Histogram equalization and matching - Digital Image Processing

Digital Image Processing, 2nd ed. www.imageprocessingbook.com
© 2002 R. C. Gonzalez & R. E. Woods
Chapter 3

Image Enhancement in the
Spatial Domain

Gray-level Transformation

Histogram Processing
• The histogram of a digital image with intensity levels in the
range [0,L-1] is a discrete function h(rk) = nk, where rk is
the kth intensity value and nk is the number of pixels in the
image with intensity rk.
• A normalized histogram is given by p(rk) = nk/MN for
k=0,1,2,….,L-1.

Histogram Processing

Histogram Equalization
• Histogram equalization:
– To improve the contrast of an image
– To transform an image in such a way that the transformed image has a
nearly uniform distribution of pixel values
• Transformation:
– Assume r has been normalized to the interval [0,1], with r = 0
representing black and r = 1 representing white
– The transformation function satisfies the following conditions:
• T(r) is single-valued and monotonically increasing in the interval
•
1
0 
r
1
0
for
1
)
(
0 


 r
r
T
1
0
)
( 

 r
r
T
s

• Histogram equalization is based on a transformation of the
probability density function of a random variable.
• Let pr(r) and ps(s) denote the probability density function of
random variable r and s, respectively.
• If pr(r) and T(r) are known, then the probability density
function ps(s) of the transformed variable s can be obtained
• Define a transformation function
where w is a dummy variable of integration and the right side
of this equation is the cumulative distribution function of
random variable r.



r
r dw
w
p
r
T
s
0
)
(
)
(
ds
dr
r
p
s
p r
s )
(
)
( 

• Given transformation function T(r), 

r
r dw
w
p
r
T
0
)
(
)
(
1
)
(
1
)
(
)
(
1
)
(
1
)
(
)
(
)
(
0

















r
p
r
p
dr
dw
w
p
d
r
p
dr
ds
r
p
ds
dr
r
p
s
p
r
r
r
r
r
r
r
s

 ps(s) now is a uniform probability density function.
 T(r) depends on pr(r), but the resulting ps(s) always is uniform.

• In discrete version:
– The probability of occurrence of gray level rk in an image is
n : the total number of pixels in the image
nk : the number of pixels that have gray level rk
L : the total number of possible gray levels in the image
– The transformation function is
–
1
,...,
2
,
1
,
0
)
(
)
(
0 0




  
 
L
k
n
n
r
p
r
T
s
k
j
k
j
j
j
r
k
k
1
,...,
2
,
1
,
0
)
( 

 L
k
n
n
r
p k
r

Histogram Equalization Example
Intensity # pixels
0 20
1 5
2 25
3 10
4 15
5 5
6 10
7 10
Total 100
Accumulative Sum of Pr
20/100 = 0.2
(20+5)/100 = 0.25
(20+5+25)/100 = 0.5
(20+5+25+10)/100 = 0.6
(20+5+25+10+15)/100 = 0.75
(20+5+25+10+15+5)/100 = 0.8
(20+5+25+10+15+5+10)/100 = 0.9
(20+5+25+10+15+5+10+10)/100 = 1.0
1.0

Histogram Equalization Example (cont.)
Intensity
(r)
No. of Pixels
(nj)
Acc Sum
of Pr
Output value Quantized
Output (s)
0 20 0.2 0.2x7 = 1.4 1
1 5 0.25 0.25*7 = 1.75 2
2 25 0.5 0.5*7 = 3.5 3
3 10 0.6 0.6*7 = 4.2 4
4 15 0.75 0.75*7 = 5.25 5
5 5 0.8 0.8*7 = 5.6 6
6 10 0.9 0.9*7 = 6.3 6
7 10 1.0 1.0x7 = 7 7
Total 100

(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd
Edition.

Histogram Equalization (cont.)
Edition.

Histogram Equalization (cont.)
Original
image
After histogram equalization, the image
become a low contrast image
Edition.

