integer. N must be 0, 4, 6, or 8. When N is greater than 0, the disk-shaped structuring element is approximated by a sequence of N periodic-line structuring elements. When N equals 0, no approximation is used, and the structuring element members comprise all pixels whose centers are no greater than R away from the origin. If N is not specified, the default value is 4. 14-348 strel Note Morphological operations run much faster when the structuring element uses approximations (N>0) than when it does not (N=0). However, structuring elements that do not use approximations (N=0) are not suitable for computing granulometries. Sometimes it is necessary for strel to use two extra line structuring elements in the approximation, in which case the number of decomposed structuring elements used is N+2. Origin SE= 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 R=31 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 SE = strel('line',LEN,DEG) creates a flat, linear structuring element, where LEN specifies the length, and DEG specifies the angle (in degrees) of the line, as measured in a counterclockwise direction from the horizontal axis. LEN is approximately the distance between the centers of the structuring element members at opposite ends of the line. Origin SE= 1 1 1 1 1 LEN=9 1 1 1 1 DEG=0 Origin 0 0 1 0 1 0 0 3 0 1 LE N = DEG = 45 SE = strel('octagon',R) creates a flat, octagonal structuring element, where R specifies the distance from the structuring element origin to the sides of the 14-349 strel octagon, as measured along the horizontal and vertical axes. R must be a nonnegative multiple of 3. SE= Origin 0 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 R=3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 SE = strel('pair',OFFSET) creates a flat structuring element containing two members. One member is located at the origin. The second member's location is specified by the vector OFFSET. OFFSET must be a two-element vector of integers. SE= Origin 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 OFFSET= [2 2] SE = strel('periodicline',P,V) creates a flat structuring element containing 2*P+1 members. V is a two-element vector containing integer-valued row and column offsets. One structuring element member is located at the origin. The other members are located at 1*V, -1*V, 2*V, -2*V, ..., P*V, -P*V. Origin SE= P=2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 V= [1 -2] SE = strel('rectangle',MN) creates a flat, rectangle-shaped structuring element, where MN specifies the size. MN must be a two-element vector of nonnegative integers. The first element of MN is the number rows in the 14-350 strel structuring element neighborhood; the second element is the number of columns. SE= Origin 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 MN=[3 5] SE = strel('square',W) creates a square structuring element whose width is W pixels. W must be a nonnegative integer scalar. Origin SE= 1 1 1 1 1 1 1 1 1 W=3 Notes For all shapes except 'arbitrary', structuring elements are constructed using a family of techniques known collectively as structuring element decomposition. The principle is that dilation by some large structuring elements can be computed faster by dilation with a sequence of smaller structuring elements. For example, dilation by an 11-by-11 square structuring element can be accomplished by dilating first with a 1-by-11 structuring element and then with an 11-by-1 structuring element. This results in a theoretical performance improvement of a factor of 5.5, although in practice the actual performance improvement is somewhat less. Structuring element decompositions used for the 'disk' and 'ball' shapes are approximations; all other decompositions are exact. Methods This table lists the methods supported by the STREL object. getheight Get height of structuring element getneighbors Get structuring element neighbor locations and heights getnhood Get structuring element neighborhood 14-351 strel Example Algorithm getsequence Extract sequence of decomposed structuring elements isflat Return true for flat structuring element reflect Reflect structuring element translate Translate structuring element se1 se2 se3 se4 = = = = strel('square',11) strel('line',10,45) strel('disk',15) strel('ball',15,5) % 11-by-11 square % line, length 10, angle 45 degrees % disk, radius 15 % ball, radius 15, height 5 The method used to decompose diamond-shaped structuring elements is known as “logarithmic decomposition” [1]. The method used to decompose disk structuring elements is based on the technique called “radial decomposition using periodic lines” [2], [3]. For details, see the MakeDiskStrel subfunction in toolbox/images/images/@strel/strel.m. The method used to decompose ball structuring elements is the technique called “radial decomposition of sphere” [2]. See Also imdilate, imerode References [1] Rein van den Boomgard and Richard van Balen, “Methods for Fast Morphological Image Transforms Using Bitmapped Images,” Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing, vol. 54, no. 3, May 1992, pp. 252-254. [2] Rolf Adams, “Radial Decomposition of Discs and Spheres,” Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing, vol. 55, no. 5, September 1993, pp. 325-332. [3] Ronald Jones and Pierre Soille, “Periodic lines: Definition, cascades, and application to granulometrie,” Pattern Recognition Letters, vol. 17, 1996, 1057-1063. 14-352 stretchlim Purpose 14stretchlim Find limits to contrast stretch an image Syntax LOW_HIGH = stretchlim(I,TOL) LOW_HIGH = stretchlim(RGB,TOL) Description LOW_HIGH = stretchlim(I,TOL) returns a pair of intensities that can be used by imadjust to increase the contrast of an image. TOL = [LOW_FRACT HIGH_FRACT] specifies the fraction of the image to saturate at low and high intensities. If TOL is a scalar, TOL = LOW_FRACT, and HIGH_FRACT = 1 - LOW_FRACT, which saturates equal fractions at low and high intensities. If you omit the argument, TOL defaults to [0.01 0.99], saturating 2%. If TOL = 0, LOW_HIGH = [min(I(:)) max(I(:))]. LOW_HIGH = stretchlim(RGB,TOL) returns a 2-by-3 matrix of intensity pairs to saturate each plane of the RGB image. TOL specifies the same fractions of saturation for each plane. Class Support Example The input image can be of class uint8, uint16, or double. The output intensities returned, LOW_HIGH, are of class double and have values between 0 and 1. I = imread('pout.tif'); J = imadjust(I,stretchlim(I),[]); imshow(I), figure, imshow(J) 14-353 stretchlim See Also 14-354 brighten, histeq, imadjust subimage Purpose 14subimage Display multiple images in the same figure Syntax subimage(X,map) subimage(I) subimage(BW) subimage(RGB) subimage(x,y,...) h = subimage(...) Description You can use subimage in conjunction with subplot to create figures with multiple images, even if the images have different colormaps. subimage works by converting images to truecolor for display purposes, thus avoiding colormap conflicts. subimage(X,map) displays the indexed image X with colormap map in