Image Enhancement - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 99

The interval [(), ()]

hghg
A

U L

can be interpreted as the possible frequency of
occurrence of the intensity level around ' g ' with the lower bound represent-
ing the minimum possible frequency of occurrence of the grey level and
the upper bound representing the maximum frequency of occurrence of
the grey level. Thus, hesitancy histogram is the length of the interval. Now
for hesitance histogram equalization, similar to the conventional histogram
equalization method, the cumulative density function of the hesitancy his-
togram is used for grey-level transformation so that the resulting image will
possess a uniform hesitancy histogram:

A

g

∑

(5.21)

H

gL hg
A

ʹ=− ( 1 ) ()

k = 0

where g and g ′ are the original and transformed grey levels of the image,
respectively. A contrast-enhanced image is formed.

5.5 Image Enhancement Using Type II Fuzzy Set
In this section, image enhancement using Type II fuzzy set is discussed. It
considers the membership function in ordinary fuzzy set as fuzzy or vague,
and so the membership function lies in an interval range with upper and
lower membership levels. Thus, Type II fuzzy set represents the uncertainty
in a different and better way than type I fuzzy set, and so better enhance-
ment results may be expected. There is very little research on medical image
enhancement using Type II fuzzy set, and these are discussed in this section.

5.5.1 Type II Fuzzy Enhancement (Method I)

Chaira suggested a Type II fuzzy enhancement method using fuzzy t -conorm
by Chaira discussed in Chapter 3. Though there are many t -conorms in the
literature, the t -conorms that are algebraic in nature perform better. The rea-
son behind this is that the operators that belong to a conditional class may
not lead to realization of perfect t -norm and t -conorm though they are com-
putationally simple. The operators in the algebraic class that do not contain
the min and max operators reveal the actual value.
The original image is initially fuzzified with the membership function μ A ( g )
using Equation 5.1. Then using Type II fuzzy set, two levels are computed:

upper α

μ () [()] g = μ g

(5.22)

lower 1 / α

μ () [()] g = μ g , 0 < α ≤ 1