IDL Wavelet Toolkit User's Guide
IDL Wavelet
Toolkit User’s
Guide
IDL Version 7.0
November 2007 Edition
Copyright © ITT Visual Information Solutions
All Rights Reserved
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Contents
Chapter 1
Introduction to the IDL Wavelet Toolkit ................................................. 5
What Is the IDL Wavelet Toolkit? .................................................................................... 6
IDL Wavelet Toolkit Architecture .................................................................................... 9
Chapter 2
Using the IDL Wavelet Toolkit .............................................................. 11
Starting the Toolkit ..........................................................................................................
Menu Description ............................................................................................................
Preferences ......................................................................................................................
Dataset Viewer ................................................................................................................
Importing Data ................................................................................................................
Wavelet Viewer ...............................................................................................................
Wavelet Power Spectrum ...............................................................................................
Multiresolution Analysis .................................................................................................
Denoise Tool ...................................................................................................................
IDL Wavelet Toolkit
12
14
18
20
25
28
32
39
41
3
4
Adding User Tools ........................................................................................................... 45
Chapter 3
Theory and Examples .......................................................................... 47
Wavelet Transform .......................................................................................................... 48
Wavelet Power Spectrum ................................................................................................. 49
Denoise ............................................................................................................................ 51
Multiresolution Analysis .................................................................................................. 54
Bibliography .................................................................................................................... 55
Chapter 4
IDL Wavelet Toolkit Reference ............................................................. 57
List of Commands by Functionality ................................................................................ 58
WV_APPLET .................................................................................................................. 60
WV_CW_WAVELET ..................................................................................................... 62
WV_CWT ........................................................................................................................ 66
WV_DENOISE ................................................................................................................ 69
WV_DWT ........................................................................................................................ 74
WV_FN_COIFLET ......................................................................................................... 78
WV_FN_DAUBECHIES ................................................................................................ 80
WV_FN_GAUSSIAN ...................................................................................................... 82
WV_FN_HAAR .............................................................................................................. 85
WV_FN_MORLET ......................................................................................................... 87
WV_FN_PAUL ............................................................................................................... 90
WV_FN_SYMLET .......................................................................................................... 93
WV_IMPORT_DATA ..................................................................................................... 96
WV_IMPORT_WAVELET ............................................................................................ 99
WV_PLOT3D_WPS ...................................................................................................... 101
WV_PLOT_MULTIRES ............................................................................................... 104
WV_PWT ...................................................................................................................... 107
WV_TOOL_DENOISE ................................................................................................. 109
Index .................................................................................................... 113
Contents
IDL Wavelet Toolkit
Chapter 1
Introduction to the IDL
Wavelet Toolkit
This chapter discusses the following topics:
What Is the IDL Wavelet Toolkit? . . . . . . . . . 6
IDL Wavelet Toolkit
IDL Wavelet Toolkit Architecture . . . . . . . . 9
5
6
Chapter 1: Introduction to the IDL Wavelet Toolkit
What Is the IDL Wavelet Toolkit?
The IDL Wavelet Toolkit consists of a set of graphical user interfaces (GUI) and IDL
routines for wavelet analysis of multi-dimensional data.
Motivation
Wavelet analysis is becoming a popular technique for data and image analysis. By
decomposing a signal using a particular wavelet function, one can construct a picture
of the energy within the signal as a function of both spatial dimension (or time) and
wavelet scale (or frequency). The wavelet transform is used in numerous fields such
as geophysics (seismic events), medicine (EKG and medical imaging), astronomy
(image processing), and computer science (object recognition and image
compression). The technique is flexible and robust, yet it is fast enough to be used in
real-time image processing.
A set of standard wavelet techniques have been developed which make it possible for
the average user to apply the wavelet method with confidence. Recent advances in
significance testing and cross-wavelet analysis have also enhanced the acceptability
of wavelet analysis within the scientific community. Nevertheless, the calculation of
the wavelet transform and the display of the output requires considerable experience.
Users
The IDL Wavelet Toolkit is designed for a wide audience, ranging from the casual
user who wishes to explore the possibilities of wavelet analysis, to the scientist or
engineer who wants to produce robust and complex results.
Potential users and their applications include:
•
Students— introduction to wavelets, graphical analysis;
•
Engineers— data analysis, signal processing, data compression;
•
Scientists— data analysis, filtering and denoising, cross-wavelet;
•
Computer scientists— image compression, speed of operations;
•
Mathematicians— explore wavelet families, test out new analysis techniques.
Applications
Examples of specific applications are:
What Is the IDL Wavelet Toolkit?
IDL Wavelet Toolkit
Chapter 1: Introduction to the IDL Wavelet Toolkit
7
•
Time-series analysis— time-scale power spectrum, noise filtering,
multiresolution analysis;
•
Self-similar series— fractals, long-memory processes;
•
Turbulence— detection of coherent structures;
•
Signal processing— filtering and denoising;
•
Image processing— edge detection, compression, enhancement.
Features
The IDL Wavelet Toolkit has the following features:
Wavelet Applet
The Toolkit Applet lets you manage your projects, import data and wavelets,
visualize the results, and add your own user tools.
Continuous Wavelet Transform
Allows you to compute the continuous wavelet transform on one-dimensional
vectors. This routine is written in IDL .pro code.
Discrete Wavelet Transform
Allows you to compute the discrete wavelet transform (partial or full) on multidimensional data. These routines are written in C and contained in the IDL wavelet
dlm.
Wavelet Functions
The Toolkit comes with several wavelet functions that are accessible both inside the
Applet and from your own programs. You can easily add your own wavelet functions
to the Toolkit.
3D Wavelet Power Spectrum
Callable from within the Applet and from your own programs, the visualizer plots the
wavelet power as a three-dimensional surface, with optional contour lines. You can
rotate, translate, and find the power at a particular location.
Multiresolution Analysis
Stand alone or callable from the Applet, this routine produces plots for the smooth
(low pass), detail (band-pass), and rough (high-pass) components of your data.
IDL Wavelet Toolkit
What Is the IDL Wavelet Toolkit?
8
Chapter 1: Introduction to the IDL Wavelet Toolkit
Denoise Tool
This widget tool enables you to denoise your vector or image array by thresholding
(hard or soft) either by cumulative power or coefficient number.
Dataset Viewer
Manage the datasets within each project by importing new data, viewing data values,
and customizing the data fields.
Import Data
You can import data from a variety of file formats: ASCII, binary, image (BMP,
JPEG, PNG, PPM, SRF, TIFF, DICOM), and WAV audio. Image files can be either
indexed color (8- or 16-bit) or TrueColor (24-bit). You can also import data directly
from the IDL> command prompt.
User Tools
You can extend the functionality of the IDL Wavelet Toolkit by adding your own
tools.
What Is the IDL Wavelet Toolkit?
IDL Wavelet Toolkit
Chapter 1: Introduction to the IDL Wavelet Toolkit
9
IDL Wavelet Toolkit Architecture
File Organization
The Toolkit consists of the following components:
•
Source (.pro) files in the wavelet directory;
•
A bitmaps subdirectory with button bitmaps;
•
The data subdirectory with sample data files;
•
The Online help manual in the IDL help directory;
•
The DLM (Dynamically Loadable Module) in the IDL bin directory.
Note
You are encouraged to view the source files for details on implementation and
technique. You are also welcome to modify the source files, however, it is strongly
encouraged that you copy the files to your own directory first. By modifying the
IDL !PATH variable you can ensure that your routines are compiled first. See
“!PATH” (IDL Reference Guide) for more information.
Structure
The IDL Wavelet Toolkit consists of three layers. The topmost layer is the Wavelet
Applet, which allows you to import data and wavelet functions, and access various
visualization and tool routines. The middle layer is the set of compound widgets and
widget tools for visualization and analysis. These tools are accessible both from the
Wavelet Applet and from your own routines. The lowest layer are the wavelet API
(application programming interface) that consist of the wavelet functions, the wavelet
transform, and the import data routine.
IDL Wavelet Toolkit
IDL Wavelet Toolkit Architecture
10
IDL Wavelet Toolkit Architecture
Chapter 1: Introduction to the IDL Wavelet Toolkit
IDL Wavelet Toolkit
Chapter 2
Using the IDL Wavelet
Toolkit
This chapter discusses the following topics:
Starting the Toolkit . . . . . . . . . . . . . . . . . . .
Menu Description . . . . . . . . . . . . . . . . . . . .
Preferences . . . . . . . . . . . . . . . . . . . . . . . . .
Dataset Viewer . . . . . . . . . . . . . . . . . . . . . . .
Importing Data . . . . . . . . . . . . . . . . . . . . . . .
IDL Wavelet Toolkit
12
14
18
20
25
Wavelet Viewer . . . . . . . . . . . . . . . . . . . . . .
Wavelet Power Spectrum . . . . . . . . . . . . . .
Multiresolution Analysis . . . . . . . . . . . . . .
Denoise Tool . . . . . . . . . . . . . . . . . . . . . . . .
Adding User Tools . . . . . . . . . . . . . . . . . . .
28
32
39
41
45
11
12
Chapter 2: Using the IDL Wavelet Toolkit
Starting the Toolkit
To start the IDL Wavelet Toolkit type the following at the IDL> command prompt:
wv_applet
This action compiles the wv_applet routines and starts up the main window, shown
in the following figure. For other startup options see “WV_APPLET” on page 60.
The window consists of Menu Items, the Toolbar, the Dataset Viewer, and a Status
Bar at the bottom.
Menu Items
The menu items, located at the top of the IDL Wavelet Toolkit window, allow you to
perform various actions. These menu items are described in the next section.
Toolbar
The toolbar is divided into five sections: File, Import, Edit, Visualize, and Help. The
toolbar buttons allow you to easily access various menu items.
When you position the mouse pointer over a toolbar the Status Bar displays a
description of its function.
Dataset Viewer
The variables contained in your dataset are displayed in the dataset table, described in
“Dataset Viewer” on page 20.
Status Bar
The Status Bar displays descriptions of the Toolbar buttons and the status of various
actions such as Open, Import, and Save. The Status Bar also provides warnings if, for
example, you select “Visualize...” without selecting a variable.
Starting the Toolkit
IDL Wavelet Toolkit
Chapter 2: Using the IDL Wavelet Toolkit
13
Figure 2-1: The main Wavelet Toolkit window.
IDL Wavelet Toolkit
Starting the Toolkit
14
Chapter 2: Using the IDL Wavelet Toolkit
Menu Description
The main window has five items: File Menu, Edit Menu, Visualize Menu, Tools
Menu, and Help Menu. Each menu and its submenus is described below.
File Menu
The File menu accesses and manipulates files.
New Applet
This menu item or button starts a new Wavelet Toolkit applet with an empty dataset.
Open Dataset...
This menu item or button closes the current dataset and allows you to open up a
different dataset. Wavelet datasets have the default filename suffix .sav and are
written in IDL SAVE format. For more information, see “SAVE” (IDL Reference
Guide).
If the previous dataset has not been saved, then you will be prompted to save the
previous dataset first.
Save
Select this menu item or button to save the current dataset and preferences. If the
dataset has not yet been saved, then you are prompted for a filename with the Save As
dialog.
Save As...
This menu item allows you to choose a new filename for the current dataset using the
Save As dialog, and then saves the dataset to this file.
Import...
Select this menu item to import an external data file into the current dataset. Details
on allowable file formats and import options can be found in “Dataset Viewer” on
page 20. You can also import data from the current IDL session using the
WV_IMPORT_DATA procedure.
Preferences...
This menu item opens up a Preferences dialog in which you can customize your
interaction with the Wavelet Toolkit. The Default button restores the built-in default
Menu Description
IDL Wavelet Toolkit
Chapter 2: Using the IDL Wavelet Toolkit
15
options for all of the preferences. The OK button keeps all of the changes to
Preferences. The Cancel button discards all of the changes.
Exit
This menu item will close the current Wavelet Toolkit applet. Other Wavelet applets
(either started from the command line or via the “New Applet” menu item) are
unaffected.
If you have made changes to the current dataset, then you will be prompted to save
the dataset before exiting.
Edit Menu
The Edit Menu manipulates the Dataset Viewer.
Move Variable Left
Select this menu item or button to move the currently-selected variable to the left.
Move Variable Right
Select this menu item or button to move the currently-selected variable to the right.
View Data Values
This menu item or button displays the values for the currently-selected variable.
Delete Variable
Select this menu item or button to delete the currently-selected variable or variables.
You are asked for confirmation before the variables are removed.
Visualize Menu
The Visualize Menu contains methods to graphically display and manipulate the
wavelet transform.
Wavelets
This menu item or button starts up the wavelet compound widget, which allows you
display the available wavelet functions and their properties. You can also start the
wavelet viewer using the WV_CW_WAVELET function from the IDL> command
prompt. The wavelet widget is described in “Wavelet Viewer” on page 28.
IDL Wavelet Toolkit
Menu Description
16
Chapter 2: Using the IDL Wavelet Toolkit
Wavelet Power Spectrum
This menu item or button starts the three-dimensional viewer for the wavelet power
spectrum, using the currently-selected variable. You can also start the viewer using
the WV_PLOT3D_WPS function from the IDL> command prompt. For more
information, see “WV_PLOT3D_WPS” on page 101. The wavelet power spectrum
viewer is described in “Wavelet Power Spectrum” on page 32.
Multiresolution Analysis
This menu item or button starts the viewer for multiresolution analysis of the
currently-selected variable. You can also start the viewer using the
WV_PLOT_MULTIRES function from the IDL> command prompt. The
Multiresolution viewer is described in “Multiresolution Analysis” on page 39.
Tools Menu
The Tools Menu contains built-in and user-defined tools.
Denoise
This menu item starts the widget for denoising, filtering, and compression of the
currently-selected variable. You can also start the viewer from the IDL> command
prompt by using the WV_TOOL_DENOISE function. The Denoise tool is described
in “Denoise Tool” on page 41.
Other user tools...
If you have added other tools then they will be displayed here. The currently-selected
variable will be passed to the tool function. See “Adding User Tools” on page 45.
Help Menu
The Help Menu provides various help functions.
IDL Help
This menu item will start up the IDL Online Help manual.
IDL Wavelet Toolkit Help
This menu item or button will start up the online help manual for the IDL Wavelet
Toolkit.
Menu Description
IDL Wavelet Toolkit
Chapter 2: Using the IDL Wavelet Toolkit
17
Wavelet Readme
This menu item will display the Readme file included with the Toolkit.
Wavelet Release Notes
This menu item will display the Release Notes file included with the Toolkit.
About IDL Wavelet Toolkit...
Select this menu item to display information about the current version of IDL and the
IDL Wavelet Toolkit.
IDL Wavelet Toolkit
Menu Description
18
Chapter 2: Using the IDL Wavelet Toolkit
Preferences
The Preferences dialog, under the File Menu, allows you to set various default
preferences and options for the currently active dataset.
Note
The Preferences are saved within each dataset rather than in a separate preferences
file; each dataset can therefore have its own set of preferences. Note, however, that
opening a new dataset may change the current preferences. These new preferences
will remain in effect until changed either via the Preferences window or by opening
a different dataset.
