On characterization of discrete triangles by discrete moments

Abstract

For a given real triangle T its discretization on a discrete point set S consists of points from S which fall into T. If the number of such points is finite, the obtained discretization of T will be called discrete triangle. In this paper we show that the discrete moments having the order up to 3 characterize uniquely the corresponding discrete triangle if the discretizationing set S is fixed. Of a particular interest is the case when S is the integer grid, i.e., S = Z2. Then the discretization of a triangle T is called digital triangle. It turns out that the proposed characterization preserves a coding of digital triangles from an integer grid of a given size, say m × m within an O(logm) amount of memory space per coded digital triangle. That is the theoretical minimum.