Histogram Matching

Histogram Matching : Algorithm



r
r dw
w
p
r
T
s
0
)
(
)
(
Concept : from Histogram equalization, we have
We get ps(s) = 1
We want an output image to have PDF pz(z)
Apply histogram equalization to pz(z), we get



z
z du
u
p
z
G
v
0
)
(
)
(
We get pv(v) = 1
Since ps(s) = pv(v) = 1 therefore s and v are equivalent
Therefore, we can transform r to z by
r T( ) s G-1
( ) z

Histogram Matching : Algorithm (cont.)
s = T(r) v = G(z)
z = G-1
(v)
1
2
3
4
Edition.

Histogram Matching Example
Intensity
(r)
# pixels
0 20
1 5
2 25
3 10
4 15
5 5
6 10
7 10
Total 100
Input image histogram
Intensity
( z )
# pixels
0 5
1 10
2 15
3 20
4 20
5 15
6 10
7 5
Total 100
Desired Histogram
Example
User define
Original
data

r (nj) SPr s
0 20 0.2 1
1 5 0.25 2
2 25 0.5 3
3 10 0.6 4
4 15 0.75 5
5 5 0.8 6
6 10 0.9 6
7 10 1.0 7
Histogram Matching Example (cont.)
1. Apply Histogram Equalization to both tables
z (nj) SPz v
0 5 0.05 0
1 10 0.15 1
2 15 0.3 2
3 20 0.5 4
4 20 0.7 5
5 15 0.85 6
6 10 0.95 7
7 5 1.0 7
sk = T(rk) vk = G(zk)

r s
0 1
1 2
2 3
3 4
4 5
5 6
6 6
7 7
2. Get a map
v z
0 0
1 1
2 2
4 3
5 4
6 5
7 6
7 7
sk = T(rk) zk = G-1
(vk)
r  s v  z
s  v
We get
r z
0 1
1 2
2 2
3 3
4 4
5 5
6 5
7 6
z # Pixels
0 0
1 20
2 30
3 10
4 15
5 15
6 10
7 0
Actual Output
Histogram

z # Pixels
0 0
1 20
2 30
3 10
4 15
5 15
6 10
7 0
Intensity
(r)
# pixels
0 20
1 5
2 25
3 10
4 15
5 5
6 10
7 10
Total 100
Intensity
( z )
# pixels
0 5
1 10
2 15
3 20
4 20
5 15
6 10
7 5
Total 100
r z
0 1
1 2
2 2
3 3
4 4
5 5
6 5
7 6

Desired histogram
Transfer function
Actual histogram
Edition.

Original
image
After
histogram
equalization
After
histogram
matching

Try Yourself
Intensity
(r)
# pixels
0 25
1 5
2 20
3 10
4 5
5 5
6 20
7 10
Total 100
Input image histogram
Intensity
( z )
# pixels
0 5
1 15
2 15
3 20
4 20
5 10
6 10
7 5
Total 100
Desired Histogram
Example
User define
Original
data

Local Enhancement : Local Histogram Equalization
Concept: Perform histogram equalization in a small neighborhood
Orignal image After Hist Eq.
After Local Hist Eq.
In 7x7 neighborhood
Edition.

Quiz

• Who was awarded the 2014 PEN Pinter prize?
 PEN International is a worldwide association of writers, founded in London in
1921 to promote friendship and intellectual co-operation among writers
everywhere. The association has autonomous International PEN centers in over 100
countries.
 English PEN is the founding centre of PEN International, the worldwide writers’
association.
 The PEN Pinter Prize comprise an annual literary award launched in 2009 by
English PEN in honour of the late Nobel Literature Prize-winning playwright
Harold Pinter.
 The award is given to "a British writer or a writer resident in Britain.

Isamu Akasaki Hiroshi Amano Shuji Nakamura

• Digital Image Processing”, C.Rafeal Gonzalez
and E.Richard Woods, , Pearson Education
2007, Page 120-138

Assessment
• Define histogram.
• What is histogram equalization?
• Consider an image matrix given below:
Perform histogram equalization.
• Explain the histogram specification technique
with an example.

Histogram equalization and matching - Digital Image Processing

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