the current axes. subimage(I) displays the intensity image I in the current axes. subimage(BW) displays the binary image BW in the current axes. subimage(RGB) displays the truecolor image RGB in the current axes. subimage(x,y...) displays an image using a nondefault spatial coordinate system. h = subimage(...) returns a handle to an image object. Class Support Example The input image can be of class logical, uint8, uint16, or double. load trees [X2,map2] = imread('forest.tif'); subplot(1,2,1), subimage(X,map) subplot(1,2,2), subimage(X2,map2) 14-355 subimage See Also imshow subplot in the MATLAB Function Reference 14-356 tformarray Purpose 14tformarray Spatial transformation of a multidimensional array Syntax B = tformarray(A,T,R,TDIMS_A,TDIMS_B,TSIZE_B,TMAP_B,F) Description B = tformarray(A,T,R,TDIMS_A,TDIMS_B,TSIZE_B,TMAP_B,F) applies a spatial transformation to array A to produce array B. The tformarray function is like imtransform, but is intended for problems involving higher-dimensioned arrays or mixed input/output dimensionality, or requiring greater user control or customization. (Anything that can be accomplished with imtransform can be accomplished with a combination of maketform, makeresampler, findbounds, and tformarray; but for many tasks involving 2-D images, imtransform is simpler.) This table provides a brief description of all the input arguments. See the following section for more detail about each argument. (Click on an argument in the table to move to the appropriate section.) Argument Description A Input array or image T Spatial transformation structure, called a TFORM, typically created with maketform R Resampler structure, typically created with makeresampler TDIMS_A Row vector listing the input transform dimensions TDIMS_B Row vector listing the output transform dimensions TMAP_B Output array size in the transform dimensions TSIZE_B Array of point locations in output space; can be used as an alternative way to specify a spatial transformation F Array of fill values A can be any nonsparse numeric array, and can be real or complex. 14-357 tformarray T is a TFORM structure that defines a particular spatial transformation. For each location in the output transform subscript space (as defined by TDIMS_B and TSIZE_B), tformarray uses T and the function tforminv to compute the corresponding location in the input transform subscript space (as defined by TDIMS_A and size(A)). If T is empty, tformarray operates as a direct resampling function, applying the resampler defined in R to compute values at each transform space location defined in TMAP_B (if TMAP_B is non-empty), or at each location in the output transform subscript grid. R is a structure that defines how to interpolate values of the input array at specified locations. R is usually created with makeresampler, which allows fine control over how to interpolate along each dimension, as well as what input array values to use when interpolating close the edge of the array. TDIMS_A and TDIMS_B indicate which dimensions of the input and output arrays are involved in the spatial transformation. Each element must be unique, and must be a positive integer. The entries need not be listed in increasing order, but the order matters. It specifies the precise correspondence between dimensions of arrays A and B and the input and output spaces of the transformer, T. length(TDIMS_A) must equal T.ndims_in, and LENGTH(TDIMS_B) must equal T.ndims_out. For example, if T is a 2-D transformation, TDIMS_A = [2 1], and TDIMS_B = [1 2], then the column dimension and row dimension of A correspond to the first and second transformation input-space dimensions, respectively. The row and column dimensions of B correspond to the first and second output-space dimensions, respectively. TSIZE_B specifies the size of the array B along the output-space transform dimensions. Note that the size of B along nontransform dimensions is taken directly from the size of A along those dimensions. If, for example, T is a 2-D transformation, size(A) = [480 640 3 10], TDIMS_B is [2 1], and TSIZE_B is [300 200], then size(B) is [200 300 3]. TMAP_B is an optional array that provides an alternative way of specifying the correspondence between the position of elements of B and the location in output transform space. TMAP_B can be used, for example, to compute the result of an image warp at a set of arbitrary locations in output space. If TMAP_B is not empty, then the size of TMAP_B takes the form: 14-358 tformarray [D1 D2 D3 ... DN L] where N equals length(TDIMS_B). The vector [D1 D2 ... DN] is used in place of TSIZE_B. If TMAP_B is not empty, then TSIZE_B should be []. The value of L depends on whether or not T is empty. If T is not empty, then L is T.ndims_out, and each L-dimension point in TMAP_B is transformed to an input-space location using T. If T is empty, then L is length(TDIMS_A), and each L-dimensional point in TMAP_B is used directly as a location in input space. F is a double-precision array containing fill values. The fill values in F may be used in three situations: • When a separable resampler is created with makeresampler and its padmethod is set to either 'fill' or 'bound'. • When a custom resampler is used that supports the 'fill' or 'bound' pad methods (with behavior that is specific to the customization). • When the map from the transform dimensions of B to the transform dimensions of A is deliberately undefined for some points. Such points are encoded in the input transform space by NaNs in either TMAP_B or in the output of TFORMINV. In the first two cases, fill values are used to compute values for output locations that map outside or near the edges of the input array. Fill values are copied into B when output locations map well outside the input array. See makeresampler for more information about 'fill' and 'bound'. F can be a scalar (including NaN), in which case its value is replicated across all the nontransform dimensions. F can also be a nonscalar, whose size depends on size(A) in the nontransform dimensions. Specifically, if K is the J-th nontransform dimension of A, then size(F,J) must be either size(A,K) or 1. As a convenience to the user, tformarray replicates F across any dimensions with unit size such that after the replication size(F,J) equals size(A,K). For example, suppose A represents 10 RGB images and has size 200-by-200-by-3-by-10, T is a 2-D transformation, and TDIMS_A and TDIMS_B are both [1 2]. In other words, tformarray will apply the same 2-D transform to each color plane of each of the 10 RGB images. In this situation you have several options for F: 14-359 tformarray • F can be a scalar, in which case the same fill value is used for each color plane of all 10 images. • F can be a 3-by-1 vector, [R G B]'. Then R, G, and B will be used as the fill values for the corresponding color planes of each of the 10 images. This can be interpreted as specifying an RGB “fill color,” with the same color used for all 10 images. • F can be a 1-by-10 vector. This can be interpreted as