Cantor, Fuzzy, Near, and Rough Sets in Image Analysis - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 1

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Cantor, Fuzzy, Near, and Rough Sets
in Image Analysis

1.1 Introduction ............................................... 1-1

1.2 Cantor Set ................................................. 1-2

1.3 Near Sets .................................................. 1-2

Near Sets and Rough Sets • Basic Near Set Approach •
Near Sets, Psychophysics and Merleau-Ponty • Visual
Acuity Tolerance • Sets of Similar Images • To l erance
Near Sets • Near Sets in Image Analysis
1.4 Fuzzy Sets .................................................

James F. Peters
Computational Intelligence Laboratory,
Electrical & Computer Engineering, Rm.
E2-390 EITC Bldg., 75A Chancellor's Circle,
University of Manitoba, Winnipeg R3T 5V6
Manitoba Canada

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Notion of a Fuzzy Set • Near Fuzzy Sets • Fuzzy Sets in
Image Analysis
1.5 Rough Sets ................................................ 1-9

Sample Non-Rough Set • Sample Rough Set • Rough
Sets in Image Analysis
1.6 Conclusion ................................................. 1-11
Acknowledgements ............................................. 1-11
Bibliography ..................................................... 1-12

Sankar K. Pal
Machine Intelligence Unit, Indian Statistical
Institute,Kolkata, 700 108, India

1.1 Introduction

The chapters in this topic consider how one might utilize fuzzy sets, near sets, and rough sets, taken
separately or taken together in hybridizations, in solving a variety of problems in image analysis. A
brief consideration of Cantor sets (Cantor, 1883, 1932) provides a backdrop for an understanding
of several recent types of sets useful in image analysis. Fuzzy, near and rough sets provide a wide
spectrum of practical solutions to solving image analysis problems such as image understanding,
image pattern recognition, image retrieval and image correspondence, mathematical morphology,
perceptual tolerance relations in image analysis and segmentation evaluation. Fuzzy sets result from
the introduction of a membership function that generalizes the traditional characteristic function.
The notion of a fuzzy set was introduced by L. Zadeh in 1965 (Zadeh, 1965). Sixteen years later,
rough sets were introduced by Z. Pawlak in 1981 (Pawlak, 1981a). A set is considered rough
whenever the boundary between its lower and upper approximation is non-empty. Of the three forms
of sets, near sets are newest, introduced in 2007 by J.F. Peters in a perception-based approach to the
study of the nearness of observable objects in a physical continuum (Peters and Henry, 2006; Peters,
2007c,a; Peters, Skowron, and Stepaniuk, 2007; Henry and Peters, 2009b; Peters and Wasilewski,
2009; Peters, 2010).
This chapter highlights a context for three forms of sets that are now part of the computational
intelligence spectrum of tools useful in image analysis and pattern recognition. The principal con-

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