# Image enhacement

AI and Robotics

Nov 6, 2013 (4 years and 6 months ago)

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EE 4780

Image Enhancement

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

3

Image Enhancement by Point Processing

Intensity Transformation

4

Image Enhancement by Point Processing

Contrast Stretching

5

Image Enhancement by Point Processing

Contrast Stretching

( ) log(1 )
T r c r
 

6

Image Enhancement by Point Processing

Intensity Transformation

Matlab exercise

7

Image Enhancement by Point Processing

Intensity Transformation

8

Image Enhancement by Point Processing

Intensity Transformation

9

Image Enhancement by Point Processing

Gray
-
Level Slicing

10

Image Enhancement by Point Processing

Histogram

0

255

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

 
 
( )
p r
r

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

12

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
 
 

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

 

 
 

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
 

15

Image Enhancement by Point Processing

Histogram Equalization

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Local Histogram Processing

Histogram processing can be applied locally.

21

Image Subtraction

The background is subtracted out, the arteries appear bright.

(,) (,) (,)
g x y f x y h x y
 

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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

 
 
 
 
 
 
 
 
 
 
 

23

Image Averaging

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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

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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:

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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.)

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Spatial Filtering

Original

3x3
averaging
filter

Salt&Pepper

3x3
median
filter

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Spatial Filtering

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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

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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

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Wiener Filter

Noisy,

=10

Denoised

(3x3neighborhood)

Mean Squared Error is 56

wiener2

in Matlab

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Spatial Filtering

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

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Spatial Filtering

High
-
boost or high
-
frequency
-
emphasis filter

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

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Spatial Filtering

High
-
boost or high
-
frequency
-
emphasis filter

High pass = Original

Low pass

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

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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
  
 
 
 
 
 
  
 

39

Spatial Filtering

High
-
boost or high
-
frequency
-
emphasis filter