specifying a different fill value for each of 10 images, with that fill value being used for all three color planes. • F can be a 3-by-10 matrix, which can be interpreted as supplying a different RGB fill-color for each of the 10 images. Class Support A can be any nonsparse numeric array, and can be real or complex. It can also be of class logical. Example Create a 2-by-2 checkerboard image where each square is 20 pixels wide, then transform it with a projective transformation. Use a pad method of 'circular' when creating a resampler, so that the output appears to be a perspective view of an infinite checkerboard. Swap the output dimensions. Specify a 100-by-100 output image. Leave TMAP_B empty, since TSIZE_B is specified. Leave the fill value empty, since it won't be needed. I = checkerboard(20,1,1); figure; imshow(I) T = maketform('projective',[1 1; 41 1; 41 41; 1 41],... [5 5; 40 5; 35 30; -10 30]); R = makeresampler('cubic','circular'); J = tformarray(I,T,R,[1 2],[2 1],[100 100],[],[]); figure; imshow(J) See Also 14-360 findbounds, imtransform, makeresampler, maketform tformfwd Purpose 14tformfwd Apply forward spatial transformation Syntax X = tformfwd(U,T) Description X = tformfwd(U,T) applies the forward transformation defined in T to each row of U, if U is a matrix. T is a TFORM struct created with maketform, fliptform, or cp2tform. The argument U is an M-by-T.ndims_in matrix. Each row of U represents a point in T’s input space. The argument X is an M-by-T.ndims_out matrix. Each row of X represents a point in T’s output space, and is the forward transformation of the corresponding row of U. If U is an (N+1)-dimensional array (including an implicit trailing singleton dimension if T.ndims_in is 1), tformfwd applies the transformation to each point U(P1,P2,...,PN,:). The value size(U,N+1) must equal T.ndims_in. The returned array, X, is an (N+1)-dimensional array containing the results of each transformation in X(P1,P2,...PN,:). The value size(X,I) equals size(U,I) for I=1,...,N and size(X,N+1) equals T.ndims_out. Example Create an affine transformation that maps the triangle with vertices (0,0), (6,3), (-2,5) to the triangle with vertices (-1,-1), (0,-10), (-2,5). u = [0 0; 6 3; -2 5]; x = [-1 -1; 0 -10; 4 4]; tform = maketform('affine',u,x); Validate the mapping by applying tformfwd to u. tformfwd(u,tform) See Also % Result should equal x cp2tform, fliptform, maketform, tforminv 14-361 tforminv Purpose 14tforminv Apply inverse spatial transformation Syntax U = tforminv(X,T) Description U = tforminv(X,T) applies the inverse transformation defined in T to each row of X, if X is a matrix. T is a TFORM struct created with maketform, fliptform, or cp2tform. The argument X is an M-by-T.ndims_out matrix. Each row of X represents a point in T’s output space. U is an M-by-T.ndims_out matrix. Each row of U represents a point in T’s input space, and is the inverse transformation of the corresponding row of X. If X is an (N+1)-dimensional array (including an implicit trailing singleton dimension if T.ndims_out is 1), then tforminv applies the transformation to each point X(P1,P2,...,PN,:). size(X,N+1) must equal T.ndims_in. U is an (N+1)-dimensional array containing the results of each transformation in U(P1,P2,...PN,:). size(U,I) equals size(X,I) for I = 1,...,N and size(U,N+1) equals T.ndims_in. Example Create an affine transformation that maps the triangle with vertices (0,0), (6,3), (-2,5) to the triangle with vertices (-1,-1), (0,-10), (-2,5). u = [0 0; 6 3; -2 5]; x = [-1 -1; 0 -10; 4 4]; tform = maketform('affine',u,x); Validate the mapping by applying tforminv to x. tforminv(x,tform) See Also 14-362 % Result should equal u cp2tform, tforminv, maketform, fliptform translate Purpose 14translate Translate structuring element Syntax SE2 = translate(SE,V) Description SE2 = reflect(SE,V) translates a structuring element SE in N-D space. V is an N-element vector containing the offsets of the desired translation in each dimension. Class Support SE and SE2 are STREL objects; V is a vector of double precision values. Example Dilating with a translated version of strel(1) is a way to translate the input image in space. This example translates the cameraman.tif image down and to the right by 25 pixels. I = imread('cameraman.tif'); se = translate(strel(1), [25 25]); J = imdilate(I,se); imshow(I), title('Original') figure, imshow(J), title('Translated'); See Also strel, reflect 14-363 truesize Purpose 14truesize Adjust display size of an image Syntax truesize(fig,[mrows mcols]) truesize(fig) Description truesize(fig,[mrows ncols]) adjusts the display size of an image. fig is a figure containing a single image or a single image with a colorbar. [mrows ncols] is a 1-by-2 vector that specifies the requested screen area in pixels that the image should occupy. truesize(fig) uses the image height and width for [mrows ncols]. This results in the display having one screen pixel for each image pixel. If you omit the figure argument, truesize works on the current figure. Remarks If the'TruesizeWarning' toolbox preference is 'on', truesize displays a warning if the image is too large to fit on the screen. (The entire image is still displayed, but at less than true size.) If 'TruesizeWarning' is 'off', truesize does not display the warning. Note that this preference applies even when you call truesize indirectly, such as through imshow. See Also imshow, iptsetpref, iptgetpref 14-364 uint8 Purpose 14uint8 Convert data to unsigned 8-bit integers uint8 is a MATLAB built-in function. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-365 uint16 Purpose 14uint16 Convert data to unsigned 16-bit integers uint16 is a MATLAB built-in function. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-366 warp Purpose 14warp Display an image as a texture-mapped surface Syntax warp(X,map) warp(I,n) warp(BW) warp(RGB) warp(z,...) warp(x,y,z,...) h = warp(...) Description warp(X,map) displays the indexed image X with colormap map as a texture map on a simple rectangular surface. warp(I,n) displays the intensity image I with gray scale colormap of length n as a texture map on a simple rectangular surface. warp(BW) displays the binary image BW as a texture map on a simple rectangular surface. warp(RGB) displays the RGB image in the array RGB as a texture map on a simple rectangular surface. warp(z,...) displays the image on the surface z. warp(x,y,z...) displays the image on the surface (x,y,z). h = warp(...) returns a handle to a texture mapped surface. Class Support The input image can be of class logical, uint8, uint16, or double. Remarks Texture-mapped surfaces generally render more slowly than images. Example This example texture maps an image of a test pattern onto a cylinder. [x,y,z] = cylinder; I = imread('testpat1.tif'); warp(x,y,z,I); 14-367 warp See Also imshow image, imagesc, surf in the MATLAB Function Reference 14-368 watershed Purpose 14watershed Find image watershed regions Syntax L = watershed(A) L = watershed(A,CONN) Description L = watershed(A) computes