# 影像處理基礎 - 雲林科技大學

```第二章

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2.1 影像的組成
2.2 影像取樣與量化
2.3 影像解析度與縮放
2.4 基本灰階轉換
3
Image Formation

An analog image is described by the spatial
distribution of brightness or gray-levels that reflect a
distribution of detected energy.


The image can be displayed using a medium such as
paper or film.
The image may show
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
A black-and-white image with gray-levels representing.
A true color image with red, green, and blue
components.
4


y
Pin-Hole Imaging
 The pin-hole imaging method is used in many
imaging systems.
 The light from the object plane enters into the image
plane through a pin-hole.
 The pin-hole is called the focal plane.
x
z
f(x1,y1)
Pin-hole
g(x2,y2)
z1
Object Plane
z2
Focal Plane
Image Plane
5


If a point in the object plane is considered to have (x1, y1, -z1)
coordinates mapped into the image plane as (x2, y2, z2) coordinates,
then
Magnification
factor
y
z 2 x1
z 2 y1
x2  
and y2  
z1
z1
x
z
f(x1,y1)
Pin-hole
g(x2,y2)
z1
Object Plane
z2
Focal Plane
Image Plane
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Digital Image Acquisition Process
7


Image Sampling and Quantization

To create a digital image, we need to convert
the continuous sensed data into digital form.
This involves two processes:
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Sampling

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Digitizing the coordinate values
Quantization

Digitizing the amplitude values
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9

10

11
Representing Digital Images

Representing Digital Images

The result of sampling and quantization is a matrix of
real numbers.
 f (0,0)
 f (1,0)
f ( x, y )  

:

 f ( M  1,0)
 a0,0
 a
1, 0
A
 :

aM 1,0
f (0,1)
...
f (1,1)
...
:
:
f ( M  1,1) ...
a0,1
a1,1
:
aM 1,1
f (0, N  1) 
f (1, N  1) 

:

f ( M  1, N  1)
... a0, N 1 
... a1, N 1 

...
:

... aM 1, N 1 
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Representing Digital Images
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Representing Digital Images
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Spatial Resolution
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
The smallest discernible detail in an image.
Line pair
Size: 1024*1024
14
Representing Digital Images
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Representing Digital Images
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Gray-Level
Resolution


The smallest
discernible change in
gray level.
The # of gray levels is
usually an integer
power of 2.
16
Representing Digital Images
False contouring
17
Color Image Formation
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
Three basic colors, red, green and blue
(RGB) could be used as three variables
for representing color images.
When combined together, the red, green
and blue intensities can produce a
selected color at a spatial location in the
image.
18
Color Models
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The RGB Color Model


Each color appears in its primary spectral
components of red, green and blue .
The number of bits used to represent each pixel in
RGB space is called the pixel depth.
Schematic of the RGB color cube.
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Color Models (cont.)
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Example

Generating the hidden face planes and a cross
section of the RGB color cube.
RGB 24-bit color cube
The three hidden surface planes
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Color Models (cont.)

Example (cont.)
Generating the RGB image of the cross-sectional color plane (127, G, B)21
Color Models (cont.)

Safe RGB color, All-system-safe color, Safe Web
color, Safe browser color


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216 colors are common to most systems
Each of the 216 safe colors is formed from three RGB
values, each value can only be 0, 51, 102, 153, 204, or
255.
The values 000000 and FFFFFF represent black and
white, respectively.
The RGB safe-color cube
22
Color Models (cont.)
Valid values of each RGB component in a safe color
The 216 safe RGB colors
All the grays in the 256-color RGB system
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Color Models (cont.)

RGB to Graylevel


Gray = R*0.299 + G*0.587 + B*0.114
Gray = (R*30 + G*59 + B*11 + 50) / 100
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24
Color Models (cont.)
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The CMY color model

Cyan, magenta, and yellow are the secondary colors of light
 When cyan is illuminated with white light, no red light is
reflected from the surface.
 Most devices that deposit colored pigments on paper, such
as color printers and copiers, require CMY data input or
perform an RGB to CMY conversion internally.
 C  1  R 
 M   1  G 
    
 Y  1  B 
 R  1  C 
G   1   M 
    
 B  1  Y 
(6.2-1)
The assumptions is that all color values have been
normalized to range [0,1].
The CMYK color model
 In order to produce true black, a fourth color black is added


25
Image Enhancement
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The principal objective of enhancement is
to process an image so that the result is
more suitable than the original image for
a specific application.
Image enhancement approaches

Spatial domain methods


Based on direct manipulation of pixels in an
image.
Frequency domain methods

Based on modifying the Fourier transform of an
image.
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Image Enhancement (cont.)

Spatial domain


Refers to the aggregate of pixels composing an
image.
