Ch 3. Image Enhancement in the Spatial Domain_okay.ppt

It's about digital image processing

1. Ch3. Image enhancement in spatial domain
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Ch3. Image Enhancement in
the Spatial Domain

2. Ch3. Image enhancement in spatial domain
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Contents
3.1 Background
3.2 Some basic gray level transformations
3.3 Histogram processing
3.4 Enhancement using arithmetic/logic operations
3.5 Basics of spatial filtering
3.6 Smoothing spatial filters
3.7 Sharpening spatial filters
3.8 Combining spatial enhancement methods

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3.1 Background
Two categories in image enhancement
approaches
1)Spatial domain processing
1)Based on direct manipulation of pixels in an image plane
itself
2)Frequency domain processing
•Based on modifying the Fourier transform of an image
Spatial domain processing
–
–Where, f(x,y): input image, g(x,y): processed image, T:
an operator on f, defined over some neighborhood of
(x,y) ),(),( yxfTyxg 

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Neighborhood of (x,y)
–Use a square or rectangular subimage area centered at
(x,y):
–Mask (filters, kernels, templates, windows): mask
processing or filtering

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–In case of 1x1 (that is, a single pixel): point processing
•
•Where, r: gray level of f(x,y), s: gray level of g(x,y)
•examples:
Contrast stretching: Fig 3.2(a)
Thresholding: Fig 3.2(b))(rTs

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3.2 Some basic gray level transformations
3 types of gray-
level transformation
functions
1)Linear: negative and
identity
transformations
2)Logarithmic: log and
inverse-log
transformations
3)Power-law: nth
power and nth root
transformations

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Image negatives
–Function:
–Reverse the intensity levels of an image:
positive negativelevelsgray of num where LrLs ,1

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Log transformations
–General form:
–Expand the values of dark pixels while compressing the
higher-level values.
–The opposite is true of the inverse log transformation
–Characteristic: compress the dynamic range of images
with large variations in pixel values0)1log(  rcrcs const, where,

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–example: Fourier spectrum
0 ~ 1.5x10
6
0 ~ 255

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Power-law transformations
–Basic form:const postive and where,  

ccrs

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–Gamma correction:
•Intensity-to-voltage
response of CRT:
=1.8~2.5
•If =2.5, s= r
1/2.5
= r
0.4
–Example 3.1 & Fig 3.8
–Example 3.2 & Fig 3.9

12. Ch3. Image enhancement in spatial domain
Example: Gamma Transformations

13. Ch3. Image enhancement in spatial domain
Example: Gamma Transformations

14. Ch3. Image enhancement in spatial domain
Example: Gamma Transformations

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Piecewise-linear
transformation functions
–Contrast stretching:
•Increase the dynamic range
of the gray levels
•If r
1=s
1and r
2=s
2, then
identity function
•If r
1= r
2, s
1=0 and s
2=L-1,
then thresholding function
•In general, r
1r
2and s
1s
2,
so the function is single
valued and monotonically
increasing

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–Gray-level slicing
•Highlighting a specific range of gray level in an image

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Highlight the major
blood vessels and study
the shape of the flow of
the contrast medium (to
detect blockages, etc.)
Measuring the actual
flow of the contrast
medium as a function of
time in a series of
images

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–Bit-plane slicing:
•Highlighting the specific bit planes
•The higher-order bits contain the majority of the visually
significant data
•Useful for image compression
•Bit-plane 7 is a thresholded (binary) image with 127
•Fig 3.13 & 3.14

19. Ch3. Image enhancement in spatial domain
Bit-plane Slicing

20. Ch3. Image enhancement in spatial domain
Bit-plane Slicing

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3.3 Histogram processing
–Histogram function of a digital image:
•Discrete function h(r
k)=n
k, where r
k: k th gray level, n
k:
number of pixels with gray level r
k
–Normalized histogram
•p(r
k)=n
k/n, for k=0,1,…,L-1, where n : total number of
pixels
•Sum of all components of the normalized histogram is
equal to 1
–Fig 3.15
•High dynamic range uniform distribution
histogram equalization

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3.3.1 Histogram equalization
–Assume that gray levelris continuous and normalized
to [0,1], that is,
s= T(r)0 r1
–Assume that the transformation function T(r) satisfies
the following conditions:
(a)T(r) is single-valued and monotonically increasing in the
interval 0 r1
(b) 0 T(r)1 for 0 r1

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–If the inverse transformation function
r = T
-1
(s)0 s1 satisfies the above conditions,
then
–If we use CDF(Cumulative Distribution Function) of r for
the transformation function, that is,
–Then, the above equation satisfies both conditions of (a)
and (b).PDF is p() where,)()(
)(
1
sTr
rs
ds
dr
rpsp








 

r
rdwwprTs
0
)()(

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)(
)(
0
rp
dwwp
dr
d
dr
rdT
dr
ds
r
r
r







 101
)(
1
)(
)()(



s
rp
rp
ds
dr
rpsp
r
r
rs

If we use CDF for the transformation function, histogram of
the transformed image becomes uniform.
Histogram equalization
Uniform density
-Thus,
[Leibniz’s rule]

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–In digital image, gray level is discrete. Thus,
and, transformation function for histogram equalization
is
–also, inverse transformation function is
–Example 3.3, Fig 3.17, and Fig 3.181,,2,1,0)(  Lk
n
n
rp
k
kr  1,,2,1,0
)()(
0
0






Lk
n
n
rprTs
k
j
j
k
j
jrkk
 1,,2,1,0)(
1


LksTr
kk 

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3.3.3 Local enhancement
Histogram processing for entire image: global
Local histogram processing:
–Define a square or rectangular neighborhood and move the
center of this area from pixel to pixel. At each location,
histogram equalization or histogram matching is obtained
–Another approach is to utilize non-overlapping regions, but it
usually produces an undesirable checkerboard effect.

