Image enhacement

peachpuceAI and Robotics

Nov 6, 2013 (3 years and 9 months ago)

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EE 4780

Image Enhancement

Bahadir K. Gunturk

2

Image Enhancement


The objective of image enhancement is to process an
image so that the result is more
suitable

than the
original image for a specific application.


There are two main approaches:


Image enhancement in spatial domain: Direct manipulation
of pixels in an image


Point processing: Change pixel intensities


Spatial filtering


Image enhancement in frequency domain: Modifying the
Fourier transform of an image



Bahadir K. Gunturk

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Image Enhancement by Point Processing


Intensity Transformation



Bahadir K. Gunturk

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Image Enhancement by Point Processing


Contrast Stretching



Bahadir K. Gunturk

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Image Enhancement by Point Processing


Contrast Stretching



( ) log(1 )
T r c r
 
Bahadir K. Gunturk

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Image Enhancement by Point Processing


Intensity Transformation



Matlab exercise

Bahadir K. Gunturk

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Image Enhancement by Point Processing


Intensity Transformation



Bahadir K. Gunturk

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Image Enhancement by Point Processing


Intensity Transformation



Bahadir K. Gunturk

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Image Enhancement by Point Processing


Gray
-
Level Slicing



Bahadir K. Gunturk

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Image Enhancement by Point Processing


Histogram

0

255

Number of pixels with intensity
( )
Total number of pixels
r
p r
 

 
 
( )
p r
r
Bahadir K. Gunturk

11

Histogram Specification

( )
s T r


Intensity mapping




Assume


T(r)

is single
-
valued and monotonically increasing.






The original and transformed intensities can be
characterized by their probability density functions (PDFs)

0 ( ) 1 and 0 1
T r r
   
( )
r
p r
( )
s
p s
Bahadir K. Gunturk

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Histogram Specification

1
( )
( ) ( )
s r
r T s
dr
p s p r
ds


 

 
 

The relationship between the PDFs is

0
( ) ( )
r
r
w
s T r p w dw

 

0
( ) ( )
r
r r
w
ds d
p w dw p r
dr dr

 


Consider the mapping

Cumulative distribution function of
r

1
( )
1
( ) ( ) 1, 0 1
( )
s r
r
r T s
p s p r s
p r


 
   
 
 
Histogram equalization!

( ) ( ) 1
s r
p s ds p r dr
 
 
Bahadir K. Gunturk

13

Image Enhancement by Point Processing


Histogram Equalization

Number of pixels with intensity
( ) 255
Total number of pixels
i r
T r round
 


 
 
0 255
r
 
0
255 ( )
r
i
round p i

 

 
 

0
Number of pixels with intensity
255
Total number of pixels
r
i
i
round

 

 
 

Bahadir K. Gunturk

14

Image Enhancement by Point Processing


Histogram Equalization Example

Intensity 0 1 2 3 4 5 6 7

Number of pixels 10 20 12 8 0 0 0 0















Intensity 0 1 2 3 4 5 6 7

Number of pixels 0 10 0 0 20 0 12 8

(0) 10/50 0.2
p
 
(1) 20/50 0.4
p
 
(2) 12/50 0.24
p
 
(3) 8/50 0.16
p
 
( ) 0/50 0, 4,5,6,7
p r r
  
0
( ) 7 ( )
r
i
T r round p i

 

 
 





(0) 7* (0) 7*0.2 1
T round p round
  






(1) 7* (0) (1) 7*0.6 4
T round p p round
   






(2) 7* (0) (1) (2) 7*0.84 6
T round p p p round
    




(3) 7* (0) (1) (2) (3) 7
T round p p p p
    
( ) 7, 4,5,6,7
T r r
 
Bahadir K. Gunturk

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Image Enhancement by Point Processing


Histogram Equalization

Bahadir K. Gunturk

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Local Histogram Processing


Histogram processing can be applied locally.

Bahadir K. Gunturk

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Image Subtraction

The background is subtracted out, the arteries appear bright.

(,) (,) (,)
g x y f x y h x y
 
Bahadir K. Gunturk

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Image Averaging

(,) (,) (,)
g x y f x y n x y
 
Original

image

Noise

Corrupted

image

Assume
n(x,y)

a white noise with mean=0, and variance



2 2
(,)
E n x y

If we have a set of noisy images

(,)
i
g x y
The noise variance in the average image is

1
1
(,) (,)
M
ave i
i
g x y g x y
M





2
2 2
2
1 1
1 1 1
(,) (,)
M M
i i
i i
E n x y E n x y
M M M

 
 
 
 
 
 
 
 
 
 
 
Bahadir K. Gunturk

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Image Averaging

Bahadir K. Gunturk

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Spatial Filtering

1 1 1
1
1 1 1
9
1 1 1
 
 
 
 
 
1 1 1
1 8 1
1 1 1
 
 

 
 
 
A low
-
pass filter

A high
-
pass filter

Bahadir K. Gunturk

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Spatial Filtering


Median Filter

10 20 10
25 10 75
90 85 100
 
 
 
 
 
Sort: (10 10 10 20 25 75 85 90 100)

100 100 100 100 10 10 10 10 10


Example

Original signal:

100 103 100 100 10 9 10 11 10

Noisy signal:


101 101 70 40 10 10 10

Filter by [ 1 1 1]/3:


100 100 100 10 10 10 10

Filter by 1x3
median filter:

Bahadir K. Gunturk

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Spatial Filtering


Median filters are nonlinear.


Median filtering reduces noise without blurring edges and
other sharp details.


Median filtering is particularly effective when the noise
pattern consists of strong, spike
-
like components. (Salt
-
and
-
pepper noise.)

Bahadir K. Gunturk

27

Spatial Filtering

Original

3x3
averaging
filter

Salt&Pepper
noise added

3x3
median
filter

Bahadir K. Gunturk

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Spatial Filtering

Bahadir K. Gunturk

29

Wiener Filter

W
X
Y


Y
X
w
x
x
2
2
2
ˆ





Wiener Filter

Original

image

Noise

Noisy

image

Noise variance

Signal variance

Bahadir K. Gunturk

30

Wiener Filter

2
2
2
ˆ
w
x
y




2
x

is estimated by

Since variance is nonnegative, it is modified as

]
,
0
max[
ˆ
2
2
2
w
x
y




2 2 2
2
1
ˆ
max[0,]
x i w
i
y
N
 
 

Estimate signal variance locally:

N

N

Bahadir K. Gunturk

31

Wiener Filter

Noisy,

=10

Denoised

(3x3neighborhood)

Mean Squared Error is 56

wiener2

in Matlab

Bahadir K. Gunturk

35

Spatial Filtering

1 1 1
1 8 1
1 1 1
  
 
 
 
 
 
  
 
Bahadir K. Gunturk

36

Spatial Filtering


High
-
boost or high
-
frequency
-
emphasis filter


Sharpens the image but does not remove the low
-
frequency
components unlike high
-
pass filtering

Bahadir K. Gunturk

37

Spatial Filtering


High
-
boost or high
-
frequency
-
emphasis filter



High pass = Original


Low pass



High boost = (Original) + K*(High pass)


Bahadir K. Gunturk

38

Spatial Filtering

1 1 1
1 8 1
1 1 1
  
 
 
 
 
 
  
 
A high
-
pass filter

A high
-
boost filter

1 1 1
1 9 1
1 1 1
  
 
 
 
 
 
  
 
Bahadir K. Gunturk

39

Spatial Filtering


High
-
boost or high
-
frequency
-
emphasis filter

Bahadir K. Gunturk

40

Spatial Filtering