Thresholding of Medical Images - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 133

For standard Cauchy distribution, γ = 1 and a = 0, and the distribution
becomes

1

fx (,,) 01 =

2

π ( 1 + x )

To find the membership values of the image, the distribution is modified.
Considering

1
const

γ=

and
const = 1/( f max − f fmin ), where f max and f fmin are the maximum and minimum
grey values of the image, respectively, and substituting the values, the dis-
tribution becomes

) = ⎜

⎞

const
const

const 1

fxa (;,) γ =

⎟

( ( )

( )

2

π

( )

2

π 1 + ⋅ xa − 1 + const ⋅ xa −

⎠

In order to make the membership values feasible, the constant term
const/ ( ) is not taken into account. So, the function that is used to derive
the membership function is

1

(6.28)

fxa (,) =

( )

2

( )

1 + const ⋅ xa −

In segmentation, t -conorm is used to form a new membership function using
the two membership functions - upper and lower membership levels in
Type II fuzzy set.
The image is initially fuzzified using the modified Cauchy membership
function in Equation 6.28.
Then the upper and lower membership functions of interval Type II fuzzy
set for the object and background regions are computed as

1 / α

lower

μ () () g = ⎣ μ g ⎦

ij ij

α

upper

μ () () g = ⎣ μ g ⎦

ij ij

where

1

μ () g = , object

( )

2

1 + const ⋅ ( xm − )

O

1

= , background

( )

2

1 + const ⋅ ( xm − )

B

m O and m B are computed using Equation 6.6, respectively.