Thresholding of Medical Images - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 126

If φ 1 and φ 2 are two automorphisms in a unit interval, then

( )

( )

− 1 − 1

REFxy (,) = ϕ 1 − | () ( | ϕ x − ϕ y with c x ( ) = ϕ 1 − ϕ ( ) x

1 2 2 2 2

is a restricted equivalence function. ' c ' is a strong negation, where c : [0, 1] →
[0, 1] is a negation if it satisfies the following properties [3]:

1. c (0) = 1, c (1) = 0

2. c ( x ) ≤ c ( y ), iff x ≥ y

and automorphism in an interval [ a , b ] is a continuously strictly increasing
function φ : [ a , b ] → [ a , b ] such that φ( a ) = a , φ( b ) = b .

6.5.1 Calculation of Membership Function

Let us consider φ 2 ( x ) = x . From the definition of restricted equivalence
function,

− 1 − 1

REFxy (,) = ϕ ( |() 1 − ϕ x − ϕ ( )|) y = −− ϕ ( | ) 1 x y

1 2 2 1

ϕ 1 1 1
Considering φ 1 ( x ) = ln[ x ( e − 1) + 1], with e = exp(1) and using inverse func-
tion, we get

−

So, REFxy (,) = ( | ). − x y −

x

() (

e
e

−
−

1
1

)

ϕ 1 1

−

x =

( )

Thus,

1 −− || xy

( e − 1 )

(6.19)

REFxy (,) =

( e − 1 )

This equation is used to define the membership function.
The membership function denotes the belongingness of a pixel to a region.
So, the smaller the difference between the grey level of any pixel a ij and the
mean of the region to which the pixel belongs, the greater the membership
value and vice versa.
Let us define the membership function μ: [0, 1] as

μ( x ) = REF ( x , y )