Operate directly on these pixels
Spatial domain process will
be denoted by
g(x,y)=T[f(x,y)]
where f(x,y): input image
g(x,y): processed image
T: an operator
mask
filter
kernel
template
windows
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Image Enhancement (cont.)
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Gray-Level (intensity) transformation Function
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

s=T(r)
where T is gray-level transformation function
Processing technologies:

Point processing


Enhancement at any point in an image depends only on the gray level at that
point.
Mask processing or filtering

Use a function of the values of f in a predefined neighborhood of (x,y) to
determine the value of g at (x,y)
Contrast
stretching
thresholding
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Image Enhancement (cont.)

Some basic Gray Level
Transforms
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s = T(r)
r : the gray level value before
process
s: the gray level value after
process
Values of the transformation
function typically are stored in
a one-dimensional array and
the mapping from r to s are
implemented via table lookups.
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Some Basic Gray Level Transforms (cont.)
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Image Negatives
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
Reversing the intensity levels of an image
Photographic Negative
s=L-1-r
Suited for enhancing white or gray detail embedded in dark
regions of an image
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Some Basic Gray Level Transforms (cont.)
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Log Transformations
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
s=c log (1+r)
Maps a narrow range of low gray-level values in the input
image into a wider range of output levels.
To expand the values of dark pixels in an image while
compressing the higher-level values
A Fourier spectrum
with values in the
range 0 to 1.5x106.
c=1, the range 31
of
values : 0 to 6.2.
Some Basic Gray Level Transforms (cont.)
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Power-Law Transformations
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s=crg
To account for an offset
s= c (r +e )r
where c and `g are positive
constants
Power-law curves with
fractional values of r map a
narrow range of dark input
values into a wider range of
output values, with the
opposite being true for
higher values of input levels.
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Some Basic Gray Level Transforms (cont.)
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Gamma Correction

The process used to correct this power-law response
phenomena
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Some Basic Gray Level Transforms (cont.)
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Example 3.1
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MR image of fractured human spine
Contrast
manipulation
c=1,
g=0.6
Fracture
dislocation
c=1,
g=0.4
The best enhancement
in terms of contrast and
discernable detail was
obtained.
c=1,
g=0.3

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Some Basic Gray Level Transforms (cont.)
c=1,
g=3.0
Washed-out
appearance
c=1,
g=5.0
c=1,
g=4.0
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Some Basic Gray Level Transforms (cont.)
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Picewise-Linear Transformation
Function

Contrast Stretching
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To increase the dynamic
range of the gray levels in the
image being processed.
Linear function

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If r1=s1 and r2=s2
Thresholding

If r1=r2, s1=0 and s2=L-1
Control
points
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Some Basic Gray Level Transforms (cont.)
Gray-level Slicing
Highlighting a specific
range of gray levels in an
image.
To display a high
value for all gray levels
in the range of interest
and a low value for all
other gray levels.
Brightens the desired
range of gray levels but
preserves the
background and graylevel in the image.
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Some Basic Gray Level Transforms (cont.)
38
Histogram Processing
Histogram
h(rk)= nk
rk is the kth gray-level
nk is the number of pixels in the
image having gray-level k
Normalized Histogram
p(rk)=nk/n
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Histogram Processing (cont.)

Histogram Equalization
s  T (r )
0  r 1
Assume that the transformation function T(r) satisfies the follows
(a) T(r) is a single-valued and monotonically increasing
(b) 0<=T(r)<=1 for 0<=r <=1
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Histogram Processing (cont.)

Condition (a) is changed to
T(r) is a strictly monotonically increasing function in the
interval 0<=r <L-1
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Histogram Processing (cont.)

The probability of occurrence of gray level rk in an image is
approximated by
n
pr (rk )  k
k  0,1,2,...,L  1
n

The discrete version of the transformation function given as
sk  T (rk ) 


k
nj
j 0
n

k
 pr (r j )
j 0
k  0,1,2,...,L  1
A processed image is obtained by mapping each pixel with level rk in
the input image into a corresponding pixel with level sk in the output
image.
Histogram equalization automatically determines a transformation
function that seeks to produce an output image that has a uniform histogram.
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Histogram Processing (cont.)
s  T r   L  1 pr wdw
r
0
43
Histogram Processing (cont.)
44
Histogram Processing (cont.)

Example 3.3Histogram equalization
45
Histogram Processing (cont.)
46
Local Enhancement
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Global enhancement
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Local enhancement

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The pixels are modified by a transformation function based
on the gray-level content of an entire image.
To design transformation functions based on the gray-level
distribution in the neighborhood of every pixel in the image.
Local enhancement Procedure
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Step 1: Define a square neighborhood
Step 2: Move the center of this area from pixel by pixel
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Calculate the histogram of the points in the neighborhood.
Apply the histogram equalization or specification
Assign new gray level to the center pixel
Step 3: Moved to an adjacent pixel location.
Repeat Step 2 until end of the image
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Histogram Processing (cont.)