Default Directory
Set this option to your working directory. The Wavelet Toolkit will start all file open
or save dialogs in this directory. This directory may be overridden if “Remember
Current Directory” is set.
Remember Current Directory
Set this option to cause the Wavelet Toolkit to store the directory selected within any
file open or save dialogs, and to use this directory for future dialogs. If this option is
not set, the “Default Directory” will be used.
Confirm Exit
If this option is set, the Wavelet Toolkit will ask you for confirmation when you exit
the Toolkit.
Compress Save Files
Set this option to use file compression when saving dataset files. Compressed files
will occupy less disk space than uncompressed files, but may be slower to save and
open.
Stride Factor
When importing large data arrays, the IDL Wavelet Toolkit will automatically
calculate the X and Y stride values by dividing the length of vector arrays by the
“Vector” stride factor, and each dimension of two-dimensional arrays by the “Array”
stride factor. After the data is imported, you may change the X and Y stride values on
the Dataset Viewer. The minimum stride factor is 2.
Preferences
IDL Wavelet Toolkit
Chapter 2: Using the IDL Wavelet Toolkit
19
Tip
To force the stride values to always be set to 1 when importing data, set the stride
factors to a value larger than the maximum dimension for your data.
“Defaults” Button
Press this button to restore all of the preferences to their default settings.
IDL Wavelet Toolkit
Preferences
20
Chapter 2: Using the IDL Wavelet Toolkit
Dataset Viewer
Your dataset can consist of several different variables, each with a different data
format. The Dataset Viewer, located in the middle of the Wavelet Toolkit applet,
allows you to organize and manipulate your dataset.
The variables are assigned a number and a name derived from the Variable name.
You can sort the variables using the Move Variable Left and Move Variable Right
buttons.
Variable Information
Each variable contains a one-dimensional vector or two-dimensional array of data
values. The data values can be of any numeric type, such as BYTE, INTEGER,
FLOAT, etc.
The variable also has several descriptor fields which you can modify, described
below and summarized in the table below. To modify a field, double-click with the
left-mouse button on the field. After editing the field, press the <Return> key to keep
your changes or, click outside of the table to discard your changes.
Type
This string shows the numeric type and the array size of the data. It is not modifiable
by the user.
Title
This string contains the overall name of the variable. The Title field is used to label
the Wavelet Power Spectrum and Multiresolution widgets. The default is the null
('') string.
Variable
This string provides a short name for the variable. The Variable is used to label plots,
and for the labels in the Dataset Viewer. For a one-dimensional vector (e.g. a time
series), the Variable is equivalent to Ytitle. The default is either the name of the
import file, or 'Data' if imported from the IDL> command prompt.
Units
This string gives the units of the variable, and is used to label various plots. For a
one-dimensional vector (e.g. a time series), the Units is equivalent to the Yunits. The
default is the null ('') string.
Dataset Viewer
IDL Wavelet Toolkit
Chapter 2: Using the IDL Wavelet Toolkit
Field
Type
21
Example (1D vector)
Example (2D array)
Title
STRING
Wave audio recording
IEEE Test Image
Variable
STRING
Channel1
IEEEtest
Units
STRING
Amplitude
intensity
Xname
STRING
Time
X
Xunits
STRING
seconds
pixels
Xstart
STRING
0
0
Dx
STRING
1d0/22050
1
Yname
STRING
Y
Yunits
STRING
pixels
Ystart
STRING
0
Dy
STRING
1
Xoffset
LONG
0
0
Xcount
LONG
16384
256
Xstride
LONG
1
2
Yoffset
LONG
0
Ycount
LONG
256
Ystride
LONG
2
Source
STRING
wavelet/data/hello.wav
wavelet/data/IEEEtest.tif
Notes
STRING
Voice saying ‘hello’
IEEE test image
Table 2-1: Data fields in the Dataset Viewer .
Xname
This string is the name of the independent variable for the first data dimension (“X”),
and is used to label the x-axis. The default is the null ('') string.
IDL Wavelet Toolkit
Dataset Viewer
22
Chapter 2: Using the IDL Wavelet Toolkit
Xunits
This string gives the units of X. The default is the null ('') string.
Xstart
This string gives the value of the first X coordinate. The default is '0'. Xstart can
contain complicated mathematical expressions, although the result must be a scalar
number.
Dx
This string gives the sampling interval between the X coordinates. The default is '1'.
Dx can contain complicated mathematical expressions, although the result must be a
scalar number.
Yname
This string is the name of the independent variable for the second data dimension
(“Y”), and is used to label the y-axis (for a one-dimensional variable this is actually
equivalent to the name of the dependent Variable). The default is the null ('') string.
Yunits
This string gives the units of Y. The default is the null ('') string.
Ystart
This string gives the value of the first Y coordinate. The default is '0'. Ystart can
contain complicated mathematical expressions, although the result must be a scalar
number.
Dy
This string gives the sampling interval between the Y coordinates. The default is '1'.
Dy can contain complicated mathematical expressions, although the result must be a
scalar number.
Xoffset
The offset along the first data dimension at which to start. The default is 0L.
Xcount
The number of data points to use along the first data dimension. The default is the
size of the first dimension.
Dataset Viewer
IDL Wavelet Toolkit
Chapter 2: Using the IDL Wavelet Toolkit
23
Xstride
The sampling interval along the first data dimension. The default is 1L.
Yoffset
This long integer gives the offset along the second data dimension at which to start.
The default is 0L.
Ycount
This long integer gives the number of data points to use along the second data
dimension. The default is the size of the second dimension.
Ystride
This long integer gives the sampling interval along the second data dimension. The
default is 1L.
Source
This string describes the original source or location of the data. The default is either
the full filename (if the data was from a file) or ’Imported’ (if the data was from
the IDL> command prompt).
Notes
You can enter miscellaneous information into the Notes string. The default is the null
('') string.
Mathematical Expressions
For Xstart, Dx, Ystart, and Dy, it is highly recommended that whenever possible you
enter mathematical expressions, rather than converting to numbers. For example, in
the above table, the sampling rate for hello.wav is 22050 Hz. One could have entered
Dx as 0.00004535 rather than ‘1d0/22050’. Nevertheless, the latter is not only more
accurate (limited only by your computer’s precision) but is also much more
informative. (Note that the ‘1d0’ forces the computation to be done in double
precision.)
You may also enter IDL functions in these strings. For example, if your X coordinate
was in Julian days, starting from say 29 February 2000, you could set
Xstart = 'JULDAY(2,29,2000)'.
IDL Wavelet Toolkit
Dataset Viewer
24
Chapter 2: Using the IDL Wavelet Toolkit
Selecting Variables
To select a particular variable for visualization or some other action, click the mouse
on any field for that variable, or click the mouse on the Table row label to highlight
the entire row.
To select multiple variables for deletion, click the mouse on any field and drag down
to select the list of variables, or click once on the row label, scroll down and hold the
<Shift> key while clicking on the last row label.
Dataset Viewer
IDL Wavelet Toolkit
Chapter 2: Using the IDL Wavelet Toolkit
25
Importing Data
You can import data in multiple file formats into the IDL Wavelet Toolkit.
ASCII Files
Select this menu item or button to import data from an ASCII text file. After choosing
the file using the Select Import File dialog, you can specify the particular format for
the ASCII_TEMPLATE dialog. See “ASCII_TEMPLATE” (IDL Reference Guide)
for more information.
The ASCII_TEMPLATE routine handles ASCII files consisting of an optional header
of a fixed number of lines, followed by columnar data. The procedure consists of
three steps:
1. “Define Data Type/Range”— Specify whether the data is in fixed width
columns or separated by commas or spaces. The first 50 lines are displayed.
Choose the first line of data and click on the Next > button;
2. “Define Fields”— Choose the number of fields per line and then click Next >;
3. “Field Specification”— You can change the names and data types for the
various fields. The Field names can also be changed once the data is imported
into the Toolkit. Click on the Finish button to import the data into the Wavelet
Toolkit.
Once the data is successfully imported, you can change the default names for the
variable Title, Variable, etc.
Independent Variable
For ASCII files with multiple columns, if the first column is determined to be
monotonically increasing in value, or is assigned the field name “TIME” within the
ASCII_TEMPLATE, then it is assumed to be the “independent variable.” In this case
the remaining columns are then imported as the “dependent variables.”
Note
You may change the name “TIME” after the data has been imported into the
Wavelet Toolkit.
Two-Dimensional Arrays
By default, each column within the file will be imported into the IDL Wavelet
Toolkit as a separate variable. To import a two-dimensional array of data, you should
IDL Wavelet Toolkit
Importing Data
26
Chapter 2: Using the IDL Wavelet Toolkit
use the Group All button within the ASCII_TEMPLATE dialog to connect all of the
columns into one field.
Binary Files
Select this menu item or button to import data from a binary data file. After choosing
the file using the Select Import File dialog, you can specify the particular format for
the file using the BINARY_TEMPLATE dialog. See “BINARY_TEMPLATE” (IDL
Reference Guide) for more information.
The BINARY_TEMPLATE routine handles raw binary files consisting of headers
and multiple data fields. The dialog consists of a Binary Template window where you
can define various fields within the file. Each field will be imported into the Wavelet
Toolkit as a separate variable.
Image Files
Select this menu item or button to import an image file. The function
DIALOG_READ_IMAGE is used to select the image file. For files with multiple
images you can choose the particular image you wish to import. See
“DIALOG_READ_IMAGE” (IDL Reference Guide) for more information.
For TrueColor (24-bit) images, you will then be asked how you wish to convert the
three channels into a single two-dimensional image. You have the option to scale the
data into an intensity from 0–255, quantize the 24-bit colors down to 256 colors, or
split the three channels into separate red, green, and blue images.
WAV Audio Files
Select this menu item or button to import a .WAV (RIFF) audio file as a onedimensional vector. The file must be in uncompressed PCM format. Multiple
channels are imported as separate variables, one for each channel.
IDL Command Line
You can also import data directly from the IDL> command prompt using the
WV_IMPORT_DATA command:
WV_IMPORT_DATA, variable
where variable is either a data vector or array, or a structure of data tags (see
“WV_IMPORT_DATA” on page 96 for tag information).
Importing Data
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Chapter 2: Using the IDL Wavelet Toolkit
27
If there is more than one Wavelet Toolkit applet currently running, then variables are
entered into the one that was most-recently active.
IDL Wavelet Toolkit
Importing Data
28
Chapter 2: Using the IDL Wavelet Toolkit
Wavelet Viewer
The Wavelet Viewer is accessible from the Visualize menu or button, and can also be
started from the IDL> command prompt using the WV_CW_WAVELET function:
wId = WV_CW_WAVELET()
For more information, see “WV_CW_WAVELET” on page 62.
The Wavelet Viewer consists of a graph of the currently-selected wavelet function, a
selection area for the wavelet function, and an information area, shown in the
following figure:
Figure 2-2: The Wavelet Viewer.
Wavelet and Scaling Functions
The wavelet consists of two components, the scaling function which describes the
low-pass filter for the wavelet transform, and the wavelet function which describes
the band-pass filter for the transform.
Changing Wavelets
The droplist contains the names of all currently-available wavelets. The Family
refers to the overall properties of the wavelet, while the Order determines the
particular wavelet within each family.
Wavelet Viewer
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29
Wavelet Information
After you select a wavelet family and order, the following information will be
displayed:
Discrete/Continuous
Discrete wavelet functions are used with the discrete wavelet transform, which
provides the most compact representation of the data. The discrete transform is very
fast and is best suited for image processing, filtering, and large arrays.
Continuous wavelet functions are used to approximate the continuous wavelet
transform, which provides a highly-redundant transformation of the data. The
continuous wavelet transform is much smoother than the discrete transform and is
better suited for time-series analysis on small arrays (less than 20000 data points).
Orthogonal/Nonorthogonal
Orthogonal wavelet functions will have no overlap with each other (zero correlation)
when computing the wavelet transform, while nonorthogonal wavelets will have
some overlap (nonzero correlation). Using an orthogonal wavelet, you can transform
to wavelet space and back with no loss of information.
Nonorthogonal wavelet functions tend to artificially add in energy (due to the
overlap) and require renormalization to conserve the information.
In general, discrete wavelets are orthogonal while continuous wavelets are
nonorthogonal.
Symmetry
This flag describes the symmetry of the wavelet function about the midpoint.
Symmetric wavelets show no preferred direction in “time,” while asymmetric
wavelets give unequal weighting to different directions.
Compact Support
This value measures the effective width of the wavelet function. A narrow wavelet
function such as the Daubechies order 2 (compact support=3) is fast to compute, but
the narrowness in “time” implies a very large width in “frequency.” Conversely,
wavelets with large compact support such as the Daubechies order 24 (compact
support=47) are smoother, have finer frequency resolution and are usually more
efficient at denoising.
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Wavelet Viewer
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Vanishing Moments
An important property of a wavelet function is the number of vanishing moments,
which describes the effect of the wavelet on various signals. A wavelet such as the
Daubechies 2 with vanishing moment=2 has zero mean and zero linear trend. When
the Daubechies 2 wavelet is used to transform a data series, both the mean and any
linear trend are filtered out of the series. A higher vanishing moment implies that
more moments (quadratic, cubic, etc.) will be removed from the signal.
Regularity
The regularity gives an approximate measure of the number of continuous derivatives
that the wavelet function possesses. The regularity therefore gives a measure of the
smoothness of the wavelet function with higher regularity implying a smoother
wavelet.
e-Folding Time (Continuous Wavelets Only)
The e-folding time is a measure of the wavelet width, relative to the wavelet scale s.
Using the wavelet transform of a spike, the e-folding time is defined as the distance at
which the wavelet power falls to 1/e^2, where e = 2.71828. Larger e-folding time
implies more spreading of the wavelet power.
User-Defined Wavelets
You can easily extend the IDL Wavelet Toolkit by adding more wavelet functions.
These wavelet functions should follow the same calling mechanism as the built-in
wavelet functions such as “WV_FN_DAUBECHIES” on page 80. In addition, your
wavelet function should begin with the prefix 'wv_fn_'.
1. Let’s say you would like to add a wavelet function called “Spline” giving the
Daubechies “Spline” wavelets. To do this, first create a wavelet function to
return the wavelet coefficients and the information structure:
FUNCTION wv_fn_spline, Order, Scaling, Wavelet, Ioff, Joff
; compute coefficients here...
...
; find support, moments, and regularity
...
info = {family:'Spline', $
order_name:'Order', $
order_range:[1,5,1], $
order:order, $
discrete:1, $
orthogonal:1, $
Wavelet Viewer
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Chapter 2: Using the IDL Wavelet Toolkit
31
symmetric:0, $
support:support, $
moments:moments, $
regularity:regularity}
RETURN, info
END
2. Save this function in a file 'wv_fn_spline.pro' that is accessible from your
current IDL path.
3. Now start the Wavelet Toolkit with your new wavelet function:
WV_APPLET, WAVELETS='Spline'
Or, if you are already running the Wavelet Toolkit:
WV_IMPORT_WAVELET, 'Spline'
Your new wavelet function should appear in the list of current wavelet functions, and
should be accessible from any of the wavelet tools.