a label matrix identifying the watershed regions of the input matrix A, which can have any dimension. The elements of L are integer values greater than or equal to 0. The elements labeled 0 do not belong to a unique watershed region. These are called watershed pixels. The elements labeled 1 belong to the first watershed region, the elements labeled 2 belong to the second watershed region, and so on. By default, watershed uses 8-connected neighborhoods for 2-D inputs and 26-connected neighborhoods for 3-D inputs. For higher dimensions, watershed uses the connectivity given by conndef(ndims(A),'maximal'). L = watershed(A,CONN) specifies the connectivity to be used in the watershed computation. CONN may have any of the following scalar values. Value Meaning Two-dimensional connectivities 4 4-connected neighborhood 8 8-connected neighborhood Three-dimensional connectivities 6 6-connected neighborhood 18 18-connected neighborhood 26 26-connected neighborhood Connectivity may be defined in a more general way for any dimension by using for CONN a 3-by-3-by- ...-by-3 matrix of 0’s and 1’s. The 1-valued elements define neighborhood locations relative to the center element of CONN. Note that CONN must be symmetric about its center element. 14-369 watershed Class Support A can be a numeric or logical array of any dimension, and it must be nonsparse. The output array L is of class double. Example 2-D Example: 1 Make a binary image containing two overlapping circular objects. center1 = -10; center2 = -center1; dist = sqrt(2*(2*center1)^2); radius = dist/2 * 1.4; lims = [floor(center1-1.2*radius) ceil(center2+1.2*radius)]; [x,y] = meshgrid(lims(1):lims(2)); bw1 = sqrt((x-center1).^2 + (y-center1).^2) <= radius; bw2 = sqrt((x-center2).^2 + (y-center2).^2) <= radius; bw = bw1 | bw2; figure, imshow(bw,'n'), title('bw') 2 Compute the distance transform of the complement of the binary image. D = bwdist(~bw); figure, imshow(D,[],'n'), title('Distance transform of ~bw') 3 Complement the distance transform, and force pixels that don’t belong to the objects to be at -Inf. D = -D; D(~bw) = -Inf; 4 Compute the watershed transform and display it as an indexed image. L = watershed(D); rgb = label2rgb(L,'jet',[.5 .5 .5]); figure, imshow(rgb,'n'), title('Watershed transfrom of D'); 3-D Example: 1 Make a 3-D binary image containing two overlapping spheres. center1 = -10; center2 = -center1; dist = sqrt(3*(2*center1)^2); radius = dist/2 * 1.4; lims = [floor(center1-1.2*radius) ceil(center2+1.2*radius)]; [x,y,z] = meshgrid(lims(1):lims(2)); 14-370 watershed bw1 = sqrt((x-center1).^2 + (y-center1).^2 + ... (z-center1).^2) <= radius; bw2 = sqrt((x-center2).^2 + (y-center2).^2 + ... (z-center2).^2) <= radius; bw = bw1 | bw2; figure, isosurface(x,y,z,bw,0.5), axis equal, title('BW') xlabel x, ylabel y, zlabel z xlim(lims), ylim(lims), zlim(lims) view(3), camlight, lighting gouraud 2 Compute the distance transform. D = bwdist(~bw); figure, isosurface(x,y,z,D,radius/2), axis equal title('Isosurface of distance transform') xlabel x, ylabel y, zlabel z xlim(lims), ylim(lims), zlim(lims) view(3), camlight, lighting gouraud 3 Complement the distance transform, force nonobject pixels to be -Inf, and then compute the watershed transform. D = -D; D(~bw) = -Inf; L = watershed(D); figure, isosurface(x,y,z,L==2,0.5), axis equal title('Segmented object') xlabel x, ylabel y, zlabel z xlim(lims), ylim(lims), zlim(lims) view(3), camlight, lighting gouraud figure, isosurface(x,y,z,L==3,0.5), axis equal title('Segmented object') xlabel x, ylabel y, zlabel z xlim(lims), ylim(lims), zlim(lims) view(3), camlight, lighting gouraud Algorithm watershed uses a variation of the Vincent and Soille algorithm [1]. For details of the variation, see toolbox/images/images/private/watershed_vs.h. See Also bwlabel, bwlabeln, bwdist, regionprops 14-371 watershed Reference 14-372 [1] Luc Vincent and Pierre Soille, “Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations,” IEEE Transactions of Pattern Analysis and Machine Intelligence, vol. 13, no. 6, June 1991, pp. 583-598. wiener2 Purpose 14wiener2 Perform two-dimensional adaptive noise-removal filtering Syntax J = wiener2(I,[m n],noise) [J,noise] = wiener2(I,[m n]) Description wiener2 lowpass filters an intensity image that has been degraded by constant power additive noise. wiener2 uses a pixel-wise adaptive Wiener method based on statistics estimated from a local neighborhood of each pixel. J = wiener2(I,[m n],noise) filters the image I using pixel-wise adaptive Wiener filtering, using neighborhoods of size m-by-n to estimate the local image mean and standard deviation. If you omit the [m n] argument, m and n default to 3. The additive noise (Gaussian white noise) power is assumed to be noise. [J,noise] = wiener2(I,[m n]) also estimates the additive noise power before doing the filtering. wiener2 returns this estimate in noise. Class Support The input image, I, is a two-dimensional image of class uint8, uint16, or double. The output image, J, is of the same size and class as I. Example Degrade and then restore an intensity image using adaptive Wiener filtering. I = imread('saturn.tif'); J = imnoise(I,'gaussian',0,0.005); K = wiener2(J,[5 5]); imshow(J) figure, imshow(K) Algorithm wiener2 estimates the local mean and variance around each pixel 14-373 wiener2 1 µ = ---------NM a ( n 1, n 2 ) n 1, n 2 ∈ η 1 σ = ---------NM 2 ∑ ∑ 2 a ( n 1, n 2 ) – µ 2 n 1, n 2 ∈ η where η is the N-by-M local neighborhood of each pixel in the image A. wiener2 then creates a pixel-wise Wiener filter using these estimates 2 2 σ –ν b ( n 1, n 2 ) = µ + ------------------ ( a ( n 1, n 2 ) – µ ) 2 σ where ν2 is the noise variance. If the noise variance is not given, wiener2 uses the average of all the local estimated variances. See Also filter2, medfilt2 Reference [1] Lim, Jae S. Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: Prentice Hall, 1990. pp. 536-540. 14-374 ycbcr2rgb Purpose 14ycbcr2rgb Convert YCbCr values to RGB color space Syntax rgbmap = ycbcr2rgb(ycbcrmap) RGB = ycbcr2rgb(YCBCR) Description rgbmap = ycbcr2rgb(ycbcrmap) converts the YCbCr values in the colormap ycbcrmap to the RGB color space. If ycbcrmap is m-by-3 and contains the YCbCr luminance (Y) and chrominance (Cb and Cr) color values as columns, then rgbmap is returned as an m-by-3 matrix that contains the red, green, and blue values equivalent to those colors. RGB = ycbcr2rgb(YCBCR) converts the YCbCr image YCBCR to the equivalent truecolor image RGB. Class Support If the input is a YCbCr image, it can be of class uint8, uint16, or double; the output image is of the same class as the input image. If the input is a colormap, the input and output colormaps are both of class double. See Also ntsc2rgb, rgb2ntsc, rgb2ycbcr 14-375 zoom Purpose 14zoom Zoom in and out on an image zoom is a MATLAB function. To get help for this function, select MATLAB Help from the Help menu and view the online function reference page. 14-376 Index Numerics 16-bit image files 2-16, 2-18 1-bit image files 3-8 24-bit