Medical Image Processing - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 32

are not considered as they are the complement of the membership degree.
An intuitionistic fuzzy set A in X may be represented as

Axx = {( , (), (), ())|} μν x xxX π ∈

A A A

with the condition

π () () () x + μ x + ν x = 1

A A A

It is obvious that

0 ≤ π A x () , or each ≤ 1 x X ∈

2.6.2 Type II Fuzzy Set

Again, if we explore properly, we find that the membership function defined
in the ordinary or type I fuzzy set theory is imprecise and uncertain. The
membership function in type I fuzzy set is considered to be 'fuzzy', and thus
the new fuzzy set is named as Type II fuzzy set, introduced by Zadeh [23].
Type II fuzzy set accounts this uncertainty by considering another degree
of freedom for better representation of uncertainty where the membership
functions are themselves fuzzy [14]. So, if the membership function of type I
fuzzy set is blurred, then Type II fuzzy set is obtained. Type I fuzzy set can
be defined by assigning upper and lower membership degrees to each ele-
ment, and the membership function does not have a single value for every
element but an interval - lower and upper intervals - which are written as

upper

α

μ =

=

[()]
[()] /

μ x

x 1

(2.1)

lower α

μ μ

where α ∈ [0, 1]. So, a more practical form of representing Type II fuzzy set
is written as

A

Type II =

{, (), ()| x μμ x x xX ∈ }

U L

and

<< ∈ [

]

μ () () (), x μ x μ x μ 01 ,

L U

So, in Type II fuzzy set, the membership function lies in an interval, and
this may be very useful in medical image processing where the fuzziness is
present in an interval.