Ch 3. Image Enhancement in the Spatial Domain_okay.ppt

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

-2-
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

-3-
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)
 
)
,
(
)
,
( y
x
f
T
y
x
g 

-4-
 Neighborhood of (x,y)
– Use a square or rectangular subimage area centered at
(x,y):
– Mask (filters, kernels, templates, windows): mask
processing or filtering

-5-
– 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)
)
(r
T
s 

-6-
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

-7-
 Image negatives
– Function:
– Reverse the intensity levels of an image:
positive  negative
levels
gray
of
num
where 


 L
r
L
s ,
1

-8-
 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 values
0
)
1
log( 


 r
c
r
c
s const,
where
,

-9-
– example: Fourier spectrum
0 ~ 1.5x106  0 ~ 255

-10-
 Power-law transformations
– Basic form: const
postive
and
where
, 
 

c
cr
s

-11-
– Gamma correction:
• Intensity-to-voltage
response of CRT:
=1.8~2.5
• If =2.5, s = r1/2.5 = r0.4
– Example 3.1 & Fig 3.8

Example: Gamma Transformations

-15-
 Piecewise-linear
transformation functions
– Contrast stretching:
• Increase the dynamic range
of the gray levels
• If r1=s1 and r2=s2, then
identity function
• If r1= r2, s1=0 and s2=L-1,
then thresholding function
• In general, r1 r2 and s1s2,
so the function is single
valued and monotonically
increasing

-16-
– Gray-level slicing
• Highlighting a specific range of gray level in an image

5/11/2024
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

-18-
– 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

Bit-plane Slicing

-21-
3.3 Histogram processing
– Histogram function of a digital image:
• Discrete function h(rk)=nk, where rk : k th gray level, nk :
number of pixels with gray level rk
– Normalized histogram
• p(rk)=nk/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

-22-

-23-
3.3.1 Histogram equalization
– Assume that gray level r is 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

-24-
– 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
s
T
r
r
s
ds
dr
r
p
s
p












r
r dw
w
p
r
T
s
0
)
(
)
(

-25-
)
(
)
(
)
(
0
r
p
dw
w
p
dr
d
dr
r
dT
dr
ds
r
r
r










1
0
1
)
(
1
)
(
)
(
)
(





s
r
p
r
p
ds
dr
r
p
s
p
r
r
r
s
If we use CDF for the transformation function, histogram of
the transformed image becomes uniform.
 Histogram equalization
 Uniform density
- Thus,
[Leibniz’s rule]

-26-
– 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 3.18
1
,
,
2
,
1
,
0
)
( 

 L
k
n
n
r
p k
k
r 
1
,
,
2
,
1
,
0
)
(
)
(
0
0









L
k
n
n
r
p
r
T
s
k
j
j
k
j
j
r
k
k

1
,
,
2
,
1
,
0
)
(
1


 
L
k
s
T
r k
k 

5/11/2024 27

-31-
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.

-32-
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
i
n
r
p
r
m
r
m
r
p
m
r
r









:
of
value
mean
the
is
where

i
i r
r
p level
gray
of
histogram
normalized
:
)
(
1
0 
 0
1 






1
0
2
2 )
(
)
(
)
(
L
i
i
i r
p
m
r
r
 )
(
2
r

 measure of average gray level
 measure of average contrast

-35-
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

-36-

-37-
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)
)
,
(
)
,
(
)
,
( y
x
h
y
x
f
y
x
g 


-38-

-39-
3.42 Image averaging
 Consider a noisy image g(x,y), original image f(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
)
,
(
)
,
(
)
,
( y
x
y
x
f
y
x
g 


2
)
,
(
2
)
,
(
1
1
)
,
(
)}
,
(
{
)
,
(
1
)
,
(
y
x
y
x
g
K
i
i
K
y
x
f
y
x
g
E
y
x
g
K
y
x
g


 

 

and
that
follows
it
then
)
,
(
)
,
(
1
y
x
y
x
g
K


 

-40-
 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
(
















y
x
f
w
y
x
f
w
y
x
f
w
y
x
f
w
y
x
f
w
R



-41-

-42-
 In general, linear filtering with m x n mask
– w(s,t): convolution mask, or convolution kernel
2
/
)
1
(
,
2
/
)
1
(
)
,
(
)
,
(
)
,
(






 

 

n
b
m
a
t
y
s
x
f
t
s
w
y
x
g
a
a
s
b
b
t
where

-43-
 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
9
9
2
2
1
1
i
i
i z
w
z
w
z
w
z
w
R 

-44-
 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

45

-46-
 Smoothing linear filters
– General implementation for weighted average filtering
– Example 3.9 & Fig 3.35, Fig 3.36



 


 



 a
a
s
b
b
t
a
a
s
b
b
t
t
s
w
t
y
s
x
f
t
s
w
y
x
g
)
,
(
)
,
(
)
,
(
)
,
(

47

48
Example: Gross Representation of Objects

-49-
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

50
Example: Use of Median Filtering for Noise
Reduction

-51-
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

-52-
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
( x
f
x
f
x
f





)
(
2
)
1
(
)
1
(
2
2
x
f
x
f
x
f
x
f








-53-

-54-
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 2nd-order derivative in x, y direction)
2
2
2
2
2
y
f
x
f
f







)
,
(
2
)
1
,
(
)
1
,
(
)
,
(
2
)
,
1
(
)
,
1
(
2
2
2
2
y
x
f
y
x
f
y
x
f
y
f
y
y
x
f
y
x
f
y
x
f
x
f
x
















:
direction
:
direction

-55-
– 2-D Laplacian is obtained by summing two components
  )
,
(
4
)
1
,
(
)
1
,
(
)
,
1
(
)
,
1
(
2
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
f 










-56-
– Adding the original and Laplacian images (superimpose)












positive
is
mask
Laplacian
the
of
t
coefficien
center
the
if
negative
is
mask
Laplacian
the
of
t
coefficien
center
the
if
)
,
(
)
,
(
)
,
(
)
,
(
)
,
( 2
2
y
x
f
y
x
f
y
x
f
y
x
f
y
x
g

-57-

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