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Wavelet Viewer
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Chapter 2: Using the IDL Wavelet Toolkit
Wavelet Power Spectrum
The wavelet transform converts the data array into a series of wavelet coefficients,
each of which represents the amplitude of the wavelet function at a particular location
within the array and for a particular wavelet scale.
The Wavelet Power Spectrum viewer, shown in the following figure, allows you to
visualize the wavelet power as a three-dimensional surface plot, where the height of
the surface represents the magnitude of the wavelet coefficients.
Figure 2-3: The Wavelet Power Spectrum 3D viewer.
Wavelet Power Spectrum
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Chapter 2: Using the IDL Wavelet Toolkit
33
File Menu
Open State...
This menu item opens a previously saved “state” file into a new window.
Save State...
This menu item saves the current state of the Wavelet Power Spectrum into a file.
Export To...
•
Bitmap File [Windows only]: The bitmap file saves the current image as a
bitmap.
•
Vector Metafile [Windows only]: The vector metafile produces a scalable
image file, but may not be able to accurately reproduce the 3D geometry.
•
Bitmap Pict [Macintosh only]: The bitmap pict saves the current image as a
bitmap.
•
Bitmap Postscript: The bitmap postscript format saves the current image as a
bitmap.
•
Vector Postscript: The vector postscript format takes less disk space than
bitmap, and is scalable, but may not be able to accurately reproduce the 3D
geometry.
•
VRML: The Virtual Reality Markup Language produces a three-dimensional
output file suitable for web publication.
Note
It is not always possible to translate the complicated 3D geometry produced by IDL
object graphics into equivalent VRML code.
Print
This menu item will output the image to a printer.
Close
This menu item closes the Wavelet Power Spectrum viewer.
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Wavelet Power Spectrum
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Chapter 2: Using the IDL Wavelet Toolkit
Edit Menu
Undo
This menu item will undo the previous rotation, scaling, or translation of the model.
Copy To Clipboard
This menu item makes a copy of the current graphics image and places it on the
system clipboard.
View Menu
Color Table
Selecting this item brings up the XLOADCT color table editor. You can then choose
different color tables for the graphics image. See “XLOADCT” (IDL Reference
Guide) for more information.
Drag Quality
This submenu has three different settings that affect the drawing speed during object
manipulations:
•
Low— only the axes are exposed for graphics manipulation such as rotation
and translation;
•
Medium— low resolution graphics are used for graphics manipulation;
•
High— full resolution is used for all graphics manipulations
Wavelet Options
If you select this menu item, the Wavelet Options panel will be hidden. Select this
menu item again to show the panel.
View Options
If you select this menu item, the View Options panel will be hidden. Select this menu
item again to show the panel.
Help Menu
This menu contains Help items for the Wavelet Power Spectrum and for IDL.
Wavelet Power Spectrum
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Chapter 2: Using the IDL Wavelet Toolkit
35
Wavelet Options
You can change the current wavelet family or the order. The plot will be
automatically updated.
Note
For two-dimensional input data, only the discrete wavelet functions are available.
View Options
3D
Turn this button off to rotate the image so it appears flat. Turn this button on to rotate
the image to a three-dimensional perspective. For vector data, this button also
controls whether the data series and global wavelet plot are flat or vertical.
Note
The surface will remain three-dimensional; only the viewpoint is changed.
Color Bar
Turn this button off to remove the color bar at the bottom. Turn this button on to
restore the color bar.
Data Plot [One-dimensional only]
Turn this button off to remove the data series plot at the back. Turn this button on to
restore the plot.
Global [One-dimensional only]
Turn this button off to remove the plot of the global wavelet. Turn the button on to
restore the plot.
Zero Phase Lines [Complex wavelet functions only]
Turn this button on to add the zero wavelet phase lines to the surface plot.
Energy Scaling
These buttons control the scaling of the wavelet magnitude in the Z-direction.
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Wavelet Power Spectrum
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Chapter 2: Using the IDL Wavelet Toolkit
Power
The power is the absolute-value-squared of the wavelet coefficients. The height of
each point measures the contribution to the total energy.
This scaling emphasizes large peaks and sharp discontinuities, and de-emphasizes
low-amplitude background noise.
Magnitude
The magnitude is the absolute value of the wavelet coefficients, and provides a
measure of the relative amplitude of each point.
This scaling reduces the weighting given to large peaks and can bring out finer-detail
features.
Decibels
The power can also be displayed in decibels, normalized relative to the mean of the
wavelet power spectrum.
Since decibels are a logarithmic scale, the smallest wavelet coefficients are given just
as much weight as the largest coefficients. This scaling is most useful for data that
contain a broad range of energy, or that contain a single sharp spike embedded in
small-amplitude noise.
db Cutoff
You can specify the lower cutoff for the Decibel plot. The default is –50 db.
Surface Style
There are seven different surface plots from which to choose:
•
Points— places colored dots at each location/height;
•
Mesh— creates an unfilled surface plot;
•
Surface— creates a shaded filled surface;
•
XZ Lines— draws lines parallel to the X-axis, one for each Y location;
•
YZ Lines— draws lines parallel to the Y-axis, one for each X location;
•
Lego— draws a lego-block plot with mesh sides;
•
Lego filled— draws a lego-block plot with solid sides.
You can also use the buttons to remove or add a “Skirt” around the surface, make the
surface “Flat”, or change to a “Gray” palette.
Wavelet Power Spectrum
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Chapter 2: Using the IDL Wavelet Toolkit
37
Contour Lines
You can choose to include contour lines at the top of the plot, the bottom, or three
dimensional.
Color Contours
You can also put color contours at the top, bottom, or 3D. The color contours can be
either open or filled. The color palette is the same as that used for the surface plot.
Tip
To produce a shaded surface with contours, make the surface “Shaded”, set the
“Gray” button, and select “3D” color contours.
Significance
The statistical significance of each point in the wavelet power spectrum can be
plotted as a three-dimensional sheet, or as contours on the top, bottom, or 3D. Points
in the wavelet power spectrum that lie above the sheet (or within the contours) are
said to be “significant at the xx% level,” where xx is your chosen percentage. You
can choose the significance level as 10%, 5%, 1%, or 0.1%.
Note
The significance level is given by the chi-square function with one degree of
freedom for real wavelet functions, or two degrees of freedom for complex wavelets
(such as the Morlet). This significance is relative to the wavelet power spectrum of
a random dataset (assuming Gaussian “white noise”).
Power Display
The graphics window contains the three-dimensional image and a color palette.
If you move the mouse cursor over points in the image, the current location and
power will be displayed in the Status Bar.
Rotation, Translation, Stretching
To rotate the image, click on the image while holding down the left mouse button,
and drag the mouse pointer to rotate the image about the midpoint.
To translate the image, click on the image while holding down the right mouse button
(on the Macintosh hold down the command key also), and drag the mouse pointer.
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Wavelet Power Spectrum
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Chapter 2: Using the IDL Wavelet Toolkit
To stretch the image, click on the image while holding the middle mouse button (on
Windows hold down the Ctrl key also; on Macintosh hold down the Option key).
Drag the mouse pointer right/left to stretch/shrink in the X-direction, drag the pointer
up/down to stretch/shrink in the Y-direction.
Wavelet Power Spectrum
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Chapter 2: Using the IDL Wavelet Toolkit
39
Multiresolution Analysis
Multiresolution Analysis uses the wavelet transform to decompose a data series in a
cascade from the smallest scales to the largest. At each scale there are three
components: the Smooth (or low-pass filtered) data series, the Details (or band-pass)
data series, and the Rough (or high-pass).
For one-dimensional vectors, this can be viewed as a hierarchy of x-y plots, as shown
in the following figure:
For two-dimensional arrays, the multiresolution analysis gives a series of images.
File Menu
Page Setup
This menu item sets up the page height and width for exporting and printing.
Export Postscript
Export the image to a postscript file.
Printer Setup
This menu item allows you to set up the printer via the Printer Dialog.
Print
This menu item prints the image.
Close
This menu item closes the Multiresolution viewer.
Wavelet Options
You can change the current wavelet family, or the order. The plot will be updated
automatically.
IDL Wavelet Toolkit
Multiresolution Analysis
40
Chapter 2: Using the IDL Wavelet Toolkit
Figure 2-4: Multiresolution Analysis of the “Chirp” Variable
Multiresolution Analysis
IDL Wavelet Toolkit
Chapter 2: Using the IDL Wavelet Toolkit
41
Denoise Tool
You can use the Denoise Tool to explore different techniques for removing noise and
compressing data using the wavelet transform.
The Denoise Tool is shown in the following figure. The plots and options are
described below.
Figure 2-5: The Denoise Tool
IDL Wavelet Toolkit
Denoise Tool
42
Chapter 2: Using the IDL Wavelet Toolkit
File Menu
Open State...
This menu item opens a previously saved “state” file into a new window.
Save State...
This menu item saves the current state of the Denoise Tool into a file.
Close
This menu item closes the Denoise Tool viewer.
Original Data
This window displays a graph of the original one-dimensional vector or twodimensional image. For images, all values are converted to an intensity (0–255) and a
grayscale color palette is used.
Filtered Data
This window displays the data after filtering using the wavelet function and options
given on the right. For images, all values are converted to an intensity (0–255) and a
grayscale color palette is used.
Wavelet Coefficients
The filtered coefficients are displayed as a two-dimensional image using a
logarithmic energy scaling. The method is as follows:
1. Find the maximum value “Pm” of the original, unfiltered, wavelet power
(absolute-value squared of the wavelet coefficients);
2. Square the filtered wavelet coefficients to get wavelet power, then take the
base-10 logarithm of each;
3. Scale this logarithmic power from the range [–10 Log10(Pm), Log10(Pm)] into
the range [32, 255]. Values greater than zero but less than –10 Log10(Pm) are
set equal to 32.
4. Set all values removed by the filter to zero (0).
5. Display the image using a grayscale color palette.
Denoise Tool
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Chapter 2: Using the IDL Wavelet Toolkit
43
Using the above method, all retained coefficients will appear in the image, shaded
from dark gray (32) to white (255). Coefficients that have been removed will be
black.
Coefficient Power
This graph shows the wavelet power for each coefficient, sorted into decreasing
order, and scaled so that the total power is 100%. The wavelet power is also shown as
a cumulative plot, where each point represents the sum of all of the previous points.
Both curves are plotted on a logarithmic x-axis, so that the largest coefficients are
easily visible.
The dashed line shows the current cutoff value that you have selected.
Wavelet Options
You can change the current wavelet family or the order. Since all of the denoise
options remain constant, you can compare the effects of different wavelet orders and
families.
Denoise Options
Cumulative Power
The slider bar allows you to set the cutoff threshold for cumulative power.
Coefficients to the right of the dotted line in the Coefficient Power graph will be
excluded. The # Coeffs box is adjusted accordingly.
Note
At low cumulative power you may notice that the slider adjusts itself in uneven
increments. This is designed so that at least one additional coefficient is either
discarded (as the slider moves left) or retained (as the slider moves right). These
jumps in power correspond to the discrete steps in the coefficient power graph.
Number of Coefficients
You can specify the exact number of coefficients that you wish to retain. The
cumulative power slider bar will be adjusted accordingly.
Hard Threshold
The hard threshold removes all discarded wavelet coefficients by setting them to zero
and computing the inverse wavelet transform. For details see “Denoise” on page 51.
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Denoise Tool
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Chapter 2: Using the IDL Wavelet Toolkit
Soft Threshold
The soft threshold also sets all discarded wavelet coefficients to zero. However, it
also linearly reduces the magnitude of the each retained wavelet coefficient by an
amount equal to the largest discarded coefficient. For details see “Denoise” on
page 51.
Results Window
This text window contains the following output results:
Threshold
The threshold is the actual wavelet power (in the variable’s units squared) that is used
for the cutoff value.
Percent of Coefficients
This is the percent number of coefficients used in the reconstruction. The smaller the
percent coefficients the more efficient the filter.
RMS Difference
This is the root-mean-square difference between the original data (upper-left plot)
and the filtered data (upper-right plot) in the variable units. A smaller number implies
a more accurate reconstruction.
Percent Difference
This is the percent difference between the original and filtered data, and is equal to
100% x (RMS difference/StdDev) where StdDev is the standard deviation of the
original data. The smaller the percent difference, the more accurate the
reconstruction.
Function Call
The text under Function Call contains the actual IDL code used to call the
WV_DENOISE function. See “WV_DENOISE” on page 69 to copy this code into
your own programs to call the denoise function directly.
Denoise Tool
IDL Wavelet Toolkit
Chapter 2: Using the IDL Wavelet Toolkit
45
Adding User Tools
You can extend the capabilities of the IDL Wavelet Toolkit by adding your own userdefined tool functions. These wavelet functions should follow the same calling
mechanism as the built-in tool functions such as “WV_TOOL_DENOISE” on
page 109. In addition, your tool function should begin with the prefix 'wv_tool_'.
1. Let’s say you want to add a wavelet tool called “Edge Detect” that uses the
wavelet transform to detect edges in images. To do this, first create a tool
function that accepts a data array and possibly other variable parameters:
FUNCTION wv_tool_edgedetect, $
Array ; 1D vector or 2D array
[,X] ; X coordinates of array
[,Y] ; Y coordinates of array
[, GROUP_LEADER=group_leader]
[, TITLE=title] [, UNITS=units]
[, XTITLE=xtitle] [, XUNITS=xunits]
[, YTITLE=ytitle] [, YUNITS=yunits]
[, XOFFSET=xoffset] [, YOFFSET=yoffset]
; start the edge detection applet...
...
; return the Widget ID for the applet
RETURN, wID
END
2. Save this function in a file wv_tool_edgedetect.pro that is accessible
from your current IDL path.
3. Now start the Wavelet Toolkit with your new wavelet function:
WV_APPLET, TOOLS=['Edge Detect']
Your new tool should appear in the Tools Menu. The actual function name is
constructed by removing all white space from the name and attaching a prefix of
WV_TOOL_.
Note
At a minimum, your tool function must accept a data Array. All other parameters
(such as X and Y) and keywords (GROUP_LEADER, TITLE, etc.) are optional. The
IDL Wavelet Toolkit will pass in only those parameters and keywords that are
usable by your tool function.
IDL Wavelet Toolkit
Adding User Tools
46
Adding User Tools
Chapter 2: Using the IDL Wavelet Toolkit
IDL Wavelet Toolkit
Chapter 3
Theory and Examples
This chapter discusses the following topics:
Wavelet Transform . . . . . . . . . . . . . . . . . . . . 48
Wavelet Power Spectrum . . . . . . . . . . . . . . . 49
Denoise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
IDL Wavelet Toolkit
Multiresolution Analysis . . . . . . . . . . . . . . 54
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 55
47
48
Chapter 3: Theory and Examples
Wavelet Transform
Background
Wavelet analysis is a technique to transform an array of N numbers from their actual
numerical values to an array of N wavelet coefficients.