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3.3.4 Use of histogram statistics for
image enhancement
Some statistical parameters obtainable directly from
the histogram for image enhancement
nth moment:
– and . The second moment is
variance of r ( ))(
)()()(
1
0
1
0
i
L
i
i
L
i
i
n
in
rprm
rm
rpmrr









: of value mean the is where
 ii rrp levelgray of histogram normalized :)( 1
0 0
1 



1
0
2
2
)()()(
L
i
ii
rpmrr )(
2
r
measure of average gray level
measure of average contrast

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3.4 Enhancement using
arithmetic/logic operations
Arithmetic/logic operations: pixel-by-pixel basis
Arithmetic operations
–Image subtraction
–Image addition
–Image multiplication
–Image division
Logic operations
–NOT: negative transformation
–AND/OR
•Used for masking
•Masking is ROI(region of interest) processing

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3.4.1 Image subtraction
Difference between f(x,y) and h(x,y):
Subtraction is the enhancement of difference between
images.
Fig 3.28:
Example 3.7, Fig 3.29
Comments on implementation
–The difference image can range –255~255 => need to scale to
0~255
•Add 255 and then divide by 2
•(diff image –min)x255/max; max: maximum of (diff image –min)
image
Image subtraction can be used for segmentation (changes)),(),(),( yxhyxfyxg 

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3.42 Image averaging
Consider a noisy image g(x,y), original imagef(x,y), noise
(x,y)
At every (x,y), assume that noise is uncorrelated and has
zero average, then averaging K different noisy images
Example 3.8, Fig 3.30, 3.31),(),(),( yxyxfyxg  2
),(
2
),(
1
1
),()},({
),(
1
),(
yxyxg
K
i
i
K
yxfyxgE
yxg
K
yxg
 


and
that follows it then ),(),(
1
yxyxg
K
 

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3.5 Basics of spatial filtering
Response of linear filtering with 3x3 mask at (x,y)
–Fig 3.32)1,1()1,1(),1()0,1(),()0,0(
),1()0,1()1,1()1,1(


yxfwyxfwyxfw
yxfwyxfwR



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3.5 Basics of spatial filtering
In general, linear filtering with mx nmask
–w(s,t):convolution mask, or convolution kernel2/)1(,2/)1(
),(),(),(


 
nbma
tysxftswyxg
a
as
b
bt
where

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Fig 3.33: 3x3 spatial filter mask
Nonlinear spatial filters
–Filtering operation is based conditionally on the values
of the pixels in the neighborhood under consideration
–Example: median filtering
Handling methods for boarder pixels
1)Not process the pixels at a distance no less than (n-
1)/2 pixels from the border
2)“padding”the image by adding rows and columns of
0’s


9
1
992211
i
ii
zwzwzwzwR 

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3.6 Smoothing spatial filters
Smoothing linear filters
–Used for blurring and for noise reduction
–Averaging filter, or lowpass filter
–Undesirable side effect: blur edges
–Fig 3.34: (a)box filter, (b)weighted average

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3.6 Smoothing spatial filters
Smoothing linear filters
–General implementation for weighted average filtering
–Example 3.9 & Fig 3.35, Fig 3.36

 
 


a
as
b
bt
a
as
b
bt
tsw
tysxftsw
yxg
),(
),(),(
),(

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Example: Gross Representation of Objects

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3.6.2 Order-statistics filters
Based on ordering (ranking) the pixels: nonlinear
spatial filters
Median filter
–Replace the value of a pixel by the median of the gray
levels in the neighborhood of that pixel
–advantages:
•Excellent noise reduction capabilities
•Less blurring than linear smoothing filters of similar size
•Effective in the presence of impulse noise (or salt-and
pepper noise)
–Example 3.10

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Example: Use of Median Filtering for Noise
Reduction

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3.7 Sharpening spatial filters
Sharpening:
–To highlight fine detail in an image
–To enhance detail that has been blurred
Applications
–Electronic printing
–Medical imaging
–Industrial inspection
–Autonomous guidance in military systems
Accomplished by spatial differentiation

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3.7.1 Foundation
First-order derivative of 1-D function f(x)
second-order derivative of 1-D function f(x)
Fig 3.38
–First-order derivatives
•Produce thicker edges
•Stronger response to a gray-level steps
–Second-order derivatives
•Stronger response to fine detail, such as thin lines and isolated
points
•Double response at step changes in gray-level)()1( xfxf
x
f


 )(2)1()1(
2
2
xfxfxf
x
f




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3.7.2 Laplacian
–Isotropic filter: independent of the direction rotation
invariant
Development of the method
–Lapacian: simplest isotropic derivative operator, Linear
operator
–discrete form (partial 2
nd
-order derivative in x, ydirection)2
2
2
2
2
y
f
x
f
f





 ),(2)1,()1,(
),(2),1(),1(
2
2
2
2
yxfyxfyxf
y
f
y
yxfyxfyxf
x
f
x








:direction
:direction

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–2-D Laplacian is obtained by summing two components  ),(4)1,()1,(),1(),1(
2
yxfyxfyxfyxfyxff 

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–Adding the original and Laplacian images (superimpose)
–Example 3.11 & Fig 3.40









positive is mask Laplacian
the of tcoefficien center the if
negative is mask Laplacian
the of tcoefficien center the if
),(),(
),(),(
),(
2
2
yxfyxf
yxfyxf
yxg

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