image files 2-8 4-bit image files 2-17 8-bit image files 2-16, 2-18 A adaptive filtering 14-373 adaptive filters 10-23 definition 10-2 addition of images 2-23 affine transformation definition 5-13 affine transformations 4-12 aliasing 4-6 definition 4-2 alpha channel 13-4 analyzing images edge detection 14-119 histograms 14-225 intensity profiles 14-246 pixel values 14-243 quadtree decomposition 14-317 anti-aliasing 4-6, 14-256 definition 4-2 applylut 9-53, 14-21 example 9-54 approximation definition 13-2 of an image background 1-9 area of binary images 9-51, 14-28 of image regions 10-9 arrays storing images 2-4 averaging filter 7-7, 14-139 B background of a binary image 9-2 of an intensity image 1-9 bestblk 14-23 bicubic interpolation 4-3 definition 4-2 bilinear interpolation 4-3 definition 4-2 binary image operations connected-components labeling 14-46 flood fill 14-41 lookup-table operations 14-21, 14-287 morphological operations 14-52 neighborhoods 14-21 binary images 2-8, 14-280 changing the display colors of 3-7 connected-components labeling 9-49 converting from other types 14-171 definition 2-2 displaying 3-6 Euler number 9-52, 14-39 feature measurement 9-49 flood fill operation 9-26 image area 9-51, 14-28 lookup table operations 9-53 morphological operations 9-4 object selection 14-60 packing 9-3 perimeter determination 9-17, 14-58 selecting objects in 9-51 binary masks 11-3 I-1 Index definition 11-2 demo that creates xxiii bit depth 1-bit images 3-8 screen bit depth 13-3 blind deconvolution algorithm demo xxi used for deblurring 12-14 blkproc 6-9, 14-24 example 6-10, 6-11 See also dctdemo and ipss003 block operations definition 6-2 block processing 6-2 block size 14-23 column processing 6-12 distinct blocks 6-9, 14-24 padding borders 6-6 sliding neighborhoods 6-5, 14-302 BMP 2-16 border padding 6-6 definition 6-2 border replication in image processing 7-12 boundary padding See border padding boundary ringing in image deblurring 12-20 bounding box defining regions in images 1-20 finding for a region 10-9 brightness adjustment 10-15 demo of xxi See also imadjust bwarea 9-51, 14-28 bwareaopen 14-30 bwdist 14-33 I-2 bwdist 9-38 bweuler 14-39 bwfill 14-41 bwhitmiss 14-44 bwlabel 14-46 bwlabeln 14-49 bwmorph 14-52 skeletonization example 9-17 bwpack 14-56 bwperim 14-58 bwselect 14-60 bwulterode 14-62 bwunpack 14-64 C camera read-out noise 12-11 cancer cell demo xxii Canny edge detector 10-10, 14-120 center of mass calculating for region 10-9 center pixel calculating 6-5 definition 6-2 checkerboard 14-65 chrominance in NTSC color space 13-15 in YCbCr color space 13-16 class support 2-12 See also data types closing 14-52 morphology 9-14 closing an image demo xxii cmpermute 14-67 cmunique 14-68 col2im 14-69 Index colfilt 6-12, 14-70 example 6-13, 6-15 color approximation 13-7, 14-117, 14-183, 14-334 dithering 13-13, 14-117 quantization 13-7, 14-334 reducing number of colors 13-6 color approximation definition 3-2 color cube a description of 13-7 quantization of 13-8 color planes 13-9, 13-18 of an HSV image 13-18, 13-19 of an RGB image 2-9 color reduction 13-6–13-14 color spaces converting between 2-14, 13-15, 14-305, 14-336, 14-337, 14-375 HSV 13-17 NTSC 13-15, 14-305, 14-336 RGB 13-15 YCbCr 13-16, 14-337, 14-375 colorbar 3-12 example 3-12 colorcube 13-12 colormap (matrix) creating a colormap using colorcube 13-12 colormap mapping 13-11 colormaps brightening 14-27 darkening 14-27 rearranging colors in 14-67 removing duplicate entries in 14-68 column processing 14-70 definition 6-2 in neighborhood operations 6-12 reshaping blocks into columns 14-173 reshaping columns into blocks 14-69 composite transformations 4-12 concatenation used to display intensity as RGB 3-20 conformal mapping demo xxi conformal transformations 5-13 conndef 14-73 connected component definition 9-2 connected-components labeling 9-49, 14-46 demo of xxii connectivity definition 9-2 overview 9-23 specifying custom 9-25 constant component See zero-frequency component contour plots 10-8, 14-195 text labels 10-8 contours definition 10-2 contrast adjustment decreasing contrast 10-15 demo of xxi increasing contrast 10-14 specifying limits automatically 10-16 See also imadjust contrast stretching See contrast adjustment Control Point Selection Tool appearance of control point symbols 5-27 changing view of images 5-18 saving a session 5-31 saving control points 5-30 specifying control points 5-22 I-3 Index starting 5-16 using 5-15 using point prediction 5-25 control points appearance of 5-27 prediction 5-25 saving 5-30 selecting 5-15 specifying 5-22 conv2 2-15 conv2 compared to imfilter 7-14 conversions between image types 2-12 convmtx2 14-76 convn 2-15 cp2tform using 5-13 cpcorr 14-88 cpcorr example 5-33 cpselect 14-90 cpselect See also ipexregaerial demo using 5-8 cpstruct2pairs 14-92 cropping an image 4-10, 14-197 cross-correlation demo xxi fine-tuning control point selection 5-10 improving control point selection 5-33 convn compared to imfilter 7-14 convolution convolution matrix 14-76 definition 7-4 Fourier transform and 8-12 two-dimensional 7-4 with imfilter 7-9 convolution kernel definition 7-4 coordinate systems pixel coordinates 2-28 spatial coordinates 2-29 corr2 10-9, 14-78 correlation definition 7-6 Fourier transform 8-13 with imfilter 7-9 correlation coefficient 14-78 correlation kernel definition 7-6 cp2tform 14-79 I-4 D damping for noise reduction 12-10 data types 8-bit integers (uint8) 2-4 converting between 14-118, 14-178, 14-179, 14-204, 14-238, 14-262 double-precision (double) 2-4, 14-118, 14-178, 14-179, 14-204, 14-238, 14-262 in image filtering 7-7 summary of image types and numeric classes DC component See zero-frequency component dconvreg 14-103 DCT image compression demo of xx dct2 8-16, 14-93 See also dctdemo dctdemo demo application xx dctmtx 8-17, 14-96 Index See also dctdemo deblurring avoiding boundary ringing 12-20 conceptual model 12-3 overview 12-3 overview of functions 12-5 use of frequency domain 12-19 using the blind deconvolution algorithm 12-14 using the Lucy-Richardson algorithm 12-9 with regularized filter 12-8 with the Wiener filter 12-6 deblurring demos xxi decomposition getting sequence 14-160 deconvblind 14-97 deconvblind example 12-14 See also ipexdeconvblind demo deconvlucy 14-100 deconvlucy example 12-9 See also ipexdeconvlucy demo deconvreg example 12-8 See also ipexdeconvreg demo deconvwnr 14-105 deconvwnr example 12-6 See also ipexdeconvwnr demo demos xx–xxiii list of ??