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Example

Enhancement using local histograms
Original image
Result of local
Result of global
histogram equalization histogram equalization
48
Histogram Processing (cont.)

Enhancement using local histograms
49
Use of Histogram Statistics for Image Enhancement

The global mean and variance

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The mean is a measure of average gray level in an image
The variance is a measure of average contrast in an
image.
Let r denote discrete gray-levels in the range r [0, L 1]
p(ri) denote the normalized histogram component
corresponding to the ith value of r.
The nth moment of r is defined as
L 1
 n (r )   ri  mn p(ri )
i 0
where m is the mean value of r
L 1
m   ri p(ri )
i 0
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Use of Histogram Statistics for Image Enhancement


0=1, 1=0
 The second moment is obtained by
L 1
 2 (r )   ri  m 2 p(ri )
i 0
2
  (r )




Standard deviation定義為variance的平方根(square root) 。
The global mean and variance are measured over an entire
image and are useful primarily gross adjustments of overall
intensity and contrast.
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Histogram Statistics for Image Enhancement

The local mean and variance


The local mean is a measure of average gray level in
neighborhood Sxy
The variance is a measure of contrast in the
neighborhood
mS xy 
2 
S xy
 rs,t p(rs,t )
( s ,t )S xy
 r
( s ,t )S xy
s ,t
 mS xy
 p(r
2
s ,t )
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Histogram Processing (cont.)
Example

53
Histogram Processing (cont.)

The problem is to enhance dark areas while leaving the
light area as unchanged as possible.

Consider the pixel as a point (x,y) as a candidate for processing
(1) if mS xy  k0 M G
(2) if  S xy  k 2 DG
(3) if k1DG   S xy , k1  k 2
MG: global mean
DG: global standard deviation
k0 : positive constant <1.0
k1: < 10
k2: > 1.0
Thus
 E  f ( x, y ) if (mS xy  k0 M G )( S xy  k 2 DG ) (k1DG   S xy ), k1  k 2
g ( x, y )  
otherwise
 f ( x, y )
54
Histogram Processing (cont.)

mean average

standard deviation

E，用來對

55
Histogram Processing (cont.)
56
Enhancement using Arithmetic/Logic Operations

Enhancement using Arithmetic/Logic
Operations
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
Arithmetic/Logic operations are performed
on a pixel-by-pixel basis.
Arithmetic operations: subtraction, addition,
division, multiplication.
Logic operations: AND, OR, NOT
When dealing with logic operations on grayscale images, pixel values are processed as
strings of binary numbers.
57
Enhancement using Arithmetic/Logic
Operations (cont.)
58
Enhancement using Arithmetic/Logic
Operations (cont.)
59
Enhancement using Arithmetic/Logic
Operations (cont.)

Image Subtraction


The enhancement of difference between images
The difference between two images f(x,y) and
h(x,y)
g ( x, y)  f ( x, y)  h( x, y)
60
Enhancement using Arithmetic/Logic
Operations (cont.)

Most images are displayed using 8 bits.

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
Thus, we expect image values not to be
outside the range from 0 to 255.
The value in a difference image can range
from a minimum of -255 to a maximum of
255.
How to solve this problem?
Solution 1: g’(x,y)=[g(x,y)+255]/2
 Solution 2:
g’(x,y)=g(x.y)-min(g(x,y))
g’’(x,y)=[g’(x,y)*255]/max(g’(x,y))

61
Enhancement using Arithmetic Operations (cont.)

Image averaging

Noisy image g(x,y) formed by the addition of noise h(x,y) to an
original image f(x,y)
g ( x, y)  f ( x, y) h ( x, y)

Assume that at every pair of coordinates (x,y) the noise is
uncorrelated and has zero average value.
Averaging K different noisy images
To reduce the noise content by adding a set of noisy images
The standard deviation at any point in the average image is


1 K
g ( x, y)   g i ( x, y)
K i 1
 g ( x, y ) 
1
 h ( x, y)
K
As K increases, Eq(3.4-6) indicates that the noise of the pixel values
at each location (x,y) decreases.
62
Enhancement using Arithmetic/Logic
Operations (cont.)

Example 3.8 Noise
reduction by image
averaging




The images gi(x,y) must
be registered in order to
avoid the introduction of
blurring and other
artifacts.
(a) Image of Galaxy pair
NGC 3314.
(b) Image corrupted by
additive Gaussian noise
with zero mean and a
standard deviation of 64
gray levels.
(c-f) Result of averaging
K=8, 16, 64 and 128
noisy images.
63
Enhancement using Arithmetic/Logic
Operations (cont.)
In the histograms, the mean
and standard deviation of the
difference images decrease as
K increases.
64
Reference


Rafael C. Gonzalez and Richard E. Woods, Digital Image
Processing (2nd, 3rd Edition)
Atam P. Dhawan, Medical Image Analysis, Wiley
Interscience, 2003.
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