Fundamentals of Digital Image Processing - Control of Color Imaging Systems: Analysis and Design - page 79

1

0.8

0.6

0.4

0.2

0

0 50 100 150 200 250

Code value f

FIGURE 2.45 CDF of LENA image.

2.8.3 H ISTOGRAM E QUALIZATION
Assume that r is a continuous random variable taking values between 0 and 1 with a
PDF p r ( r ) . Let T ( r ) be a single-valued and monotonically increasing function of its

argument such that

0 T ( r ) 1 for 0 r 1 ( 2 : 128 )

Then, according to the theory of functions of one random variable, the random
variable s ¼ T ( r ) has a PDF given by

p r ( r )
d T

p s ( s ) ¼

d r j r ¼ T 1

( 2 : 129 )

( s )

Now if we want the random variable s to be uniformly distributed over the interval
[

01 ] , then we set p s ( s ) ¼ 1. Substituting this into Equation 2.129 yields

p r ( r )
d T
d r

1 ¼ ( 2 : 130 )

or

d T
d r ¼ p r ( r )

( 2 : 131 )

Integrating both sides of Equation 2.131, we have

ð

r

s ¼ T ( r ) ¼ p r ( x )d x ( 2 : 132 )

0