–xxiii location of xx running xx detail rectangle in Control Point Selection Tool 5-17 dicominfo 14-107 dicomread 14-108 dicomwrite 14-111 dilate 14-115 dilation 9-4, 14-53, 14-115 grayscale 14-306 discrete cosine transform 8-16, 14-93 image compression 8-18 inverse 14-168 transform matrix 8-17, 14-96 discrete Fourier transform 8-8 discrete transform definition 8-2 display depth 13-3 See screen bit depth See screen color resolution display techniques 14-260 displaying at true size 14-364 multiple images 14-355 texture mapping 14-367 displaying images adding a colorbar 3-12 at true size 3-24 binary 3-6 binary images with different colors 3-7 directly from disk 3-11 indexed images 3-3 intensity images 3-4 multiframe images 3-13 multiple images 3-17 texture mapping 3-28 toolbox preferences for See preferences troubleshooting for 3-30 unconventional ranges of data 3-5 zooming 3-26 distance between pixels 10-3 Euclidean 10-3 I-5 Index distance transform 9-38 distinct block operations 6-9 definition 6-2 overlap 6-10, 14-24 zero padding 6-9 dither 14-117 dithering 13-13, 14-117, 14-334 example 13-13 division of images 2-27 double 2-19, 14-179, 14-193, 14-204, 14-235, 14-238, 14-262 E edge 10-10, 14-119 example 10-10 See also edgedemo edge detection 10-10 Canny method 10-10 demo of xx example 10-10 methods 14-119 Sobel method 10-10 edgedemo demo application xx edges definition 10-2 edgetaper 14-124 edgetaper avoiding boundary ringing 12-20 enhancing images intensity adjustment 10-14, 14-180 noise removal 10-20 erode 14-125 erosion 9-4, 14-53, 14-125 grayscale 14-306 Euclidean distance 10-3, 14-315 I-6 Euler number 9-52, 14-39 F fan beam projections 8-28 fast Fourier transform 8-8 higher-dimensional 14-128 higher-dimensional inverse 14-170 two-dimensional 14-127 zero padding 8-10 feature measurement 14-212 area 10-9 binary images 9-49 bounding box 1-20, 10-9 center of mass 10-9 feature measurements 1-18 feature-based logic demo of xxii fft 8-8 fft2 2-15, 8-8, 14-127 example 8-9, 8-11 fftn 2-15, 8-8 fftshift example 7-17, 8-11 file formats graphics formats 2-16 files displaying images from disk 3-11 filling a region 11-8 definition 11-2 demo of xxiii filling holes in images 9-28 filter design 7-16 frequency sampling method 7-18, 14-136 frequency transformation method 7-17, 14-144 windowing method 7-19, 14-147, 14-151 filter2 Index compared to imfilter 7-14 example 3-5, 3-12, 10-22 filtering a region 11-6 masked filtering 11-6 regions 11-2 filters adaptive 10-23, 14-373 averaging 14-139 binary masks 11-6 designing 7-16 finite impulse response (FIR) 7-16 frequency response 7-21, 8-11 imfilter 7-7 Infinite Impulse Response (IIR) 7-17 Laplacian of Gaussian 14-139 linear 7-4 median 10-21, 14-298 multidimensional 7-13 order-statistic 14-306 predefined types 14-139 Prewitt 14-139 regularized demo xxi Sobel 14-139 unsharp 14-139 unsharp masking 7-14 FIR filters 7-16 demo of xx transforming from one-dimensional to two-dimensional 7-17 firdemo demo application xx flat-field correction 12-10 flood-fill operation 9-26, 14-41 foreground of a binary image 9-2 fast Fourier transform See also Fourier transform Fourier transform 8-3 applications of the Fourier transform 8-11 centering the zero-frequency coefficient 8-11 computing frequency response 8-11 convolution and 8-12 correlation 8-13 DFT coefficients 8-9 examples of transform on simple shapes 8-7 fast convolution with 8-12 for performing correlation 8-13 frequency domain 8-3 higher-dimensional 14-128 higher-dimensional inverse 14-170 increasing resolution 8-10 padding before computation 8-10 two-dimensional 8-3, 14-127 zero-frequency component 8-3 freqspace 7-20 example 7-18, 7-20, 7-21 frequency domain 8-3 definition 8-2 frequency response computing 7-21, 8-11, 14-134 desired response matrix 7-20 frequency sampling method (filter design) 7-18, 14-136 frequency transformation method (filter design) 7-17, 14-144 freqz example 7-17 freqz2 7-21, 8-11, 14-134 example 7-18, 7-20, 7-22 See also firdemo fsamp2 7-18, 14-136 example 7-18 See also firdemo fspecial 14-139 I-7 Index fspecial creating predefined filters 7-14 ftrans2 14-144 example 7-17 See also firdemo fwind1 7-19, 14-147 example 7-20 See also firdemo fwind2 7-19, 14-151 See also firdemo graphics file formats converting from one format to another 2-20 list of formats supported by MATLAB 2-16 See also BMP, HDF, JPG, PCX, PNG, TIFF, XWD gray2ind 14-161 grayscale imagesSeeintensity images grayscale morphological operations 14-306 grayslice 14-162 graythresh 14-163 thresholding image values 1-14 G gamma correction 10-17 demo of xxi See also imadjust Gaussian convolution kernel frequency response of 8-11 Gaussian filter 14-139 Gaussian noise 10-23 geocoded images 5-7 geometric operations cropping 4-10, 14-197 definition 4-2 interpolation 4-3 resizing 4-5, 14-256 rotation 4-8, 14-258 georegistered images 5-7 getheight 14-155 getimage 14-156 example 3-11 getneighbors 14-158 getnhood 14-159 getsequence 14-160 granulometry demo xxi graphics card 13-4 I-8 H HDF 2-16 head phantom image 8-29 histeq 14-164 example 10-20 increase contrast example 10-18 See also imadjdemo and roidemo histogram equalization 10-18, 14-164 demo of xxi histograms 10-8, 14-225 definition 10-2 demo of xxi holes filling 9-26 HSV color space 13-17 color planes of 13-18, 13-19 hsv2rgb 13-17 hue in HSV color space 13-16 in NTSC color space 13-15 I idct2 14-168 Index See also dctdemo ifft 8-8 ifft2 8-8 ifftn 8-8 IIR filters 7-17 im2bw 2-13, 14-171 im2col 14-173 See also dctdemo im2double 2-19, 14-174 example 6-15 See also dctdemo and ipss003 im2uint16 2-19, 14-177, 14-177 im2uint8 2-19, 14-176 imabsdiff 14-178 imadjdemo demo application xxi imadjust 10-14, 14-180 brightening example 10-16 gamma correction and 10-17 gamma correction example 10-18 increase contrast example 10-14, 10-16 See also imadjdemo, landsatdemo, roidemo, and ipss003 image analysis contour plots 10-8 edge detection 10-10 histograms 10-8 intensity profiles 10-4 overview 10-10 pixel values 10-3 quadtree decomposition 10-11 summary statistics 10-9 image area (binary images) 14-28 image arithmetic combining functions 2-27 example 9-43 overview 2-21 truncation rules 2-22 image editing 11-8 image filtering data types 7-7 unsharp masking 7-14 with imfilter 7-7 image processing demos xx–xxiii See also demos image profiles definition 10-2 image properties definition 10-2 set by imshow 3-4 set by imshow for binary images 3-9 set by imshow for intensity images 3-6 set by imshow for RGB images 3-10 image registration demo xxi fine-tuning point placement 5-33 overview 5-4 procedure 5-4 selecting control points 5-15 specifying control point pairs 5-22 types of transformations 5-13 using control point prediction 5-25 image transformations affine 5-13 custom 4-12 demos xxii gallery of examples xxii local weighted mean 5-14 piecewise linear 5-14 polynomial 5-14 projective 5-14 supported by cp2tform 5-13 types of 5-13 using imtransform 4-12 I-9 Index image types 2-5 binary 2-8 converting between 2-12 indexed 2-5 intensity 2-7 interpolation and 4-4 multiframe images 2-11 overview 2-2 RGB 2-8 See also indexed, intensity, binary, RGB, multiframe supported by the toolbox 2-5 images adding 2-23 analyzing 10-3 arithmetic operations 2-21 as Handle Graphics objects 1-11 causes of blurring 12-3 converting to binary 14-171 data types 2-4, 14-118, 14-178, 14-179, 14-204, 14-238, 14-262 displaying 14-260 displaying multiple images 3-17, 14-355 dividing 2-27 feature measurement 1-18 filling holes in 9-28 finding image minima and maxima 9-29 getting data from axes 14-156 how MATLAB stores 2-4 image types 2-5 