Image Enhancement - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 91

where

L − 1

1

∑

( )

( )

γ DA ( ) = hg () μ () g ⋅ 1 − μ () g

α IFSopt _ A D A α ( ) D A α ( )

IFS pt _ IFS pt _

4 MN

g = 0

L − 1

1

∑

( ) ( )

= hg g () (; ) μλαπ λ + ⋅ ( ; g ) 1 − μ ((; ) g λαπλ −⋅ ( ; g )

A A opt A opt A opt A opt

4 MN

g = 0

Now differentiating γ D α ( A IFS_opt ) and equating it to 0,

L 1

−
∑ 0

d
d

γ
α

( ) ×− ( )

hg g () ( ; ) ( g ; ) ( g ; )

= μλαπ λ + ⋅ π λ

A A opt A opt A opt

g =

−
∑

L 1

( ) ×

+ hg () ⋅ 1 − μλαπ λ ( g ; ) − ⋅ ( g ; ) π ( g ; λ ) = 0

A A opt A opt A opt

g = 0

−
∑ 0

L 1

( )

⇒ hg ( ) μλαπ λπλ ( ; g ) + ⋅ (; ) g ⋅ ( ; g )

A A opt A opt A opt

g =

L − 1

∑ 0

( )

= hg () 1 − μ (; g λλαπλπλ ) −⋅ ( ; g ) ⋅ (; ) g

A A opt A opt A opt

g =

L − 1 L − 1

∑ ∑

⇒ hg g () (; ) ⋅ πλ − 2 hg g () (; ) ⋅ μλπλ ⋅ ( ; g )

A A opt A A opt A opt

g = 0 g = 0

L − 1

∑

2 hg ()

2 (; )

g

= α ⋅ ππλ

A A opt

g = 0

or

∑

L − 1

hg g () (; )( πλ μλ 12 − (; )) g

A A opt A opt

g = 0

α ʹ =

opt

∑ 1

L −

2

2 hg g () (; πλ )

A A op t

g =

As α opt may not lie in [0, 1], α opt is used to keep the parameter in [0, 1]:

0 if

if
if

α ʹ < 0

opt

α = α ʹ 0 ≤ ʹ ≤

ʹ >

α 1

opt opt opt

1 α 1

opt

where h A being the histogram of the fuzzified image A .