Each wavelet coefficient represents the closeness of the fit (or correlation) between
the wavelet function at a particular size and a particular location within the data array.
By varying the size of the wavelet function (usually in powers-of-two) and shifting
the wavelet so it covers the entire array, you can build up a picture of the overall
match between the wavelet function and your data array.
Since the wavelet functions are compact (hence the term wave-let), the wavelet
coefficients only measure the variations around a small region of the data array. This
property makes wavelet analysis very useful for signal or image processing; the
“localized” nature of the wavelet transform allows you to easily pick out features in
your data such as spikes (for example, noise or discontinuities), discrete objects (in,
for example, astronomical images or satellite photos), edges of objects, etc.
The localization also implies that a wavelet coefficient at one location is not affected
by the coefficients at another location in the data. This makes it possible to remove
“noise” of all different scales from a signal, simply by discarding the lowest wavelet
coefficients.
For a general introduction to the wavelet transform and its applications see Hubbard
(1998).
Method
The IDL Wavelet Toolkit uses the continuous and discrete wavelet transforms.
Details on the discrete wavelet transform can be found in Daubechies (1992) and
Mallat (1989). A good introduction to the DWT and multiresolution analysis is given
in Lindsay et al. (1996).
The DWT routines are based on the routines described in section 13.10 of Numerical
Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press),
and are used by permission.
An introduction to the continuous wavelet transform for time series analysis can be
found in Torrence and Compo (1998), along with a discussion of statistical
significance testing.
Wavelet Transform
IDL Wavelet Toolkit
Chapter 3: Theory and Examples
49
Wavelet Power Spectrum
Background
The wavelet coefficients yield information as to the correlation between the wavelet
(at a certain scale) and the data array (at a particular location). A larger positive
amplitude implies a higher positive correlation, while a large negative amplitude
implies a high negative correlation.
A useful way to determine the distribution of energy within the data array is to plot
the wavelet power, equivalent to the amplitude-squared. By looking for regions
within the Wavelet Power Spectrum (WPS) of large power, you can determine which
features of your signal are important and which can be ignored.
Method
Given the wavelet transform Wi of a multi-dimensional data array, Ai, where i=0...N–
1 is the index and N is the number of points, then the Wavelet Power Spectrum is
defined as the absolute-value squared of the wavelet coefficients, |Wi|2.
One-dimensional Vector
For a vector (such as a time series) the coefficients of wavelet power can be
rearranged to yield a two-dimensional picture, where the first dimension is the
independent variable (e.g. time) and the second dimension is the wavelet scale (e.g.
1/frequency).
Two-dimensional Array
The wavelet transform of a 2D array is also two-dimensional, and is arranged so that
the smallest scales are in the upper-right quadrant (assuming that index [0, 0] is in the
lower-left).
Example
Use the “Chirp” dataset that is included in the Wavelet sample file. This dataset
contains a time series with a sine wave that has an exponentially-increasing
frequency. You can use the Multiresolution Analysis viewer to examine the time
series.
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Wavelet Power Spectrum
50
Chapter 3: Theory and Examples
Try the following steps:
1. From the main window, select the Chirp dataset and start the Wavelet Power
Spectrum viewer using either the Visualize Menu or the Toolbar button. The
WPS can be seen under “Wavelet Power Spectrum” on page 32.
2. Select the Morlet wavelet function from the Family dropdown box. You
should be able to see the exponential increase in frequency as a band of high
power extending from left to right, and ranging from about Scale=256 sec. near
the beginning to Scale=16 sec. near the end of the time series.
3. To bring out the features more clearly, change the Energy Scaling dropdown
item from Power to Magnitude.
4. Notice the large peak near Scale=256 sec. This is primarily due to the
discontinuity that occurs when the dataset is wrapped around from the end
back to the beginning. Move the Order slider bar from 6 to 4 to make the peak
more narrow.
Tip
You can use your mouse to rotate, zoom in or out, or move the plot.
5. To find the chirp peaks, select the Zero Phase Lines check box.
6. Now deselect the 3D check box to view the surface from above.
Figure 3-1: The Wavelet Power Spectrum of the Chirp Signal
Wavelet Power Spectrum
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Chapter 3: Theory and Examples
51
Denoise
Background
One of the most useful applications of wavelet analysis is to remove unwanted noise
from a dataset. This noise could be due to measurement errors or instrument noise. In
image processing the “noise” might be small-scale features or artifacts.
You could try to remove noise from the signal by using a low-pass or band-pass
Fourier filter. There are two problems with this approach:
1. You need to carefully choose the width and shape of your filter, both to avoid
removing too much of your signal and to decrease “ringing” from peaks and
discontinuities, and,
2. In many cases the noise is “white,” in other words, it is distributed across all
frequencies or spatial scales.
Wavelet analysis, on the other hand, offers a scale-independent and robust method to
filter out noise. The basic technique involves computing the wavelet transform of
your data and then decreasing or discarding the smallest wavelet coefficients. The
inverse transform of these coefficients will then be a filtered version of your data.
Method
We assume that you have computed the wavelet transform Wi of a multi-dimensional
data array, Ai, where i=0...N–1 is the index and N is the number of points.
You then compute a threshold level W0. This threshold level can be based on the
percent of wavelet power that you wish to retain, the number of coefficients, or some
other method. Suggestions for choosing the threshold are given in Donoho and
Johnstone (1994). Wavelet coefficients smaller than this threshold are discarded
while those above are retained. There are two methods for thresholding:
Hard threshold
The hard threshold removes all discarded wavelet coefficients by setting them to zero
and computing the inverse wavelet transform. This can be defined as:
⎧
⎪ W
Wi = ⎨ i
⎪ 0
⎩
IDL Wavelet Toolkit
Wi > W0
Wi ≤ W0
Denoise
52
Chapter 3: Theory and Examples
where Wi is the wavelet coefficient and W0 is the chosen threshold level.
Soft Threshold
The soft threshold also sets all discarded wavelet coefficients to zero. However, it
also linearly reduces the magnitude of the each retained wavelet coefficient by an
amount equal to the largest discarded coefficient, i.e.:
⎧
⎪ sgn ( W i ) ( W i – W 0 )
Wi = ⎨
0
⎪
⎩
Wi > W 0
Wi ≤ W 0
where sgn(Wi) is the sign of Wi.
Example
We will look at a magnetic-resonance image (MRI) of the brain, and use the
Denoising widget tool to filter out unwanted speckles and compress the size of the
image.
Try the following steps:
1. In WV_APPLET, choose File → Import → Image File, and navigate to the
examples/data directory in the IDL distribution.
2. Import the file mr_brain.dcm. The file should contain a 256 x 256 unsigned
integer (UINT) image.
3. In the Dataset Viewer, change the Title field to ‘MRI Brain Image’ and the
Variable field to 'Brain'.
4. Select the Brain dataset and start up the Denoise tool from the Tools Menu.
You should see the Denoise widget, with the threshold set to 100% and all
coefficients retained.
5. Set the # coeffs threshold to 8192 points. You should then see a view similar to
that of the following figure.
Denoise
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53
Figure 3-2: The Denoise Widget for the MRI Brain Scan
Notice that you have retained 12.5% of the coefficients and have discarded 87.5%.
The black regions of the “Wavelet Coeffs” plot shows the discarded coefficients. The
percent difference between the original and filtered image is about 6%. Examining
the filtered image, you will notice that much of the speckling around the outside is
now gone. In addition, some of the small-scale features and low-contrast regions
within the image have been diminished. Finally, the dotted line on the Cumulative
Power graph indicates that although you are only retaining 12.5% of the information
you are preserving almost 100% of the variance, or power.
IDL Wavelet Toolkit
Denoise
54
Chapter 3: Theory and Examples
Multiresolution Analysis
Background
The wavelet transform can be thought of as a band-pass filter, where the location and
width in Fourier space depends on the wavelet scale. Larger scales imply a lower
frequency and small bandwidth.
In computing the wavelet transform, you change from small scales to larger scales. At
each stage you can stop and compute the inverse wavelet transform using the
remaining coefficients, while setting the small-scale coefficients to zero. You can
then build up a series of smooth (or low-passed), detailed (or band-passed), or rough
(high-passed) versions of your original data.
Method
Details on computing the multiresolution analysis can be found in Lindsay et al.
(1996).
Example
Use the “Mantle convection” dataset that is included in the Wavelet sample file. This
dataset contains an image of convection within the Earth’s mantle.
Try the following steps:
1. Select the Convection dataset and start up the Multiresolution Analysis viewer
using either the Visualize Menu or the Toolbar button.
2. As you progress from top to bottom the wavelet scale increases in powers of
two. At the smallest scale most of the image is still in the Smooth image.
Notice that the Rough image contains only the edges or discontinuities which
the small scales can pick out.
3. Change to the Haar wavelet and observe the different structure of the images.
Multiresolution Analysis
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55
Bibliography
Daubechies, I., 1992: Ten Lectures on Wavelets. Society for Industrial and Applied
Mathematics, 357 pp.
Donoho, D. L. and I. M. Johnstone, 1994: Ideal spatial adaptation by wavelet
shrinkage. Biometrika, 81, 425–455.
Hubbard, B. B., 1998: The World According to Wavelets, 2nd ed. A. K. Peters,
Wellesley, Mass., 331 pp.
Lindsay, R. W., D. B. Percival, and D. A. Rothrock, 1996: The discrete wavelet
transform and the scale analysis of the surface properties of sea ice. IEEE Trans.
Geosci. Remote Sens., 34, 771–787.
Mallat, S., 1989: Multiresolution approximation and wavelets. Trans. Amer. Math.
Soc., 315, 69–88.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992: Numerical
Recipes in C: The Art of Scientific Computing, 2nd ed. Cambridge University Press,
994 pp.
Torrence, C., and G. P. Compo, 1998: A practical guide to wavelet analysis. Bull.
Amer. Meteor. Soc., 79, 61–78.
IDL Wavelet Toolkit
Bibliography
56
Bibliography
Chapter 3: Theory and Examples
IDL Wavelet Toolkit
Chapter 4
IDL Wavelet
Toolkit Reference
This reference lists the following topics:
List of Commands by Functionality . . . . . .
WV_APPLET . . . . . . . . . . . . . . . . . . . . . . .
WV_CW_WAVELET . . . . . . . . . . . . . . . . .
WV_CWT . . . . . . . . . . . . . . . . . . . . . . . . . .
WV_DENOISE . . . . . . . . . . . . . . . . . . . . . .
WV_DWT . . . . . . . . . . . . . . . . . . . . . . . . . .
WV_FN_COIFLET . . . . . . . . . . . . . . . . . . .
WV_FN_DAUBECHIES . . . . . . . . . . . . . .
WV_FN_GAUSSIAN . . . . . . . . . . . . . . . . .
WV_FN_HAAR . . . . . . . . . . . . . . . . . . . . .
IDL Wavelet Toolkit
58
60
62
66
69
74
78
80
82
85
WV_FN_MORLET . . . . . . . . . . . . . . . . . . 87
WV_FN_PAUL . . . . . . . . . . . . . . . . . . . . . 90
WV_FN_SYMLET . . . . . . . . . . . . . . . . . . 93
WV_IMPORT_DATA . . . . . . . . . . . . . . . . 96
WV_IMPORT_WAVELET . . . . . . . . . . . . 99
WV_PLOT3D_WPS . . . . . . . . . . . . . . . . 101
WV_PLOT_MULTIRES . . . . . . . . . . . . . 104
WV_PWT . . . . . . . . . . . . . . . . . . . . . . . . . 107
WV_TOOL_DENOISE . . . . . . . . . . . . . . 109
57
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Chapter 4: IDL Wavelet Toolkit Reference
List of Commands by Functionality
Widget Commands and Visualization Tools
The following table describes the widget and visualization tools:
Command
Description
WV_APPLET
Run IDL Wavelet Toolkit GUI (graphical user
interface).
WV_CW_WAVELET
Compound widget to display and select wavelets.
WV_IMPORT_DATA
Import data from the IDL> command prompt.
WV_IMPORT_WAVELET
Import wavelet functions into the current applet.
WV_PLOT3D_WPS
Run the wavelet power spectrum GUI.
WV_PLOT_MULTIRES
Run the multiresolution analysis GUI.
WV_TOOL_DENOISE
Run the wavelet de-noising GUI.
Table 4-1: Widget Commands and Tools
Wavelet Transform
The following table describes the wavelet transform commands:
Command
Description
WV_CWT
Compute the continuous wavelet transform of an array.
WV_DENOISE
Denoise an array using the discrete wavelet transform.
WV_DWT
Compute the discrete wavelet transform of an array.
WV_PWT
Compute the partial wavelet transform of a vector.
Table 4-2: Wavelet Transform Commands
List of Commands by Functionality
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
59
Wavelet Functions
The following table describes the built-in wavelet functions:
Command
Description
WV_FN_COIFLET
Construct coiflet wavelet coefficients.
WV_FN_DAUBECHIES
Construct Daubechies wavelet coefficients.
WV_FN_GAUSSIAN
Construct the Gaussian wavelet function.
WV_FN_HAAR
Construct Haar wavelet coefficients.
WV_FN_MORLET
Construct the Morlet wavelet function.
WV_FN_PAUL
Construct the Paul wavelet function.
WV_FN_SYMLET
Construct symlet wavelet coefficients.
Table 4-3: Wavelet Basis Functions
IDL Wavelet Toolkit
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Chapter 4: IDL Wavelet Toolkit Reference
WV_APPLET
The WV_APPLET procedure runs the IDL Wavelet Toolkit graphical user interface.
Note
The IDL Wavelet Toolkit must be licensed on your system to be able to use this
procedure.
Syntax
WV_APPLET [, Input] [, ARRAY=array] [, GROUP_LEADER=widget_id]
[, /NO_SPLASH] [, TOOLS=string array] [, WAVELETS=string or string array]
Arguments
Input
Input can be either a string giving the name of a IDL Wavelet Toolkit save file, or a
one- or two-dimensional array of data. If Input is not specified, then the sample file
wv_sample.sav is opened. If Input is set to null string ('') then the IDL Wavelet
Toolkit is started with an empty dataset.
Keywords
ARRAY
Set this keyword to a one- or two-dimensional array of data to be imported into the
IDL Wavelet Toolkit upon startup. If argument Input is set to a filename then ARRAY
will be added to the list of variables.
GROUP_LEADER
The widget ID of an existing widget that serves as group leader for the newly-created
widget. When a group leader is killed, for any reason, all widgets in the group are
also destroyed.
A given widget can be in more than one group. The WIDGET_CONTROL procedure
can be used to add additional group associations to a widget. For more information
see “WIDGET_CONTROL” (IDL Reference Guide). It is not possible to remove a
widget from an existing group.
WV_APPLET
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NO_SPLASH
If this keyword is set then the splash screen will not be displayed on startup.
TOOLS
A scalar string or vector of strings giving the names of user-defined functions to be
included in the WV_APPLET Tools menu. The actual function names are constructed
by removing all white space from each name and attaching a prefix of WV_TOOL_.
WAVELETS
A scalar string or vector of strings giving the names of user-defined wavelet functions
to be included in WV_APPLET. The actual function names are constructed by
removing all white space from each name and attaching a prefix of WV_FN_.
Examples
WV_APPLET, TOOLS=['Renormalize','My Tool']
The above statement will start up the Wavelet Toolkit, and add the user tools
'Renormalize' and 'My Tool' to the Tools menu. When these are selected the
actual functions that will be called are WV_TOOL_RENORMALIZE and
WV_TOOL_MYTOOL.
Version History
5.3
Introduced
See Also
WV_CW_WAVELET, WV_IMPORT_DATA, WV_IMPORT_WAVELET,
WV_PLOT3D_WPS, WV_PLOT_MULTIRES, WV_TOOL_DENOISE
IDL Wavelet Toolkit
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Chapter 4: IDL Wavelet Toolkit Reference
WV_CW_WAVELET
The WV_CW_WAVELET function is a compound widget that lets the user select and
display wavelet functions. WV_CW_WAVELET is accessible from the Visualize
Menu of WV_APPLET.
Note
The IDL Wavelet Toolkit must be licensed on your system to be able to use this
function.
Syntax
Result = WV_CW_WAVELET( [Parent] [, /DISCRETE] [, /NO_COLOR]
[, /NO_DRAW_WINDOW] [, TITLE=string] [, UNAME=string]
[, UVALUE=value] [, VALUE=structure] [, WAVELETS=string array] )
Return Value
The returned value of this function is the widget ID of the newly-created widget.
Arguments
Parent
The widget ID of the parent widget. Omit this argument to created a top-level widget.
Keywords
DISCRETE
Set this keyword to include only discrete wavelets in the list of wavelet functions. Set
this keyword to zero to include only continuous wavelets. The default is to include all
available wavelets.
NO_COLOR
If this keyword is set, the wavelet functions will be drawn in black and white.
NO_DRAW_WINDOW
If this keyword is set, the draw window will not be included within the widget.
WV_CW_WAVELET
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TITLE
Set this keyword equal to a scalar string containing the title of the top level base.
TITLE is not used if the wavelet widget has a parent widget. If it is not specified, the
default title is “Wavelets.”
UNAME
Set this keyword to a string that can be used to identify the widget in your code. You
can associate a name with each widget in a specific hierarchy, and then use that name
to query the widget hierarchy and get the correct widget ID.
To query the widget hierarchy, use the WIDGET_INFO function with the
FIND_BY_UNAME keyword. See “WIDGET_INFO” (IDL Reference Guide) for
more information. The UNAME should be unique to the widget hierarchy because
the FIND_BY_UNAME keyword returns the ID of the first widget with the specified
name.
UVALUE
Set this keyword equal to the user value associated with the widget.
VALUE
Set this keyword to an anonymous structure of the form {FAMILY:'', ORDER:0d}
representing the initial value for the widget.
WAVELETS
A scalar string or vector of strings giving the names of user-defined wavelet functions
to be included in WV_CW_WAVELET. The actual function names are constructed
by removing all white space from each name and attaching a prefix of WV_FN_.
Widget Keywords Accepted
The WV_CW_WAVELET function also accepts the following WIDGET_BASE
keywords: ALIGN_BOTTOM, ALIGN_CENTER, ALIGN_LEFT, ALIGN_RIGHT,
ALIGN_TOP, DISPLAY_NAME, FRAME, GROUP_LEADER,
KBRD_FOCUS_EVENTS, MAP, NOTIFY_REALIZE, RESOURCE_NAME,
SCR_XSIZE, SCR_YSIZE, SPACE, TLB_FRAME_ATTR, TRACKING_EVENTS,
UNITS, XOFFSET, XSIZE, YOFFSET, YSIZE. See “WIDGET_BASE” (IDL
Reference Guide) for more information.
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Keywords to WIDGET_CONTROL and WIDGET_INFO
The widget ID returned by most compound widgets is actually the ID of the
compound widget’s base widget. This means that many keywords to the
WIDGET_CONTROL and WIDGET_INFO routines that affect or return information
on base widgets can be used with compound widgets.
In addition, you can use the GET_VALUE and SET_VALUE keywords to
WIDGET_CONTROL to obtain or set the value of the wavelet. Use the command:
WIDGET_CONTROL, id, GET_VALUE=value
to read the current wavelet. To change the current wavelet, use the command:
WIDGET_CONTROL, id, SET_VALUE=value
In both cases value is an anonymous structure, {FAMILY: '', ORDER: 0}, where
FAMILY is a string containing the name (for example ‘Daubechies’), and ORDER is
a variable giving the order number. Depending on the family, ORDER can be of type
Integer or Double.
See “Creating a Compound Widget” (Chapter 2, Widget Application Programming)
for a more complete discussion of controlling compound widgets using
WIDGET_CONTROL and WIDGET_INFO.
Widget Events Returned by the WV_CW_WAVELET
Widget
This widget generates event structures each time the family or order is changed. The
event structure has the following definition:
Event = { ID:0L, TOP:0L, HANDLER:0L, FAMILY:'', ORDER:0}
The ID field is the widget ID of the WV_CW_WAVELET widget. The TOP field is
the widget ID of the top-level widget. HANDLER is the widget ID of the widget
handler. The FAMILY field contains the family name. The ORDER field contains
the order number, and can be an Integer or a Double depending on the family.
Version History
5.3
WV_CW_WAVELET
Introduced
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65
See Also
WV_FN_COIFLET, WV_FN_DAUBECHIES, WV_FN_GAUSSIAN,
WV_FN_HAAR, WV_FN_MORLET, WV_FN_PAUL, WV_FN_SYMLET
IDL Wavelet Toolkit
WV_CW_WAVELET
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Chapter 4: IDL Wavelet Toolkit Reference
WV_CWT
The WV_CWT function returns the one-dimensional continuous wavelet transform
of the input array. The transform is done using a user-inputted wavelet function.
Syntax
Result = WV_CWT(Array, Family, Order [, /DOUBLE] [, DSCALE=scalar]
[, NSCALE=scalar] [, /PAD] [, SCALE=variable] [, START_SCALE=scalar])
Return Value
The result is a two-dimensional array of type complex or double complex, containing
the continuous wavelet transform of the input Array.
Arguments
Array
A one-dimensional array of length N, of floating-point or complex type.
Family
A scalar string giving the name of the wavelet function to use for the transform.
Order
The order number, or parameter, for the wavelet function given by Family.
Keywords
DOUBLE
Set this keyword to force the computation to be done in double-precision arithmetic.
DSCALE
Set this keyword to a scalar value giving the spacing between scale values, in
logarithmic units. The default is 0.25, which gives four subscales within each major
scale.
WV_CWT
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NSCALE
Set this keyword to a scalar value giving the total number of scale values to use for
the wavelet transform. The default is [log2(N/START_SCALE)]/DSCALE+1.
PAD
Set this keyword to force Array to be padded with zeroes before computing the
transform. Enough zeroes are added to make the total length of Array equal to the
next-higher power-of-two greater than 2N. Padding with zeroes prevents wraparound
of the Array and speeds up the fast Fourier transform.
Note
Padding with zeroes reduces, but does not eliminate, edge effects caused by the
discontinuities at the start and end of the data.
SCALE
Set this keyword to a named variable in which to return the scale values used for the
continuous wavelet transform. The SCALE values range from START_SCALE up to
START_SCALE·2^[(NSCALE–1)DSCALE].
START_SCALE
Set this keyword to a scalar value giving the starting scale, in non-dimensional units.
The default is 2, which gives a starting scale that is twice the spacing between Array
elements.
Reference
Torrence and Compo, 1998: A Practical Guide to Wavelet Analysis. Bull. Amer.
Meteor. Soc., 79, 61–78.
Example
; Assume we have monthly random data.
n = 500
dt = 1d/12 ; time sampling
seed = 999
data = RANDOMN(seed, n)
time = 1960 + dt*FINDGEN(n)
; Compute the wavelet transform and the power.
wave = WV_CWT(data, 'Morlet', 6, /PAD, SCALE=scales)
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wavePower = ABS(wave^2)
; Convert scales to time units.
scales *= dt
; Contour visualization.
ICONTOUR, wavePower, time, scales, /Y_LOG, YRANGE=[20,0.25], $
C_VALUE=FINDGEN(7)+1, $
/FILL, RGB_TABLE=39, $
YTITLE='Scale (years)', $
VIEW_TITLE='Wavelet Power'
; Insert a legend.
tool = ITGETCURRENT(TOOL=oTool)
void = oTool->DoAction('Operations/Insert/Legend')
Version History
5.4
Introduced
See Also
WV_DWT, WV_FN_GAUSSIAN, WV_FN_MORLET, WV_FN_PAUL
WV_CWT
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Chapter 4: IDL Wavelet Toolkit Reference
69
WV_DENOISE
The WV_DENOISE function uses the wavelet transform to filter (or de-noise) a
multi-dimensional array.
WV_DENOISE computes the discrete wavelet transform of Array, and then discards
wavelet coefficients smaller than a certain threshold. WV_DENOISE then computes
the inverse wavelet transform on the filtered coefficients and returns the result.
Syntax
Result = WV_DENOISE(Array [, Family, Order] [, COEFFICIENTS=value]
[, CUTOFF=variable] [, DENOISE_STATE=variable] [, /DOUBLE]
[, DWT_FILTERED=variable] [, PERCENT=value] [, THRESHOLD=value]
[, WPS_FILTERED=variable])
Return Value
The result is an array of the same dimensions as the input Array. If Array is double
precision or /DOUBLE is set then the result is double precision, otherwise the result
is single precision.
Arguments
Array
A one-dimensional array of length N, of floating-point or complex type.
Family
A scalar string giving the name of the wavelet function to use for the transform.
WV_DENOISE will construct the actual function name by removing all white space
and attaching a prefix of 'WV_FN_'.
Note
WV_DENOISE may only be used with discrete wavelets, such as
WV_FN_COIFLET, WV_FN_DAUBECHIES, WV_FN_HAAR, and
WV_FN_SYMLET.
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Order
The order number, or parameter, for the wavelet function given by Family. If not
specified the default for the wavelet function will be used.
Note
If you pass in a DENOISE_STATE structure, then Family and Order may be
omitted. In this case the values from DENOISE_STATE are used.
Keywords
COEFFICIENTS
Set this keyword to a scalar specifying the number of wavelet coefficients to retain in
the filtered wavelet transform. This keyword is ignored if keyword PERCENT is
present.
CUTOFF
Set this keyword to a named variable that, upon return, will contain the cutoff value
of wavelet power that was used for the threshold.
DENOISE_STATE
This is both an input and an output keyword. If this keyword is set to a named
variable, then on exit, DENOISE_STATE will contain the following structure:
Tag
Type
Definition
FAMILY
STRING
Name of the wavelet function used.
ORDER
DOUBLE
Order for the wavelet function.
DWT
FLT/DBLARR
Discrete wavelet transform of Array
WPS
FLT/DBLARR
Wavelet power spectrum, equal to
|DWT|^2
SORTED
FLT/DBLARR
Percent-normalized WPS, sorted
CUMULATIVE
FLT/DBLARR
Cumulative sum of SORTED
Table 4-4: The Structure Tags for DENOISE_STATE
WV_DENOISE
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Tag
71
Type
Definition
COEFFICIENTS
LONG
Number of coefficients retained
PERCENT
DOUBLE
Percent of coefficients retained
Table 4-4: The Structure Tags for DENOISE_STATE (Continued)
Note
If the DOUBLE keyword is set, then the arrays will be of type double.
Upon input, if DENOISE_STATE is set to a structure with the above form, then
DWT, WPS, SORTED, and CUMULATIVE will not be recomputed by
WV_DENOISE. This is useful if you want to make multiple calls to WV_DENOISE
using the same Array.
Warning
No error checking is made on the input values. The values should not be modified
between calls to DENOISE_STATE.
DOUBLE
Set this keyword to force the computation to be done using double-precision
arithmetic.
DWT_FILTERED
Set this keyword to a named variable in which the filtered discrete wavelet transform
will be returned.
PERCENT
Set this keyword to a scalar specifying the percentage of cumulative power to retain.
Note
If neither COEFFICIENTS nor PERCENT is present then all of the coefficients are
retained (i.e. no filtering is done).
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THRESHOLD
Set this keyword to a scalar specifying the type of threshold. The actual threshold, T,
is set using COEFFICIENTS or PERCENT. Possible values are:
Value
Description
0
Hard threshold (this is the default). The hard threshold sets all wavelet
coefficients with magnitude less than or equal to T to zero.
1
Soft threshold. The soft threshold sets all DWT[i] with magnitude less
than T to zero, and also linearly reduces the magnitude of the each
retained wavelet coefficient by T: Positive coefficients are set equal to
DWT[i] – T, while negative coefficients are set equal to DWT[i] + T.
Table 4-5: THRESHOLD Values
WPS_FILTERED
Set this keyword to a named variable in which the filtered wavelet power spectrum
will be returned.
Examples
Remove the noise from a 128 x 128 image:
image = dist(128) + 5*randomn(1, 128, 128)
; Keep only 100 out of 16384 coefficients:
denoise = WV_DENOISE(image, 'Daubechies', 2, COEFF=100, $
DENOISE_STATE=denoise_state)
window, xsize=256, ysize=155
tvscl, image, 0
tvscl, denoise, 1
xyouts, [64, 196], [5, 5], ['Image', 'Filtered'], $
/device, align=0.5, charsize=2
print, 'Percent of power retained: ', denoise_state.percent
IDL prints:
Percent of power retained:
93.151491
Change to a “soft” threshold (use DENOISE_STATE to avoid re-computing):
denoise2 = WV_DENOISE(image, COEFF=100, $
DENOISE_STATE=denoise_state, THRESHOLD=1)
WV_DENOISE
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Chapter 4: IDL Wavelet Toolkit Reference
73
Figure 4-1: Example of De-Noising an Image.
Version History
5.4
Introduced
See Also
WV_DWT, WV_TOOL_DENOISE
IDL Wavelet Toolkit
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Chapter 4: IDL Wavelet Toolkit Reference
WV_DWT
The WV_DWT function returns the multi-dimensional discrete wavelet transform of
the input Array. The transform is done by WV_PWT using a user-inputted wavelet
filter.
The length of each dimension of Array must be either a power of two (2), or must be
less than four (4). The transform is not computed over dimensions of lengths less than
four (4), but is computed over all other dimensions (for example, the wavelet
transform of an array of size [3, 256] is computed over each [1, 256] column vector).
WV_DWT is based on the routine wtn described in section 13.10 of Numerical
Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press),
and is used by permission.
Syntax
Result = WV_DWT(Array, Scaling, Wavelet, Ioff, Joff [, /DOUBLE] [, /INVERSE]
[, N_LEVELS=value])
Return Value
The result is an output array of the same dimensions as Array, containing the discrete
wavelet transform over each dimension.
Arguments
Array
The input vector or array. The length of each dimension must be either less than four
(4) or a power of two (2).
Scaling
A vector of scaling (father) coefficients, of length N.
Wavelet
A vector of wavelet (mother) coefficients, of length N.