improving contrast 1-13 multiplication 2-25 reducing number of colors 13-6, 14-183 registering 5-4 restoring blurred images 12-3 returning information about 2-18 sample images 1-2 I-10 statistical analysis of 1-21 storage classes of 2-4 subtraction 2-24 viewing as a surface plot 1-10 imapprox 13-12, 14-183 example 13-12 imbothat 14-184 imbothat example 9-42 See also ipexsegwatershed demo imclearborder 14-186 imclose 14-189 imclose See also ipexsegmicro demo using 9-14 imcomplement example 9-44 imcontour 10-8, 14-195 example 10-8 imcrop 4-10, 14-197 example 4-10 imdilate 14-200 imerode 14-205 closure example 9-15 imextendedmax 14-208 imextendedmax example 9-31 imextendedmin 14-210 imextendedmin example 9-45 imfeature 14-212 imfill 14-218 imfill example 9-28 imfilter 14-221 imfilter compared to other filtering functions 7-14 Index convolution option 7-9 correlation option 7-9 padding options 7-10 using 7-7 imfinfo 2-18 example 3-9 imhist 10-8, 14-225 example 10-9, 10-15 See also imadjdemo imhmax 14-227 imhmin 14-229 imimposemin 14-231 imlincomb example 2-27 example 4-5, 4-8 imshow 3-23, 14-260 example for binary images 3-6 example for indexed images 3-3 example for intensity images 3-4, 3-5 example for RGB images 3-10 preferences for 3-23 truesize option 3-25 imtophat 14-263 imtophat example 9-42 See also ipexsegwatershed demo imtransform 14-241, 14-265 using 4-12 immovie 14-237 imwrite example 3-16 imnoise 10-21, 14-239 example 10-23 salt & pepper example 10-22 See also nrfiltdemo and roidemo example 3-8 ind2gray 14-272 ind2rgb 2-13, 14-273 example 3-20 indexed images 2-5, 14-283 converting from intensity 14-161 converting from RGB 14-334 converting to intensity 14-272 converting to RGB 14-273 definition 2-2 displayed as intensity image 3-30 reducing number of colors 13-6 reducing number of colors in 13-12 infinite impulse response (IIR) filter 7-17 inline 6-10 See also function functions intensity adjustment 10-14, 14-180 gamma correction 10-17 histogram equalization 10-18 specifying limits automatically 10-16 See also contrast adjustment intensity images 2-7, 14-282 imopen See also ipexsegmicro demo using 9-14 impixel 10-3, 14-243 example 10-4 improfile 10-4, 14-246 example 10-6 grayscale example 10-5 imread 2-16 example for multiframe image 3-14 imreconstruct 14-250 imreconstruct example 9-20 imregionalmax 14-252 imregmin 14-254 imresize 4-5, 14-256 imrotate 14-258 I-11 Index converting from indexed 14-272 converting from matrices 14-296 converting from RGB 14-332 converting to indexed 14-161 definition 2-2 displaying 3-4 flood-fill operation 9-26 number of gray levels displayed 3-5 intensity profiles 10-4, 14-246 interpolation 4-3 bicubic 4-3 definition 4-2 bilinear 4-3 definition 4-2 default 4-4 definition 4-2 intensity profiles 10-4 nearest neighbor 4-3 definition 4-2 of binary images 4-4 of indexed images 4-4 of RGB images 4-4 tradeoffs between methods 4-3 within a region of interest 11-8 inverse Radon transform 8-26, 8-28 example 8-32 filtered backprojection algorithm 8-28 inverse transform definition 8-2 ipexconformal demo xxi ipexdeconvblind demo xxi ipexdeconvlucy demo xxi ipexdeconvreg demo xxi ipexdeconvwnr demo xxi ipexgranulometry demo xxi ipexmri demo xxi ipexnormxcorr demo xxi I-12 ipexregaerial demo xxi ipexrotate demo xxii ipexsegcell demo xxii ipexsegwatershed demo xxii ipexshear demo xxii ipextform demo xxii ipss001 demo slideshow xxii ipss002 demo slideshow xxii ipss003 demo slideshow xxii iptgetpref 3-24, 14-274 iptsetpref 3-23, 14-275 example 3-24, 3-25 iradon 8-26, 14-277 example 8-26 isbw 14-280 isflat 14-281 isgray 14-282 isind 14-283 isrgb 14-284 J JPEG 2-16 JPEG compression discrete cosine transform and 8-18 L label matrix creating 9-49 viewing as pseudo-color image 1-17, 9-50 label2rgb 14-285 labeling connected-components 9-49 levels of contours 10-8 Laplacian of Gaussian edge detector 14-120 Laplacian of Gaussian filter 14-139 Index line detection 8-24 line segment pixel values along 10-4 linear conformal transformations 5-13 linear filtering 6-6, 7-4 convolution 7-4 filter design 7-16 FIR filters 7-16 IIR filters 7-17 noise removal and 10-21 local weighted mean transformations 5-14 lookup table operations 9-53 lookup-table operations 14-287 Lucy-Richardson algorithm demo xxi used for deblurring 12-9 luminance in NTSC color space 13-15 in YCbCr color space 13-16 M magnifying See resizing images makelut 9-53, 14-287 example 9-53 marker controlled watershed segmentation example 9-41 marker image creating 9-35 definition 9-19 mask image definition 9-19 masked filtering 11-6, 14-342 definition 11-2 mat2gray 2-13, 14-296 matrices converting to intensity images 14-296 storing images in 2-4 maxima finding in images 9-29 imposing 9-34 suppressing 9-32 McClellan transform 14-144 mean2 10-9, 14-297 medfilt2 10-21, 14-298 example 10-22 See also nrfiltdemo and roidemo median filtering 10-21, 14-298 minima finding in images 9-29 imposing 9-34 suppressing 9-32 minimum variance quantization See quantization Moiré patterns 4-6 montage 14-300 example 3-15 morphological closing demo xxii morphological opening demo xxii morphological operations 9-4, 14-52 closing 14-52 diagonal fill 14-53 dilation 9-4, 14-53, 14-115 erosion 9-4, 14-53, 14-125 grayscale 14-306 opening 14-53 predefined operations 9-16 removing spur pixels 14-53 shrinking objects 14-53 skeletonization 9-17, 14-53 thickening objects 14-54 I-13 Index thinning objects 14-54 morphological reconstruction finding peaks and valleys 9-29 overview 9-19 morphology closing 9-14 definition 9-2 demos xxi opening 9-14 overview 9-1 See also morphological reconstruction watershed demo xxii mouse filling region of interest in intensity image 11-8 getting an intensity profile with 10-3 returning pixel values with 10-3 selecting a polygonal region of interest 11-3 movies creating from images 3-16, 14-237 playing 3-17 MRI data demo xxi multidimensional filters 7-13 multiframe images about 2-11 definition 2-2 displaying 3-13, 14-300 limitations 2-11 troubleshooting display of 3-30 multilevel thresholding 14-162 multiplication of images 2-25 N nearest neighbor interpolation 4-3 I-14 definition 4-2 neighborhood operations definition 6-2 neighborhoods binary image operations 14-21 definition 9-2 neighborhood operations 6-2 nlfilter 6-7, 14-302 example 6-7 noise definition 10-2 noise amplification reducing 12-10 noise removal 10-20 adaptive filtering (Weiner) and 10-23 adding noise 14-239 demo of xxiii Gaussian noise 10-23, 14-239 grain noise 10-21 linear filtering and 10-21 localvar noise 14-239 median filter and 10-21 poisson noise 14-239 salt and pepper noise 10-21, 14-239 speckle noise 14-239 nonlinear filtering 6-6 nonuniform illumination demo of correcting for xxii normalized cross-correlation 5-33 demo xxi normxcorr2 14-303 nrfiltdemo demo application xxiii NTSC color space 13-15, 14-305, 14-336 ntsc2rgb 13-15, 14-305 Index O object selection 14-60 observed image in image registration 5-15 opening 14-53 morphology 9-14 opening an image demo xxii optical transfer function (OTF) definition 12-2 order-statistic filtering 14-306 ordfilt2 14-306 orthonormal matrix 8-18 orthophoto defined 5-7 orthorectified image 5-7 OTF See optical transfer function otf2psf 14-308 otf2psf use of 12-19 outliers 10-21 overlap in block operations 6-2 in distinct block operations 6-9 P packed binary image definition 9-3 padarray 14-309 padding demo xxii padding borders block processing 6-6 options with imfilter 7-10 parallel beam projections 8-27 PCX 2-16 perimeter determination 14-58 in binary images 9-17 phantom 8-29, 14-312 piecewise linear transformations 5-14 pixel values 14-243, 14-315 along a line segment 10-4 returning using a mouse 10-3 pixels correcting for bad pixels 12-10 defining connectivity 9-23 definition 2-4 displaying coordinates of 10-3 Euclidean distance between 10-3 returning coordinates of 10-3 selecting 10-3 pixval 14-315 pixval 10-3 PNG 2-16 writing as 16-bit 2-17 point mapping for image registration 5-4 point spread function importance of in deblurring 12-3 point spread function (PSF) definition 12-2 polygon pixels inside 11-3 selecting a polygonal region of interest 11-3 polynomial transformations 5-14 predicting control point locations in image registration 5-25 preferences getting values 14-275 ImshowAxesVisible 3-24 ImshowBorder 3-24 ImshowTruesize 3-24 I-15 Index TrueSizeWarning 3-24 Prewitt edge detector 14-120 Prewitt filter 14-139 profile 10-3 projections fan beam 8-28 parallel beam 8-27 projective transformations 4-12, 5-14 PSF See point spread function psf2otf 14-316 psf2otf use of 12-19 Q qtdecomp 10-11, 14-317 example 10-12 See also qtdemo qtdemo demo application xxiii qtgetblk 14-320 See also qtdemo qtsetblk 14-322 See also qtdemo quadtree decomposition 10-11, 14-317 definition 10-2 demo of xxiii getting block values 14-320 setting block values 14-322 quantization 13-7 minimum variance quantization 14-334 trade-offs between using minimum variance and uniform quantization methods 13-11 uniform quantization 14-334 I-16 R radon 8-20, 8-26, 14-323 example 8-22 Radon transform 8-20, 14-323 center pixel 8-22 detecting lines 8-24 example 8-29 inverse 14-277 inverse Radon transform 8-26 line detection example 8-24 of the Shepp-Logan Head phantom 8-30 relationship to Hough transform 8-24 rank filtering 10-23 See also order-statistic filtering ratioing 2-27 read-out noise correcting 12-11 real orthonormal matrix 8-18 reconstruction morphological 9-19 reference image in image registration 5-15 reflect 14-325 regcorr 14-131, 14-132, 14-289 region labelling 9-49 See connected-components labelling region of interest based on color or intensity 11-5 binary masks 11-3 definition 11-2 demo of xxiii filling 11-8, 14-340 filtering 11-6, 14-342 polygonal 11-3 selecting 11-3, 11-4, 14-339, 14-344 region property measurement 10-9 regional maxima Index definition 9-3, 9-29 imposing 9-34 suppressing 9-32 regional minima definition 9-3, 9-29 imposing 9-34 suppressing 9-32 regionprops 10-9, 14-326 using 1-18 regionprops example 9-47 registering an image 5-4 regularized filter demo xxi used for deblurring 12-8 replication to avoid border effect 7-12 resizing images 4-5, 14-256 anti-aliasing 4-6 resolution screen color resolution 13-3 See also bit depth 13-3 RGB color cube a description of 13-7 quantization of 13-8 RGB images 2-8, 14-284 converting from indexed 14-273 converting to indexed 14-334 converting to intensity 14-332 definition 2-3 displaying 3-10 intensities of each color plane 10-7 reducing number of colors 13-6 rgb2gray 2-14, 14-332 rgb2hsv 13-17 example 13-18 rgb2ind 2-14, 13-7, 14-334 colormap mapping example 13-12 example 13-9, 13-10, 13-12, 13-13 minimum variance quantization example 13-10 specifying a colormap to use 13-11 uniform quantization example 13-9 rgb2ntsc 13-15, 14-336 example 13-15 rgb2ycbcr 13-16 example 13-16 Richardson-Lucy algorithm See Lucy-Richardson ringing in image deblurring 12-20 Roberts edge detector 14-120 roicolor 11-5, 14-339 roidemo demo application xxiii, 11-3 roifill 11-8, 14-340 example 11-8 See also roidemo roifilt2 11-6, 14-342 contrast example 11-6 inline example 11-6 See also roidemo roipoly 11-3, 11-4, 14-344 example 11-3 See also roidemo rotating an image 4-8, 14-258 rotation angle demo of finding xxii S salt and pepper noise 10-21 sample images 1-2 sampling I-17 Index handling undersampled images 12-11 saturation in HSV color space 13-16 in NTSC color space 13-15 scale demo of finding xxii screen bit depth 3-18, 13-3 definition 13-2 See also ScreenDepth property screen color resolution 13-3 definition 13-2 ScreenDepth 13-3 segmentation demo xxii shearing demo xxii Shepp-Logan head phantom 8-29 shrinking See resizing images Signal Processing Toolbox hamming function 7-20 skeletonization 9-17 sliding neighborhood operations 6-5, 14-302 center pixel in 6-5 padding in 6-6 Sobel edge detector 14-119 Sobel filter 14-139 spatial coordinates 2-29 spatial domain definition 8-2 statistical properties mean 14-297 of image objects 1-21 standard deviation 14-346 std2 10-9, 14-346 storage classes I-18 converting between 2-19 of images 2-3 strel 14-347 stretchlim 14-353 adjusting image contrast 1-13 structuring elements 9-7 creating 9-8 decomposition sequence 14-160 definition 9-3 determining composition 14-202 subimage 3-20, 3-21, 14-355 subplot 3-20 subtraction of images 2-24, 9-43 of one image from another 1-12 sum 2-15 surf viewing images 1-10 T template matching 8-13 texture mapping 3-28, 14-367 tform 14-293 tformarray 14-357 tformfwd 14-361 tforminv 14-362 thresholding to create a binary image 1-14, 14-171 to create indexed image from intensity image 14-162 TIFF 2-16 tomography 8-26 transformation matrix 7-17 transforms 8-1 definition 8-2 discrete cosine 8-16, 14-93 Index discrete Fourier transform 8-8 Fourier 8-3, 14-127, 14-128 inverse discrete cosine 14-168 inverse Fourier 14-170 inverse Radon 8-26, 14-277 Radon 8-20, 14-323 two-dimensional Fourier transform 8-3 translate 14-363 transparency 13-4 troubleshooting for display 3-30 image displays 3-30 truecolor 13-6 truesize 3-25, 14-364 truncation rules for image arithmetic operators 2-22 typographical conventions (table) xix watershed example 9-46 watershed demo xxii watershed segmentation example 9-41 weight array in deblurring 12-10 Wiener filter deblurring with 12-6 demo xxi wiener2 10-23, 14-373 example 10-23 See also nrfiltdemo windowing method (filter design) 7-19, 14-147, 14-151 X X-ray absorption tomography 8-27 XWD 2-16 U uint16 storing images in 2-4, 2-16 uint8 storing images in 2-4, 2-16 undersampling correcting 12-11 uniform quantization See quantization unsharp filter 14-139 demo of xxiii unsharp masking 7-14 W warp 3-28, 14-367 example 3-28 watershed 14-369 Y YCbCr color space 13-16, 14-337, 14-375 ycbcr2rgb 13-16 YIQ 13-15 Z zero padding 8-12 and the fast Fourier transform 8-10 image boundaries 7-10 zero-cross edge detector 14-120 zero-frequency component 8-3 zoom 3-26 zooming Control Point Selection Tool 5-20 I-19 Index in figure window 3-26 I-20

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement