Linear-time recognition of digital polynomial curves and surfaces that have periodic differences

Abstract

Digital polynomial curves and surfaces arise from the digitization of algebraic surfaces such as lines, parabolas, planes, or paraboloids. It is known that the n-th difference of a digital polynomial curve of degree n is periodic for polynomials that have rational coefficients. In this paper we consider the following problem: Suppose we have a digital curve S whose n-th difference is known to be periodic. When is S a digital rational polynomial curve? As a solution to this problem we state a simple criterion that can be checked in linear time. As a first application of this criterion we describe a linear time algorithm for the recognition of digital straight lines. In comparison to other algorithms, the advantages of the new algorithm are its simplicity, and its ability to actually find the coefficients of the rational polynomial representing the line. We then go on to discuss the applicability of this criterion to the recognition of digital curves and surfaces of arbitrary degree.