WV_DWT
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Ioff
An integer that specifies the support offset for Scaling. To center the scaling function
over each point in Array, set Ioff to –N/2+2.
Joff
An integer that specifies the support offset for Wavelet. To center the wavelet
function over each point in Array, set Joff to –N/2+2.
Keywords
DOUBLE
Set this keyword to force the computation to be done in double-precision arithmetic.
INVERSE
If set, the inverse transform is computed. By default, the forward transform is
computed.
N_LEVELS
Set this keyword to the number of wavelet levels to compute in the pyramid
algorithm, starting with the smallest wavelet scale and progressing to larger scales. If
this keyword is not set or is set to zero, then all wavelet levels in the pyramid
algorithm are computed.
Method and Result Format
The WV_DWT function computes the wavelet coefficients using the pyramidal
algorithm (Mallat 1989).
One-Dimensional Vector
For a one-dimensional vector, the pyramid appears below:
Array elements
[ 0, 1, 2, 3,
\ /
\ /
s0,d0
s1,d1
\
/
\ /
S0,D0
\
\
IDL Wavelet Toolkit
4, 5, 6, 7,
\ /
\ /
s2,d2
s3,d3
\
/
\ /
S1,D1
/
/
8, 9, 10, 11, 12, 13, 14, 15]
\ /
\ /
\ /
\ /
s4,d4
s5,d5
s6,d6
s7,d7
\
/
\
/
\ /
\ /
S2,D2
S3,D3
\
/
\
/
WV_DWT
76
Chapter 4: IDL Wavelet Toolkit Reference
\ /
\ /
S0,D0
S1,D1
At each level of the hierarchy, the WV_PWT function is used to compute the scaling
coefficient Si and wavelet coefficient Di (where i represents the position). The
letters s, S, S and d, D, D represent increasing scale. The wavelet coefficients are
stored in Result in order from largest scales to smallest:
Result = [ S0, S1, D0, D1, D0, D1, D2, D3,
d0, d1, d2, d3, d4, d5, d6, d7 ]
Two-Dimensional Array
For a two-dimensional Array, the wavelet transform is computed using the pyramidal
algorithm along each dimension. The wavelet coefficients are stored in order with the
largest scales in the [0, 0] position. As an example, for an 8 x 8 Array, the Result is an
8 x 8 array with the following structure:
[0,0]
[[ A0B0
[ A0B1
A1B0
A1B1
C0B0
C0B1
C1B0
C1B1
c0B0
c0B1
c1B0
c1B1
c2B0
c2B1
c3B0 ],
c3B1 ],
[ A0D0
[ A0D1
A1D0
A1D1
C0D0
C0D1
C1D0
C1D1
c0D0
c0D1
c1D0
c1D1
c2D0
c2D1
c3D0 ],
c3D1 ],
[
[
[
[
A1d0
A1d1
A1d2
A1d3
C0d0
C0d1
C0d2
C0d3
C1d0
C1d1
C1d2
C1d3
c0d0
c0d1
c0d2
c0d3
c1d0
c1d1
c1d2
c1d3
c2d0
c2d1
c2d2
c2d3
c3d0
c3d1
c3d2
c3d3
A0d0
A0d1
A0d2
A0d3
],
],
],
]]
Here A and B represent the scale coefficients for the first and second dimensions,
respectively, The C and D represent the largest-scale wavelet coefficients for the first
and second dimensions, respectively, while c and d represent the small-scale wavelet
coefficients. Subscripts 0, 1, 2, 3 denote the position of the wavelet within the image.
Example
The following example shows how to compute the first three levels of the pyramid
algorithm using either the N_LEVELS keyword or WV_PWT:
; Construct a random vector.
n = 1024
x = randomn(s,n)
info = WV_FN_DAUBECHIES(2, wavelet, scaling, ioff, joff)
; Take the wavelet transform but stop at level 3.
WV_DWT
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
77
wv_dwtpartial = WV_DWT(x, wavelet, scaling, ioff, joff, $
N_LEVELS=3)
; First level of the pyramid algorithm.
wv_level1 = WV_PWT(x, wavelet, scaling, ioff, joff)
w_scaling1 = wv_level1[0:n/2-1] ; Left (scaling) half
w_wavelet1 = wv_level1[n/2:*]
; Right (wavelet) half
; Second level of the pyramid algorithm.
wv_level2 = WV_PWT(w_scaling1, wavelet, scaling, ioff, joff)
w_scaling2 = wv_level2[0:n/4-1] ; Left (scaling) half
w_wavelet2 = wv_level2[n/4:*]
; Right (wavelet) half
; Third level of the pyramid algorithm.
wv_level3 = WV_PWT(w_scaling2, wavelet, scaling, ioff, joff)
; Verify that using WV_DWT with N_LEVELS=3
; is the same as calling WV_PWT three times.
wv_partial123 = [wv_level3, w_wavelet2, w_wavelet1]
print, MAX(ABS(wv_dwtpartial - wv_partial123))
IDL prints:
0.000000
Version History
5.3
Introduced
6.1
Added N_LEVELS keyword
See Also
WV_CWT, WV_PWT, WTN
IDL Wavelet Toolkit
WV_DWT
78
Chapter 4: IDL Wavelet Toolkit Reference
WV_FN_COIFLET
The WV_FN_COIFLET function constructs wavelet coefficients for the coiflet
wavelet function.
Syntax
Result = WV_FN_COIFLET( [Order, Scaling, Wavelet, Ioff, Joff] )
Return Value
The returned value of this function is an anonymous structure of information about
the particular wavelet.
Tag
Type
Definition
FAMILY
STRING
‘Coiflet’
ORDER_NAME
STRING
‘Order’
ORDER_RANGE
INTARR(3)
[1, 5, 1] Valid order range [first, last, default]
ORDER
INT
The chosen Order
DISCRETE
INT
1 [0=continuous, 1=discrete]
ORTHOGONAL
INT
1 [0=nonorthogonal, 1=orthogonal]
SYMMETRIC
INT
2 [0=asymmetric, 1=symm., 2=near symm.]
SUPPORT
INT
6*Order – 1 [Compact support width]
MOMENTS
INT
2*Order [Number of vanishing moments]
REGULARITY
DOUBLE
The number of continuous derivatives
Table 4-6: Structure Tags for Result
Arguments
Order
A scalar that specifies the order number for the wavelet. The default is 1.
WV_FN_COIFLET
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
79
Scaling
On output, contains a vector of double-precision scaling (father) coefficients.
Wavelet
On output, contains a vector of double-precision wavelet (mother) coefficients.
Ioff
On output, contains an integer that specifies the support offset for Scaling.
Joff
On output, contains an integer that specifies the support offset for Wavelet.
Note
If none of the above arguments are present then the function will simply return the
Result structure using the default Order.
Keywords
None.
Reference
Coefficients are from Daubechies, I., 1992: Ten Lectures on Wavelets, SIAM, p. 261.
Note that Daubechies has divided by Sqrt(2), and the coefficients are reversed.
Version History
5.3
Introduced
See Also
WV_DWT, WV_FN_DAUBECHIES, WV_FN_HAAR, WV_FN_SYMLET
IDL Wavelet Toolkit
WV_FN_COIFLET
80
Chapter 4: IDL Wavelet Toolkit Reference
WV_FN_DAUBECHIES
The WV_FN_DAUBECHIES function constructs wavelet coefficients for the
Daubechies wavelet function.
Syntax
Result = WV_FN_DAUBECHIES( [Order, Scaling, Wavelet, Ioff, Joff] )
Return Value
The returned value of this function is an anonymous structure of information about
the particular wavelet.
Tag
Type
Definition
FAMILY
STRING
‘Daubechies’
ORDER_NAME
STRING
‘Order’
ORDER_RANGE
INTARR(3)
[1, 24, 2] Valid order range [first, last, default]
ORDER
INT
The chosen Order
DISCRETE
INT
1 [0=continuous, 1=discrete]
ORTHOGONAL
INT
1 [0=nonorthogonal, 1=orthogonal]
SYMMETRIC
INT
0 [0=asymmetric, 1=symm., 2=near symm.]
SUPPORT
INT
2*Order – 1 [Compact support width]
MOMENTS
INT
Order [Number of vanishing moments]
REGULARITY
DOUBLE
The number of continuous derivatives
Table 4-7: Structure Tags for Result
Arguments
Order
A scalar that specifies the order number for the wavelet. The default is 2.
WV_FN_DAUBECHIES
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
81
Scaling
On output, contains a vector of double-precision scaling (father) coefficients.
Wavelet
On output, contains a vector of double-precision wavelet (mother) coefficients.
Ioff
On output, contains an integer that specifies the support offset for Scaling.
Joff
On output, contains an integer that specifies the support offset for Wavelet.
Note
If none of the above arguments are present then the function will simply return the
Result structure using the default Order.
Keywords
None.
Reference
Coefficients for orders 1–10 are from Daubechies, I., 1992: Ten Lectures on
Wavelets, SIAM, p. 195. Note that Daubechies has multiplied by Sqrt(2). Coefficients
for orders 11–24 are from http://www.isds.duke.edu/~brani/filters.html.
Version History
5.3
Introduced
See Also
WV_DWT, WV_FN_COIFLET, WV_FN_HAAR, WV_FN_SYMLET
IDL Wavelet Toolkit
WV_FN_DAUBECHIES
82
Chapter 4: IDL Wavelet Toolkit Reference
WV_FN_GAUSSIAN
The WV_FN_GAUSSIAN function constructs wavelet coefficients for the Gaussian
wavelet function. In real space, the Gaussian wavelet function is proportional to the
m-th order derivative of a Gaussian, exp(–x2/2). The Gaussian second derivative,
(x2–1) exp(–x2/2), is often referred to as the Marr wavelet.
Syntax
Result = WV_FN_GAUSSIAN( [Order] [, Scale, N]
[, /DOUBLE] [, FREQUENCY=variable] [, /SPATIAL] [, WAVELET=variable])
The returned value of this function is an anonymous structure of information about
the particular wavelet.
Tag
Type
Definition
FAMILY
STRING
‘Gaussian’
ORDER_NAME
STRING
‘Derivative’
ORDER_RANGE
DBLARR(3)
Valid orders [first, last, default]
ORDER
DOUBLE
The chosen Order
DISCRETE
INT
0 [0=continuous, 1=discrete]
ORTHOGONAL
INT
0 [0=nonorthogonal, 1=orthogonal]
SYMMETRIC
INT
1 [0=asymmetric, 1=symm.]
SUPPORT
DOUBLE
Infinity [Compact support width]
MOMENTS
INT
1 [Number of vanishing moments]
REGULARITY
DOUBLE
Infinity [Number of continuous
derivatives]
E_FOLDING
DOUBLE
SQRT(2) [Autocorrelation e-fold distance]
FOURIER_PERIOD
DOUBLE
Ratio of Fourier wavelength to scale
Table 4-8: Structure Tags for Result
WV_FN_GAUSSIAN
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
83
Arguments
Order
A scalar that specifies the non-dimensional order parameter for the wavelet. The
default is 2.
Scale
A scalar that specifies the scale at which to construct the wavelet function.
N
An integer that specifies the number of points in the wavelet function. For Fourier
space (SPATIAL=0), the frequencies are constructed following the FFT convention:
•
For N even: 0, 1/N, 2/N, ..., (N–2)/(2N), 1/2, –(N–2)/(2N), ..., –1/N.
•
For N odd: 0, 1/N, 2/N, ..., (N–1)/(2N), –(N–1)/(2N), ..., –1/N.
For real space (/SPATIAL), the spatial coordinates are –(N–1)/2...(N–1)/2.
Note
If none of the above arguments are present then the function will simply return the
Result structure using the default Order.
Keywords
DOUBLE
Set this keyword to force the computation to be done in double-precision arithmetic.
FREQUENCY
Set this keyword to a named variable in which to return the frequency array used to
construct the wavelet. This variable will be undefined if SPATIAL is set.
SPATIAL
Set this keyword to return the wavelet function in real space. The default is to return
the wavelet function in Fourier space.
WAVELET
Set this keyword to a named variable in which to return the wavelet function.
IDL Wavelet Toolkit
WV_FN_GAUSSIAN
84
Chapter 4: IDL Wavelet Toolkit Reference
Reference
Torrence and Compo, 1998: A Practical Guide to Wavelet Analysis. Bull. Amer.
Meteor. Soc., 79, 61–78.
Examples
Plot the Gaussian wavelet function at scale=20:
n = 1000 ; pick a nice number of points
info = WV_FN_GAUSSIAN( 2, 20, n, /SPATIAL, $
WAVELET=wavelet)
plot, wavelet
Now plot the same wavelet in Fourier space:
info = WV_FN_GAUSSIAN( 2, 20, n, $
FREQUENCY=frequency, WAVELET=wave_fourier)
plot, frequency, wave_fourier, $
xrange=[-0.2,0.2], thick=2
Version History
5.4
Introduced
See Also
WV_CWT, WV_FN_MORLET, WV_FN_PAUL
WV_FN_GAUSSIAN
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
85
WV_FN_HAAR
The WV_FN_HAAR function constructs wavelet coefficients for the Haar wavelet
function.
Note
The Haar wavelet is the same as the Daubechies wavelet of order 1.
Syntax
Result = WV_FN_HAAR( [Order, Scaling, Wavelet, Ioff, Joff] )
Return Value
The returned value of this function is an anonymous structure of information about
the particular wavelet.
Tag
Type
Definition
FAMILY
STRING
‘Haar’
ORDER_NAME
STRING
‘Order’
ORDER_RANGE
INTARR(3)
[1, 1, 1] Valid order range [first, last, default]
ORDER
INT
1
DISCRETE
INT
1 [0=continuous, 1=discrete]
ORTHOGONAL
INT
1 [0=nonorthogonal, 1=orthogonal]
SYMMETRIC
INT
0 [0=asymmetric, 1=symm., 2=near symm.]
SUPPORT
INT
1 [Compact support width]
MOMENTS
INT
1 [Number of vanishing moments]
REGULARITY
DOUBLE
0d [Number of continuous derivatives]
Table 4-9: Structure Tags for Result
IDL Wavelet Toolkit
WV_FN_HAAR
86
Chapter 4: IDL Wavelet Toolkit Reference
Arguments
Order
A scalar that specifies the order number for the wavelet. The default is 1.
Scaling
On output, contains a vector of double-precision scaling (father) coefficients.
Wavelet
On output, contains a vector of double-precision wavelet (mother) coefficients.
Ioff
On output, contains an integer that specifies the support offset for Scaling.
Joff
On output, contains an integer that specifies the support offset for Wavelet.
Note
If none of the above arguments are present then the function will simply return the
Result structure using the default Order.
Keywords
None.
Reference
Daubechies, I., 1992: Ten Lectures on Wavelets, SIAM.
Version History
5.3
Introduced
See Also
WV_DWT, WV_FN_COIFLET, WV_FN_DAUBECHIES, WV_FN_SYMLET
WV_FN_HAAR
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
87
WV_FN_MORLET
The WV_FN_MORLET function constructs wavelet coefficients for the Morlet
wavelet function. In real space, the Morlet wavelet function consists of a complex
exponential modulated by a Gaussian envelope: π–1/4s–1/2 exp[i k x / s] exp[–
(x / s)2/2], where s is the wavelet scale, k is a non-dimensional parameter, and x is the
position.
