Polygonal approximation by the minimax method

Abstract

This paper is concerned with the problem of approximating digitized pictures by polygons. The digitized picture is represented by a two-dimensional array of points, and its is desired to convert the given array into a set of polygons, such that each polygon has the least number of sides and the error between the initial points and the approximated lines is less than a given constant (E). There are many other solutions to this problem, but to evaluate the error, they use either the least-squares method or the cone intersection method. In this paper, it is shown that the minimax approximation that minimizes the maximum distance between the given points and the approximated line is the best approximation for the problem. A method is presented for obtaining the minimax approximated lines from the given N points in time proportional to N ∗ log N. From the obtained lines a polygon is then found using another algorithm. The polygon satisfies the condition that the number of sides is minimum and the maximum distance between the given points and the sides is less than the given E.