Mathematical Morphology and Rough Sets - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 66

4

Mathematical Morphology and
Rough Sets

Homa Fashandi
Computational Intelligence
Laboratory,University of Manitoba, Winnipeg
R3T 5V6 Manitoba Canada

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

4.2 Basic Concepts from Topology . . . . . . . . . . . . . . . . . . . . . . 4-1

4.3 Mathematical Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3

4.4 Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5

4.5 Mathematical Morphology and Rough Sets 4-9

James F. Peters
Computational Intelligence
Laboratory,University of Manitoba, Winnipeg
R3T 5V6 Manitoba Canada

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Some Experiments

4.6 Conclusion 4-13

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Bibliography 4-14

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4.1 Introduction

This chapter focuses on the relation between mathematical morphology (MM) (Serra, 1983)
operations and rough sets (Pawlak, 1981, 1982; Pawlak and Skowron, 2007c,b,a) mainly
based on topological spaces considered in the context of image retrival (see, e.g., (Fashandi,
Peters, and Ramanna, 2009)) and the basic image correspondence problem (see, e.g., (Pe-
ters, 2009, 2010; Meghdadi, Peters, and Ramanna, 2009)). There are some obvious similar-
ities between MM operations and set approximations in rough set theory. There have been
several attempts to link MM and rough sets. Two major works have been published in this
area (Polkowski, 1993; Bloch, 2000). L. Polkowski defines hit-or-miss topology on rough
sets and proposed a scheme to approximate mathematical morphology within the general
paradigm of soft computing (Polkowski, 1993),(Polkowski, 1999). Later, I. Bloch tries to
demonstrate a direct link between MM and rough sets through relations, a pair of dual
operations and neighbourhood systems (Bloch, 2000). I.Bloch's approach is carried forward
by J.G. Stell, who defines a single framework that includes the principal constructions of
both mathematical morphology and rough sets (Stell, 2007). To make this chapter fairly
self-contained, background information on the basics of topology is presented, first. The
chapter then presents the basics of mathematical morphology. Then principles of rough
set theory are considered and the links between them are discussed. Finally, a proposed
application of the ideas from these two areas is given in terms of image retrieval.

4.2 Basic Concepts from Topology

This section introduces the basic concepts of topology (Engelking, 1989; Gemignani, 1990).
For the sake of completeness, basic definitions from topology are briefly presented in this

4-1