Image Enhancement - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 89

An image (say A ) of size M × N is initially fuzzified using the following
formula:

gg −

(5.1)

min

μ A g () =

g − g

max in

where
g is the grey level that ranges from 0 to L −a1
g min and g max are the minimum and maximum values of the grey levels of
the image, respectively

Based on the fuzzy set, the membership degree of the IF image is calcu-
lated as

λ − 1

μ (;) g λ =−− 11 ( μ ()) g

IFS A

Using standard fuzzy negation, φ( x ) = (1 − x ) λ , λ ≥ 1, the non-membership
function is given as

(5.2)

ν (;) g λ ϕμ = ( (;)) g λ

IFS IFS

or

λ

ν (;)( (;)) , g λ =− 1 μ g λ λ ≥ 1

IFS IFS

(5.3)

( ; ) (

λλ − 1 )

=− ( 1 μλ g

A

The hesitation degree is

π (;) g λ =− 1 μ ( ;) (;) g λ − ν g λ

IFS IFS IFS

As λ is not fixed for all the images, the optimum value of λ is obtained using
IF entropy. The optimum values are calculated using different entropies.
There are many types of entropies suggested by different authors. These are

1. Entropy by Burillo and Bustince [4]

1 1

M − N −

1

∑ = ∑

(5.4)

1 − ( μ ( g ) + ν ( g ))

EA ( ) = ( 1 − μ ( g ) − ν ( g )) e

Aij Aij

1 IFS A ij Aij

MN ×

j 0 i = 0

2. Entropy by Vlachos and Sergiadis [17]

1 1

−
∑ 0

N M −

2

1 2 μν ()() ()

() ()

g g + π g

Aij Aij Aij

EA MN ( ) = (5.5)

2 IFS

2 2 2

× π g + μ g + ν (() g ij

Aij A ij A

j = 0 i =