Histogram Specification
Question
Is histogram equalization always good?
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No
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WHY !
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Histogram Matching
Histogram matching (histogram specification)
— generate a processed image that has a specified histogram
Let pr (r ) and pz ( z ) denote the continous probability
density functions of the variables r and z. pz ( z ) is the
specified probability density function.
Let s be the random variable with the probability
r
s  T ( r )  ( L  1)  pr ( w) dw
0
Define a random variable z with the probability
z
G( z )  ( L  1)  pz (t ) dt  s
0
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Histogram Matching
r
s  T (r )  ( L  1)  pr ( w)dw
0
z
G ( z )  ( L  1)  pz (t )dt  s
0
1
z  G ( s)  G
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1
T (r)
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Histogram Matching: Procedure
►
Obtain pr(r) from the input image and then obtain the values of s
r
s  ( L 1) pr (w)dw
0
►
Use the specified PDF and obtain the transformation function G(z)
z
G( z)  ( L 1) pz (t )dt  s
0
►
Mapping from s to z
z  G 1 ( s)
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Example: Histogram Matching
Suppose that a 3-bit image (L=8) of size 64 × 64 pixels (MN = 4096)
has the intensity distribution shown in the following table (on the
left). Get the histogram transformation function and make the output
image with the specified histogram, listed in the table on the right.
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Example: Histogram Matching
Obtain the scaled histogram-equalized values,
s0  1, s1  3, s2  5, s3  6, s4  7,
s5  7, s6  7, s7  7.
Compute all the values of the transformation function G,
0
G ( z0 )  7 pz ( z j )  0.00  0
j 0
G ( z1 )  0.00
G ( z3 )  1.05
0
1
G ( z5 )  4.55  5
G ( z7 )  7.00  7
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G ( z2 )  0.00  0
G ( z4 )  2.45  2
G ( z6 )  5.95  6
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Example: Histogram Matching
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Example: Histogram Matching
Obtain the scaled histogram-equalized values,
s0  1, s1  3, s2  5, s3  6, s4  7,
s5  7, s6  7, s7  7.
Compute all the values of the transformation function G,
0
G ( z0 )  7 pz ( z j )  0.00  0
j 0
G ( z1 )  0.00
G ( z3 )  1.05
0
 1 s0
G ( z5 )  4.55  5
G ( z7 )  7.00  7
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s2
G ( z2 )  0.00  0
G ( z4 )  2.45  2 s1
G ( z6 )  5.95  6 s3
s4 s5 s6 s7
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Example: Histogram Matching
s0  1, s1  3, s2  5, s3  6, s4  7,
s5  7, s6  7, s7  7.
rk
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0
1
2
3
4
5
6
7
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Example: Histogram Matching
rk  zq
03
1 4
25
36
47
57
67
77
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Example: Histogram Matching
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