Image Segmentation - Bezier and Splines in Image Processing and Machine Vision - page 38

where p ( a k ,b l ) is the joint probability of occurrence of ( a k ,b l ).
Let p ( i j ) be the probability that a gray value i belongs to the object,

given that the adjacent pixel with gray value j belongs to the background,

|

p ( i | j ) = 1. Thus, for a given threshold s , the conditional entropy of the

i

object given the background, as defined by Pal and Bhandari [130] (using
(2.9)) is

B )=
i

H s ( O | p o ( i, j ) I ( p o ( i | j )) ,

∈ object j ∈ background

(2.10)

s L − 1

= p o ( i, j ) I ( p o ( i | j )) ,

i =0 j = s +1

where

t ij

p o ( i, j )= (2.11)

s L− 1

t ij

i =0 j = s +1

and

t ij
s

p o ( i | j )= (2.12)

t ij

i =0

for 0 1. Here t ij is the frequency of occurrence

of the pair ( i, j ). The conditional entropy of the background given the object
can similarly (using(2.9)) can defined as

≤ i ≤ s and s +1 ≤ j ≤ L −

H s ( B | O )= p b ( i, j ) I ( p b ( i | j ))

(2.13)

i ∈ background j ∈ object

where

t ij

p b ( i, j )= (2.14)

L− 1 s

t ij

i = s +1 j =0

and

t ij
L− 1

p b ( i | j )=

(2.15)

t ij

i = s +1

for s +1 ≤ i ≤ L − 1 and 0 ≤ j ≤ s . Then the total conditional entropy of

the partitioned image is

H T C = H s ( O | B )+ H s ( B | O ) . (2.16)

For an image, the conditional entropy of the object, given the background,
provides a measure of information about the object when we know about the