Rough Hybrid Scheme: An application of breast cancer imaging - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 83

This chapter is organized as follows: Section 5.2 gives a brief mathematics background
of fuzzy and rough sets and neural network. Section 5.3 discusses the proposed rough
hybrid scheme in detail. Experimental analysis and discussion of the results are described
in Section

5.4. Finally, conclusions are presented in Section 5.5.

5.2 Fuzzy sets, rough sets and neural networks: Brief Intro-

duction

Recently various intelligent techniques and approaches have been applied to handle the
different challenges posed by data analysis. The main constituents of intelligent systems
include fuzzy logic, neural networks, genetic algorithms, and rough sets. Each of them
contributes a distinct methodology for addressing problems in its domain. This is done in a
cooperative, rather than a competitive, manner. The result is a more intelligent and robust
system providing a human-interpretable, low cost, exact enough solution, as compared to
traditional techniques. This section provides a brief introduction into fuzzy sets, rough sets
and neural networks.

5.2.1 Fuzzy Sets

Professor Lotfi Zadeh (Zadeh, 1965) introduced the concept of fuzzy logic to present vague-
ness in linguistics, and further implement and express human knowledge and inference ca-
pability in a natural way. Fuzzy logic starts with the concept of a fuzzy set. A fuzzy set is
a set without a crisp, clearly defined boundary. It can contain elements with only a partial
degree of membership. A Membership Function (MF) is a curve that defines how each point
in the input space is mapped to a membership value (or degree of membership) between
0 and 1. The input space is sometimes referred to as the universe of discourse. Let X be
the universe of discourse and x be a generic element of X. A classical set A is defined as a
collection of elements or objects x∈X, such that each x can either belong to or not belong
to the set A, AvX. By defining a characteristic function (or membership function) on
each element x in X, a classical set A can be represented by a set of ordered pairs (x, 0)
or (x, 1), where 1 indicates membership and 0 non-membership. Unlike conventional set
mentioned above fuzzy set expresses the degree to which an element belongs to a set. Hence
the characteristic function of a fuzzy set is allowed to have value between 0 and 1, denoting
the degree of membership of an element in a given set. If X is a collection of objects denoted
generically by x, then a fuzzy set A in X is defined as a set of ordered pairs:

A ={(x, µ A (x))|x∈X}

(5.1)

µ A (x) is called the membership function of linguistic variable x in A, which maps X to the
membership space M, M = [0, 1], where M contains only two points 0 and 1, A is crisp and
µ A (x) is identical to the characteristic function of a crisp set. Triangular and trapezoidal
membership functions are the simplest membership functions formed using straight lines.
Some of the other shapes are Gaussian, generalized bell, sigmoidal and polynomial based
curves.
The adoption of the fuzzy paradigm is desirable in image processing because of the uncer-
tainty and imprecision present in images, due to noise, image sampling, lightning variations
and so on. Fuzzy theory provides a mathematical tool to deal with the imprecision and am-
biguity in an elegant and e cient way. Fuzzy techniques can be applied to different phases