Thresholding of Medical Images - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 131

6.7 Segmentation Using Type II Fuzzy Set Theory
Tizhoosh also suggested a segmentation method [23] using Type II fuzzy set,
but on artificial images. In most of the images, the object of interest is dark
and the primary membership of dark pixels is defined using a Z member-
ship function in an interval ( g max −(( g max + g min )/2)). S o,

2

⎛ ⎞

gg −

≤≤ +

g g

min max in

18 − if g g

⎜ ⎟

min

g − g 4

⎝ ⎠

max in

2

( ) ( ) −

⎛ g − g g + g 8 g ⎞

g + g

<< +

g g max m in max in

max in max min

2 if g

μ Ag

() =

⎜ ⎟

g − g 4 2

⎝ ⎠

max in

g + g

max in

0 if g >

2

(6.25)

The lower and upper membership functions are computed as

()

( )

α ij ,

() = ()

μ ij , μ gij ,

L

( )

1 /(,) α ij

() = ()

μ ij , μ gij ,

U

α for each point is computed with respect to its neighbourhood:

( ) − ( )

⎛ max gi kj k ++ , † min g ikjk + , + ⎞

() =×

k k

(6.26)

α ij F

, min 1 , ††

⎜ ⎟

L − 1

⎝ ⎠

where
k ∈ [− n , …, −1, 0, 1, …, n ], n = W /2, and W is the size of the window
F is an amplification factor that lies between [1, ∞]

With more F value, that is, around 20, weak edges also are included, and with
lesser F value, that is, around 2, strong edges are included.
This means that if the difference in the intensity of the centre and its imme-
diate neighbourhood is more, then uncertainty increases. Next, the weight of
upper and lower membership functions is computed as

( )

min gi kj k

g

++ ,

() =

k

wij ,

L

max

( )

max gi kj k

g

++ ,

() =

k

wij ,

U

max