Image Enhancement - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 98

By varying the λ value, various membership and non-membership values are
obtained and each one provides a different notion of representation of grey
level around ' g '. In order to optimally model the image grey levels, optimum
value of λ is required. Using IF e nt r o py, λ opt is calculated as

1

−
∑

L

IFEA hgEA
A

() = () (;) λ

g

g = 0

where

L − 1

∑

( )

A A

EAg (;) = h k ( ) 1 − μλνλ ( ;) (; ) k − k

g A g g

k = 0

⎛

− λλ ( 1 )

⎞

λ − 1

L − 1 L − 1

⎛ ⎞ ⎛ ⎞

⎛ kg

p

− ⎞ ⎛ kg

p

− ⎞

∑ = ∑

⎜
⎜

⎟
⎟

= hg hk () () ⎜ 1 − max, 0 1 − ⎟ −− − ⎜ 1 max, 01 ⎟

⎜ ⎟ ⎜ ⎟ A A

⎜

⎝ ⎠ ⎝ ⎠

⎝ ⎠ ⎝ ⎠

g 0 k = 0

⎝ ⎠

(5.19)

h A is the crisp histogram of the image.
The optimal value of λ corresponds to the maximum entropy value. Based
on the lower and upper membership functions, the lower (minimum) IF
histogram is written as follows.
Then lower IF histogram is

= {

}

L A

hg ij () (, ), (, μ g λ )

A g ij opt

and upper (maximum) IF histogram is

{ }

U A

hg ij () (, ), (, = 1 − ν g λ )

A g ij opt

The hesitancy histogram of an image is then given as

H L

() () (),

U

hghghg g = − = 012 , ,, , … L − 1

A A A

The normalized hesitancy histogram is then computed as

U L

hghg () () −

(5.20)

H

hg () =

A

∑ 1

L −

U L

hkhk () () −

A A

k =