Cantor, Fuzzy, Near, and Rough Sets in Image Analysis - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 9

...A fuzzy set is characterized by a membership
function which assigns to each object its grade of membership
(a number lying between 0 and 1) in the fuzzy set.
-A new view of system theory
-L.A. Zadeh, 20-21 April 1965.

Set Theory Law 2 Fuzzy Sets
Every element in a fuzzy set has a graded membership.

1.4.1 Notion of a Fuzzy Set
The notion of a fuzzy set was introduced by L.A. Zadeh in 1965 (Zadeh, 1965). In effect, a Cantor
set is a fuzzy set if, and only if every element of the set has a grade of membership assigned to it by
a specified membership function. Notice that a membership function

φ : X → [0 , 1] is a special case

of what is known as a probe function in near set theory.

1.4.2 Near Fuzzy Sets
A fuzzy set X is a near set relative to a set Y if the grade of membership of the objects in sets X

, Y is

assigned to each object by the same membership function φ and there is a least one pair of objects

x , y ∈ X × Y such that φ ( x ) − φ ( y ) 2 ≤ ε } , i.e. , the description of x is similar to the description y

within some ε .

1.4.3 Fuzzy Sets in Image Analysis
Fuzzy sets have widely used in image analysis (see, e.g. , (Rosenfeld, 1979; Pal and King, 1980,
1981; Pal, 1982; Pal, King, and Hashim, 1983; Pal, 1986, 1992; Pal and Leigh, 1995; Pal and Mitra,
1996; Nachtegael and Kerre, 2001; Deng and Heijmans, 2002; Martino, Sessa, and Nobuhara, 2008;
Sussner and Valle, 2008; Hassanien et al., 2009)). In the notion of fuzzy sets, (Pal and King, 1980,
1981) defined an image of M N dimension and L levels as an array of fuzzy singletons, each with

a value of membership function denoting the degree of having brightness or some property relative
to some brightness level l

×

1. The literature on fuzzy image analysis is
based on the realization that the basic concepts of edge, boundary, region, relation in an image do
not lend themselves to precise definition.
From set composition Law 2, it can be observed that fuzzy sets are Cantor sets.

, where l = 0 , 1 , 2 ,..., L −

1.5 Rough Sets

A new approach to classification,based on information systems theory,
given in this paper.... This approach leads to a new formulation
of the notion of fuzzy sets (called here the rough sets).
The axioms for such sets are given, which are the same
as the axioms of topological closure and interior.
-Classification of objects by means of attributes.
-Z. Pawlak, 1981.