Computing the Multivalued Shape of a Pattern Class

Abstract

The present chapter deals with the problem of determining the pattern class and its multivalued shape/boundary from sampled points (training samples). Once these are computed, some salient features of the class can then be extracted which are useful in making decisions about a course of action (e.g., identification, classification and pattern description) to be taken later. This will also reduce the storage requirement of the complete pattern class.

It may be noted that in most of the real life patter recognition problems, the complete description of a pattern class is not known. Instead, a few sampled points are usually available which are assumed to represent the class. Hence determining the pattern class and its shape from sampled points is an important problem in pattern recognition.

There are various approaches described in the literature for determining the shape of a pattern class from sampled points xc1–6. These methods are mostly heuristic in nature and they povide an exact boundary or shape of the pattern class. One of the inherent observations about these algorithms is that the boundary of the class is restricted by the sampled points, This need not be true because the resulting boundary leaves certain regions not confined in it, although it should be. So, it is necessary to extend the boundaries to some extent to handle the possible uncovered portions by the sampled points. The extended portions should have the following two properties:

• (i) As the number of sampled points increases, the extended portions should de- crease.

• (ii) The extended portions should have less possibility to be in the patter class than the portions explicitly highlighted by the sampled points.

The second property leads to define a multivalued or fuzzy (with continuum grade of belongingness) boundary of a pattern class. The basic concept of one of the existing methods xc1–6 is described below in short for illustration.