Image Processing Reference

In-Depth Information

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

. . . . . . .

Some Experiments

4.6

Conclusion

4-13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bibliography

4-14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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