# Estimating a Set in R2 with Finite Number of Points using Minimum Spanning Tree and Multi Layer Perceptron.

December 1992

## Date of Award

Winter 12-12-1993

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

Computer Science

## Department

Machine Intelligence Unit (MIU-Kolkata)

## Supervisor

Murthy, C. A. (MIU-Kolkata; ISI)

## Abstract (Summary of the Work)

1.1 Problem SpecificationThe problem discussed here is Given a collection of representative points from a set of points, how to get back the original set.Once the set is computed, some salient features of the class can then be extracted which may be useful in making decisions about a course of action ( for eg. , in identification , classification etc. ) to be taken later. This will also reduce storage requirement of the set.It may be noted that in most of the real life pattern recognition problems, the complete description of a set is not known. Instead a few sampled points are usually available. Hence determining the set and its shape from sampled points is an important problem in pattern recognition.1.2 Work Already DoneThe first method by which people tried to solve this problem was constructing convex hulls. The efficient construction of convex hulls for finite sets of points in the plane is one of the most exhaustively examined problems in computational geometry. Part of the motivation is theoretical in nature. It seems to stem from the fact that the convex hull problem, like sorting, is easy to formulate and visualize. Furthermore, the convex hull problem, again like sorting, plays an important role as a component of a large number of more complex problems. Nevertheless, much of the work is motivated by the direct applications in some more practical branches of computer science. Akl and Toussiant[2], for instance, discuss the relevance of the convex hull problem to pattern recognition. By identifying and ordering the extrene points of a point set, the convex hull serves to characterize, atleast in a rough way, the shape of such a set.Jarvis(3] presents several algorithms based on so called nearest- neighbour logic that compute what he calls the shape of a finite set of points. The shape, in Jarvis terminology, is a notion made concrete by the algorithms that he proposes for its construction. Besides this lack of any analytic definition, the inefficiency of Jarvis terminology to construct the shape is striking drawback. Fairfeild[4] introduced a notion of the shape of a finite point set based on the closest point voronoi diagram of the set. He informally links his notion of shape with human perception but presents no concrete properties of his shapes, in particular, algorithmic results.The disadvantage of the above methods is that they do not work for non-convex sets. Grenander(11] tried to give a method which finds the pattern class for non-convex sets. His method was, given a set of points, for each point include all the points within a circle of radius e e for suitable e. But the drawback with his method is that he did not give any method by which to calculate the value of e.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843266

## Control Number

ISI-DISS-1992-172