Syntax
Result = WV_FN_MORLET( [Order] [, Scale, N] [, /DOUBLE]
[, FREQUENCY=variable] [, /SPATIAL] [, WAVELET=variable])
Return Value
The returned value of this function is an anonymous structure of information about
the particular wavelet.
Tag
Type
Definition
FAMILY
STRING
‘Morlet’
ORDER_NAME
STRING
‘Parameter’
ORDER_RANGE
DBLARR(3)
[3, 24, 6] Valid orders [first, last, default]
ORDER
DOUBLE
The chosen Order
DISCRETE
INT
0 [0=continuous, 1=discrete]
ORTHOGONAL
INT
0 [0=nonorthogonal, 1=orthogonal]
SYMMETRIC
INT
1 [0=asymmetric, 1=symm.]
SUPPORT
DOUBLE
Infinity [Compact support width]
MOMENTS
INT
1 [Number of vanishing moments]
REGULARITY
DOUBLE
Infinity [Number of continuous derivatives]
E_FOLDING
DOUBLE
SQRT(2) [Autocorrelation e-fold distance]
FOURIER_PERIOD
DOUBLE
Ratio of Fourier wavelength to scale
Table 4-10: Structure Tags for Result
IDL Wavelet Toolkit
WV_FN_MORLET
88
Chapter 4: IDL Wavelet Toolkit Reference
Arguments
Order
A scalar that specifies the non-dimensional order parameter for the wavelet. The
default is 6.
Scale
A scalar that specifies the scale at which to construct the wavelet function.
N
An integer that specifies the number of points in the wavelet function. For Fourier
space (SPATIAL=0), the frequencies are constructed following the FFT convention:
•
For N even: 0, 1/N, 2/N, ..., (N–2)/(2N), 1/2, –(N–2)/(2N), ..., –1/N.
•
For N odd: 0, 1/N, 2/N, ..., (N–1)/(2N), –(N–1)/(2N), ..., –1/N.
For real space (/SPATIAL), the spatial coordinates are –(N–1)/2...(N–1)/2.
Note
If none of the above arguments are present then the function will simply return the
Result structure using the default Order.
Keywords
DOUBLE
Set this keyword to force the computation to be done in double-precision arithmetic.
FREQUENCY
Set this keyword to a named variable in which to return the frequency array used to
construct the wavelet. This variable will be undefined if SPATIAL is set.
SPATIAL
Set this keyword to return the wavelet function in real space. The default is to return
the wavelet function in Fourier space.
WAVELET
Set this keyword to a named variable in which to return the wavelet function.
WV_FN_MORLET
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
89
Reference
Torrence and Compo, 1998: A Practical Guide to Wavelet Analysis. Bull. Amer.
Meteor. Soc., 79, 61–78.
Examples
Plot the Morlet wavelet function at scale=100:
n = 1000 ; pick a nice number of points
info = WV_FN_MORLET( 6, 100, n, /SPATIAL, $
WAVELET=wavelet)
plot, float(wavelet), THICK=2
oplot, imaginary(wavelet)
Now plot the same wavelet in Fourier space:
info = WV_FN_MORLET( 6, 100, n, $
FREQUENCY=frequency, WAVELET=wave_fourier)
plot, frequency, wave_fourier, $
xrange=[-0.2,0.2], thick=2
Version History
5.4
Introduced
See Also
WV_CWT, WV_FN_GAUSSIAN, WV_FN_PAUL
IDL Wavelet Toolkit
WV_FN_MORLET
90
Chapter 4: IDL Wavelet Toolkit Reference
WV_FN_PAUL
The WV_FN_PAUL function constructs wavelet coefficients for the Paul wavelet
function. In real space, the Paul wavelet function is proportional to the complex
polynomial (1 – i x / s)^(–m–1), where s is the wavelet scale, m is a non-dimensional
parameter, and x is the position.
Syntax
Result = WV_FN_PAUL( [Order] [, Scale, N] [, /DOUBLE]
[, FREQUENCY=variable] [, /SPATIAL] [, WAVELET=variable])
Return Value
The returned value of this function is an anonymous structure of information about
the particular wavelet.
Tag
Type
Definition
FAMILY
STRING
‘Paul’
ORDER_NAME
STRING
‘Parameter’
ORDER_RANGE
DBLARR(3)
[1, 20, 4] Valid orders [first, last, default]
ORDER
DOUBLE
The chosen Order
DISCRETE
INT
0 [0=continuous, 1=discrete]
ORTHOGONAL
INT
0 [0=nonorthogonal, 1=orthogonal]
SYMMETRIC
INT
1 [0=asymmetric, 1=symm.]
SUPPORT
DOUBLE
Infinity [Compact support width]
MOMENTS
INT
1 [Number of vanishing moments]
REGULARITY
DOUBLE
Infinity [Number of continuous
derivatives]
E_FOLDING
DOUBLE
1/sqrt(2) [Autocorrelation e-fold distance]
FOURIER_PERIOD
DOUBLE
Ratio of Fourier wavelength to scale
Table 4-11: Structure Tags for Result
WV_FN_PAUL
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
91
Arguments
Order
A scalar that specifies the non-dimensional order for the wavelet. The default is 4.
Scale
A scalar that specifies the scale at which to construct the wavelet function.
N
An integer that specifies the number of points in the wavelet function. For Fourier
space (SPATIAL=0), the frequencies are constructed following the FFT convention:
•
For N even: 0, 1/N, 2/N, ..., (N–2)/(2N), 1/2, –(N–2)/(2N), ..., –1/N.
•
For N odd: 0, 1/N, 2/N, ..., (N–1)/(2N), –(N–1)/(2N), ..., –1/N.
For real space (/SPATIAL), the spatial coordinates are –(N–1)/2...(N–1)/2.
Note
If none of the above arguments are present then the function will simply return the
Result structure using the default Order.
Keywords
DOUBLE
Set this keyword to force the computation to be done in double-precision arithmetic.
FREQUENCY
Set this keyword to a named variable in which to return the frequency array used to
construct the wavelet. This variable will be undefined if SPATIAL is set.
SPATIAL
Set this keyword to return the wavelet function in real space. The default is to return
the wavelet function in Fourier space.
WAVELET
Set this keyword to a named variable in which to return the wavelet function.
IDL Wavelet Toolkit
WV_FN_PAUL
92
Chapter 4: IDL Wavelet Toolkit Reference
Reference
Torrence and Compo, 1998: A Practical Guide to Wavelet Analysis. Bull. Amer.
Meteor. Soc., 79, 61–78.
Examples
Plot the Paul wavelet function at scale=100:
n = 1000 ; pick a nice number of points
info = WV_FN_PAUL( 6, 100, n, /SPATIAL, $
WAVELET=wavelet)
plot, float(wavelet), THICK=2
oplot, imaginary(wavelet)
Now plot the same wavelet in Fourier space:
info = WV_FN_PAUL( 6, 100, n, $
FREQUENCY=frequency, WAVELET=wave_fourier)
plot, frequency, wave_fourier, $
xrange=[-0.2,0.2], thick=2
Version History
5.4
Introduced
See Also
WV_CWT, WV_FN_GAUSSIAN, WV_FN_MORLET
WV_FN_PAUL
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
93
WV_FN_SYMLET
The WV_FN_SYMLET function constructs wavelet coefficients for the Symlet
wavelet function.
Note
The Symlet wavelet for orders 1–3 are the same as the Daubechies wavelets of the
same order.
Syntax
Result = WV_FN_SYMLET( [Order, Scaling, Wavelet, Ioff, Joff] )
Return Value
The returned value of this function is an anonymous structure of information about
the particular wavelet.
Tag
Type
Definition
FAMILY
STRING
‘Symlet’
ORDER_NAME
STRING
‘Order’
ORDER_RANGE
INTARR(3)
[1, 15, 4] Valid order range [first, last, default]
ORDER
INT
The chosen Order
DISCRETE
INT
1 [0=continuous, 1=discrete]
ORTHOGONAL
INT
1 [0=nonorthogonal, 1=orthogonal]
SYMMETRIC
INT
2 [0=asymmetric, 1=symm., 2=near symm.]
SUPPORT
INT
2*Order – 1 [Compact support width]
MOMENTS
INT
Order [Number of vanishing moments]
REGULARITY
DOUBLE
The number of continuous derivatives
Table 4-12: Structure Tags for Result
IDL Wavelet Toolkit
WV_FN_SYMLET
94
Chapter 4: IDL Wavelet Toolkit Reference
Arguments
Order
A scalar that specifies the order number for the wavelet. The default is 4.
Scaling
On output, contains a vector of double-precision scaling (father) coefficients.
Wavelet
On output, contains a vector of double-precision wavelet (mother) coefficients.
Ioff
On output, contains an integer that specifies the support offset for Scaling.
Joff
On output, contains an integer that specifies the support offset for Wavelet.
Note
If none of the above arguments are present then the function will simply return the
Result structure using the default Order.
Keywords
None.
Reference
Coefficients for orders 1–10 are from Daubechies, I., 1992: Ten Lectures on
Wavelets, SIAM, p. 198. Note that Daubechies has multiplied by Sqrt(2), and for
some orders the coefficients are reversed. Coefficients for orders 11–15 are from
http://www.isds.duke.edu/~brani/filters.html.
Version History
5.3
WV_FN_SYMLET
Introduced
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
95
See Also
WV_DWT, WV_FN_COIFLET, WV_FN_DAUBECHIES, WV_FN_HAAR
IDL Wavelet Toolkit
WV_FN_SYMLET
96
Chapter 4: IDL Wavelet Toolkit Reference
WV_IMPORT_DATA
The WV_IMPORT_DATA procedure allows the user to add a variable to the
currently active WV_APPLET widget from the IDL> command prompt.
Note
The IDL Wavelet Toolkit must be licensed on your system to be able to use this
procedure.
Syntax
WV_IMPORT_DATA, Data [, MESSAGE_OUT=string] [, PARENT=variable]
Arguments
Data
A one- or two-dimensional array of data, or a structure containing the data.
Keywords
MESSAGE_OUT
A scalar string giving a message to be output to the WV_APPLET message bar.
PARENT
A long integer specifying the ID of the WV_APPLET widget in which to import the
data. The default is the most-recently active WV_APPLET widget.
Examples
To import a 1D or 2D array directly into the active WV_APPLET widget:
WV_IMPORT_DATA, Array
To import a data structure:
WV_IMPORT_DATA, {DATA: PTR_NEW(Array), $
SOURCE: ’Chandra X-Ray Observatory’, $
TITLE: ’Cygnus X-1 X-Ray Image’, $
VARIABLE: ’Cygnus X-1’, $
UNITS: ’Intensity’}
WV_IMPORT_DATA
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
97
If Data is a structure then it must include at the very least a DATA tag containing a
pointer to a one- or two-dimensional array. Recognized tags are shown in the
following table. Tags other than these will be quietly ignored.
Tag
Type
Definition
DATA
PTR->Array
Pointer to data array
TITLE
STRING
Long name of data variable
VARIABLE
STRING
Short name of data variable
UNITS
STRING
Units for variable
XNAME
STRING
Name of X coordinate
XUNITS
STRING
Units for X coordinate
XSTART
STRING
Starting value for X coord
DX
STRING
Sampling rate for X coord
YNAME
STRING
Name of Y coordinate
YUNITS
STRING
Units for Y coordinate
YSTART
STRING
Starting value for Y coord
DY
STRING
Sampling rate for Y coord
XOFFSET
LONG
Starting index of X coord to use
XCOUNT
LONG
Number of X coords to use
XSTRIDE
LONG
X sampling interval to use
YOFFSET
LONG
Starting index of Y coord to use
YCOUNT
LONG
Number of Y coords to use
YSTRIDE
LONG
Y sampling interval to use
SOURCE
STRING
Filename or contact info
NOTES
STRING
Miscellaneous notes
COLORS
PTR->Bytarr(3,256)
Pointer to color table for Data
Table 4-13: Tags Recognized by WV_IMPORT_DATA
IDL Wavelet Toolkit
WV_IMPORT_DATA
98
Chapter 4: IDL Wavelet Toolkit Reference
Version History
5.3
Introduced
See Also
WV_APPLET
WV_IMPORT_DATA
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
99
WV_IMPORT_WAVELET
The WV_IMPORT_WAVELET procedure allows the user to add wavelet functions
to the currently-active IDL Wavelet Toolkit(s).
Note
Any widgets that are currently active will not have access to the new wavelet
functions until they are restarted.
Note
The IDL Wavelet Toolkit must be licensed on your system to be able to use this
procedure.
Syntax
WV_IMPORT_WAVELET [, Wavelet] [, /RESET]
Arguments
Wavelet
A string scalar or vector giving the names of the wavelet functions. The actual
function names are constructed by removing all white space from each name and
attaching a prefix of WV_FN_.
Keywords
RESET
If set, then remove all user-defined wavelets from memory. If Wavelet is also
specified then the new wavelets will be appended onto the built-in wavelets.
Version History
5.3
IDL Wavelet Toolkit
Introduced
WV_IMPORT_WAVELET
100
Chapter 4: IDL Wavelet Toolkit Reference
See Also
WV_APPLET, WV_CW_WAVELET
WV_IMPORT_WAVELET
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
101
WV_PLOT3D_WPS
The WV_PLOT3D_WPS function runs the graphical user interface for threedimensional visualization of the wavelet power spectrum. WV_PLOT3D_WPS is
accessible from the Visualize Menu of WV_APPLET.
Note
The IDL Wavelet Toolkit must be licensed on your system to be able to use this
function.
Syntax
Result = WV_PLOT3D_WPS( Array [, X] [, Y] [, GROUP_LEADER=widget_id]
[, SURFACE_STYLE=value] [, TITLE=string] [, UNITS=string]
[, XTITLE=string] [, XUNITS=string] [, YTITLE=string] [, YUNITS=string] )
Return Value
The returned variable is the Widget ID of the newly-created widget.
Arguments
Input
Input must be either a string giving the name of a file to open, or a one- or twodimensional array of data. If set to a string, the file must contain a valid
WV_PLOT3D_WPS “saved state.”
X
An optional vector of uniformly-spaced values giving the location of points along the
first dimension of Input. The default is 0, 1, 2,..., NX–1, where NX is the size of the
first dimension. This argument is ignored if Input is a filename.
Y
An optional vector of uniformly-spaced values giving the location of points along the
second dimension of Input. The default is 0, 1, 2,..., NY–1, where NY is the size of the
second dimension. This argument is ignored if Input is a filename.
IDL Wavelet Toolkit
WV_PLOT3D_WPS
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Chapter 4: IDL Wavelet Toolkit Reference
Keywords
GROUP_LEADER
The widget ID of an existing widget that serves as "group leader" for the newlycreated widget. When a group leader is killed, for any reason, all widgets in the group
are also destroyed.
A given widget can be in more than one group. The WIDGET_CONTROL procedure
can be used to add additional group associations to a widget. See
“WIDGET_CONTROL” (IDL Reference Guide) for more information. It is not
possible to remove a widget from an existing group.
Note
The following keywords are ignored if Input is a filename. This includes the
SURFACE_STYLE, TITLE, UNITS, XTITLE, XUNITS, YTITLE, and YUNITS
keywords.
SURFACE_STYLE
Set this keyword to an integer specifying the initial style to use for the threedimensional surface. Valid values are:
•
0 = Off
•
1 = Points
•
2 = Mesh
•
3 = Shaded
•
4 = XZ lines
•
5 = YZ lines
•
6 = Lego
•
7 = Lego fill
TITLE
A scalar string giving the label to be used for the widget. The default is ‘WPS:’.
UNITS
A scalar string giving the units of Array.
WV_PLOT3D_WPS
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
103
XTITLE
A scalar string giving the label to be used for the first dimension.
XUNITS
A scalar string giving the units of X.
YTITLE
A scalar string giving the label to be used for the y-axis (for a 1D vector) or for the
second dimension (for a 2D array).
YUNITS
A scalar string giving the units of Array (for a 1D vector) or the units of Y (for a 2D
array).
Widget Keywords Accepted
The WV_PLOT3D_WPS function also accepts the following WIDGET_BASE
keywords: DISPLAY_NAME, EVENT_FUNC, FRAME,
KBRD_FOCUS_EVENTS, KILL_NOTIFY, MODAL, NOTIFY_REALIZE,
RESOURCE_NAME, SCR_XSIZE, SCR_YSIZE, SPACE, TLB_FRAME_ATTR,
TRACKING_EVENTS, UNITS, XOFFSET, XSIZE, YOFFSET, YSIZE. See
“WIDGET_BASE” (IDL Reference Guide) for more information.
Version History
5.3
Introduced
See Also
WV_APPLET
IDL Wavelet Toolkit
WV_PLOT3D_WPS
104
Chapter 4: IDL Wavelet Toolkit Reference
WV_PLOT_MULTIRES
The WV_PLOT_MULTIRES function runs the graphical user interface for
multiresolution analysis. WV_PLOT_MULTIRES is accessible from the Visualize
Menu of WV_APPLET.
Note
The IDL Wavelet Toolkit must be licensed on your system to be able to use this
function.
Syntax
Result = WV_PLOT_MULTIRES( Array [, X] [, Y] [, GROUP_LEADER=widget_id]
[, TITLE=string] [, UNITS=string] [, XTITLE=string] [, XUNITS=string]
[, YTITLE=string] [, YUNITS=string])
Return Value
The returned variable is the Widget ID of the newly-created widget.
Arguments
Array
A one- or two-dimensional array of data to be analyzed.
X
An optional vector of uniformly-spaced values giving the location of points along the
first dimension of Array. The default is 0, 1, 2,..., NX–1, where NX is the size of the
first dimension.
Y
An optional vector of uniformly-spaced values giving the location of points along the
second dimension of Array. The default is 0, 1, 2,..., NY–1, where NY is the size of
the second dimension.
WV_PLOT_MULTIRES
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
105
Keywords
GROUP_LEADER
The widget ID of an existing widget that serves as “group leader” for the newlycreated widget. When a group leader is killed, for any reason, all widgets in the group
are also destroyed.
A given widget can be in more than one group. The WIDGET_CONTROL procedure
can be used to add additional group associations to a widget. See
“WIDGET_CONTROL” (IDL Reference Guide) for more information. It is not
possible to remove a widget from an existing group.
TITLE
A scalar string giving the label to be used for the widget. The default is ‘MRes:’.
UNITS
A scalar string giving the units of Array.
XTITLE
A scalar string giving the label to be used for the first dimension.
XUNITS
A scalar string giving the units of X.
YTITLE
A scalar string giving the label to be used for the y-axis (for a 1D vector) or for the
second dimension (for a 2D array).
YUNITS
A scalar string giving the units of Array (for a 1D vector) or the units of Y (for a 2D
array).
Widget Keywords Accepted
The WV_PLOT_MULTIRES function also accepts the following WIDGET_BASE
keywords: DISPLAY_NAME, EVENT_FUNC, FRAME,
KBRD_FOCUS_EVENTS, KILL_NOTIFY, MODAL, NOTIFY_REALIZE,
RESOURCE_NAME, SCR_XSIZE, SCR_YSIZE, SPACE, TLB_FRAME_ATTR,
IDL Wavelet Toolkit
WV_PLOT_MULTIRES
106
Chapter 4: IDL Wavelet Toolkit Reference
TRACKING_EVENTS, UNITS, XOFFSET, XSIZE, YOFFSET, YSIZE. See
“WIDGET_BASE” (IDL Reference Guide) for more information.
Version History
5.3
Introduced
See Also
WV_APPLET
WV_PLOT_MULTIRES
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
107
WV_PWT
The WV_PWT function returns the partial wavelet transform of the input vector A.
The transform is done using a user-inputted wavelet filter. WV_PWT is called by
WV_DWT.
WV_PWT is based on the routine pwt described in section 13.10 of Numerical
Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press),
and is used by permission.
Syntax
Result = WV_PWT( A, Scaling, Wavelet, Ioff, Joff [, /DOUBLE] [, /INVERSE] )
Return Value
The result is an output vector of the same length as A, containing one stage of the
pyramidal algorithm (Mallat 1989).
Arguments
A
The input vector. The length must be either less than four (4) or a power of two (2).
Scaling
A vector of scaling (father) coefficients, of length N.
Wavelet
A vector of wavelet (mother) coefficients, of length N.
Ioff
An integer that specifies the support offset for Scaling. To center the scaling function
over each point in Array, set Ioff to –N/2+2.
Joff
An integer that specifies the support offset for Wavelet. To center the wavelet
function over each point in Array, set Joff to –N/2+2.
IDL Wavelet Toolkit
WV_PWT
108
Chapter 4: IDL Wavelet Toolkit Reference
Keywords
DOUBLE
Set this keyword to force the computation to be done in double-precision arithmetic.
INVERSE
If set, the inverse transform is computed. By default, the forward transform is
computed.
Method and Result Format
The WV_PWT function computes the wavelet coefficients for one level of the
pyramidal algorithm. For a one-dimensional vector with 16 elements, one level of the
pyramid appears below:
Array elements
[ 0, 1, 2, 3,
\ /
\ /
s0,d0
s1,d1
4, 5,
\ /
s2,d2
6, 7,
\ /
s3,d3
8, 9, 10, 11, 12, 13, 14, 15]
\ /
\ /
\ /
\ /
s4,d4
s5,d5
s6,d6
s7,d7
where Si and Di are the scaling and wavelet coefficients and i represents the
position. The wavelet coefficients are stored in Result in the following order:
Result = [ s0, s1, s2, s3, s4, s5, s6, s7,
d0, d1, d2, d3, d4, d5, d6, d7 ]
Version History
5.3
Introduced
See Also
WV_DWT
WV_PWT
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
109
WV_TOOL_DENOISE
The WV_TOOL_DENOISE function runs the graphical user interface for wavelet
filtering and denoising. WV_TOOL_DENOISE is accessible from the Tools Menu of
WV_APPLET.
Note
The IDL Wavelet Toolkit must be licensed on your system to be able to use this
function.
Syntax
Result = WV_TOOL_DENOISE( Array [, X] [, Y] [, GROUP_LEADER=widget_id]
[, TITLE=string] [, UNITS=string] [, XTITLE=string] [, XUNITS=string]
[, YTITLE=string] [, YUNITS=string])
Return Value
The returned variable is the Widget ID of the newly-created widget.
Arguments
Array
A one- or two-dimensional array of data to be analyzed.
X
An optional vector of uniformly-spaced values giving the location of points along the
first dimension of Array. The default is 0, 1, 2,..., NX–1, where NX is the size of the
first dimension.
Y
An optional vector of uniformly-spaced values giving the location of points along the
second dimension of Array. The default is 0, 1, 2,..., NY–1, where NY is the size of
the second dimension.
IDL Wavelet Toolkit
WV_TOOL_DENOISE
110
Chapter 4: IDL Wavelet Toolkit Reference
Keywords
GROUP_LEADER
The widget ID of an existing widget that serves as "group leader" for the newlycreated widget. When a group leader is killed, for any reason, all widgets in the group
are also destroyed.
A given widget can be in more than one group. The WIDGET_CONTROL procedure
can be used to add additional group associations to a widget. See
“WIDGET_CONTROL” (IDL Reference Guide) for more information. It is not
possible to remove a widget from an existing group.
TITLE
A scalar string giving the label to be used for the widget. The default is 'Denoise:'.
UNITS
A scalar string giving the units of Array.
XTITLE
A scalar string giving the label to be used for the first dimension.
XUNITS
A scalar string giving the units of X.
YTITLE
A scalar string giving the label to be used for the y-axis (for a 1D vector) or for the
second dimension (for a 2D array).
YUNITS
A scalar string giving the units of Array (for a 1D vector) or the units of Y (for a 2D
array).
Widget Keywords Accepted
The WV_TOOL_DENOISE function also accepts the following WIDGET_BASE
keywords: DISPLAY_NAME, EVENT_FUNC, FRAME,
KBRD_FOCUS_EVENTS, KILL_NOTIFY, MODAL, NOTIFY_REALIZE,
RESOURCE_NAME, SCR_XSIZE, SCR_YSIZE, SPACE, TLB_FRAME_ATTR,
WV_TOOL_DENOISE
IDL Wavelet Toolkit
Chapter 4: IDL Wavelet Toolkit Reference
111
TRACKING_EVENTS, UNITS, XOFFSET, XSIZE, YOFFSET, YSIZE. See
“WIDGET_BASE” (IDL Reference Guide) for more information.
Version History
5.3
Introduced
See Also
WV_APPLET, WV_DENOISE
IDL Wavelet Toolkit
WV_TOOL_DENOISE
112
WV_TOOL_DENOISE
Chapter 4: IDL Wavelet Toolkit Reference
IDL Wavelet Toolkit
Index
A
adding
Wavelet Toolkit tools, 45
confirm exit. See preferences
continuous wavelet transform, 29, 48, 66
contours in wavelet power spectrum, 37
copyrights, 2
cumulative power plot, 43
B
bandpass
multiresolution plots, 39
C
cascade plot. See multiresolution analysis
coefficient power plot, 43
coiflet. See wavelet functions
compact support, 29
compress save files. See preferences
IDL Wavelet Toolkit
D
datasets
mathematical expressions, 23
selecting variables, 24
variable information fields, 20
Daubechies. See wavelet functions
default directory. See preferences
denoising techniques
coefficient power plot, 43
coefficient threshold, 43
113
114
color scaling, 42
cumulative power threshold, 43
denoise tool, 41
hard thresholding, 43, 51
MRI, 52
soft thresholding, 44, 52
theory, 51
wavelet coefficient method, 42
WV_DENOISE function, 69
WV_TOOL_DENOISE function, 109
detail multiresolution plots, 39
DIALOG_READ_IMAGE. See importing
discrete wavelet transform, 29, 48, 74
drag quality, 34
in wavelet toolkit, 26
importing
adding wavelet functions, 99
data from command line, 96
IDL command line data, 26
structure tags, 97
user-defined wavelet functions, 30
WAV audio files, 26
L
legalities, 2
lego-style surface, 36
localization of wavelet functions, 48
low-pass multiresolution plots, 39
E
e-folding time, 30
energy scaling, 35
G
Gaussian
See also wavelet functions.
H
Haar. See wavelet functions
high-pass multiresolution plots, 39
I
IDL Wavelet Toolkit
main window, 12
menus, 14
status bar, 12
toolbar, 12
image compression. See denoising techniques
images
Index
M
Macintosh
using mouse with wavelet toolkit, 37
Mallat. See pyramidal algorithm
Marr wavelet See WV_FN_GAUSSIAN function
mathematical expressions. See datasets
Morlet. See wavelet functions
MRI denoising technique, 52
multiresolution analysis, 104
about, 54
using, 39
N
noise removal
See also denoising techniques
nonorthogonal wavelet functions, 29
O
orthogonal wavelet functions, 29
IDL Wavelet Toolkit
115
P
parameters
passing to Wavelet Toolkit functions, 45
partial wavelet transform. See wavelet transform
passing parameters, 45
Paul. See wavelet functions
PCM format, 26
percent difference, 44
plotting
multiresolution analysis, 39, 104
wavelet power spectrum, 32, 101
preferences
compress save files, 18
confirm exit, 18
current directory selection, 18
default directory, 18
restoring defaults, 19
stride factor, 18
pyramidal algorithm
result format, 75
returning, 107
importing wavelet data, 97
surfaces
style, 36
symlet. See wavelet functions
symmetry of wavelet functions, 29
T
toolkit structure, 9
tools
adding, 45
denoise function, 109
denoise tool, 41
user-defined, 16, 45
trademarks, 2
TrueColor (24-bit) images, 26
U
user-defined tools, 45
V
R
regularity of wavelet functions, 30
remember current directory. See preferences
RMS difference, 44
root-mean-square difference, 44
rough multiresolution plots, 39
S
scaling
functions, 28
starting
wavelet toolkit, 12
statistical significance testing, 48
stride factor. See preferences
structure tags
IDL Wavelet Toolkit
vanishing moments, 30
variable information
wavelet dataset viewer, 20
variable selection. See datasets
viewing
wavelet functions, 28
W
WAV audio files, 26
wavelet commands
functions, 59
transform, 58
widgets, 58
wavelet functions
coiflet, 78
Index
116
compact support, 29
Daubechies, 80
family, 64
Gaussian, 82
Haar, 85
Morlet, 87
nonorthogonal, 29
order, 64
orthogonal, 29
Paul, 90
regularity, 30
symlet, 93
symmetry, 29
user-defined, 30, 99
vanishing moments, 30
viewing, 28
wavelet power spectrum
See also WV_PLOT3D_WPS function
energy scaling, 35
plotting method, 49
rotation, translation, stretching, 37
theory, 49
viewer, 32
zero phase lines, 35
wavelet toolkit
importing data, 25
status bar, 12
wavelet transform
continuous, 29, 48, 66
discrete, 29, 48, 74
Index
partial, 107
wavelet widget commands, 58
WV_APPLET procedure, 60
WV_CW_WAVELET function
GET_VALUE, 64
reference, 62
SET_VALUE, 64
widget events generated, 64
WV_CWT function, 66
WV_DENOISE function, 69
WV_DWT function, 74
WV_FN_COIFLET function, 78
WV_FN_DAUBECHIES function, 80
WV_FN_GAUSSIAN function, 82
WV_FN_HAAR function, 85
WV_FN_MORLET function, 87
WV_FN_PAUL function, 90
WV_FN_SYMLET function, 93
WV_IMPORT_DATA procedure, 96
WV_IMPORT_WAVELET procedure, 99
WV_PLOT_MULTIRES function, 104
WV_PLOT3D_WPS function, 101
WV_PWT function, 107
WV_TOOL_DENOISE function, 109
Z
zero phase lines, 35
IDL Wavelet Toolkit
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