Representation of digital parabolas by least-square fit

Abstract

The concept of noisy straight line introduced by Melter and Rosenfeld is generalized and applied for digital parabolas. It is proved that digital parabola segments and their least square parabola fits are in one-to-one correspondence. This enables a (first known) vector space representation of a digital parabola segment. One of such representations is (x1, n, a, b, c) where x1 and n are the x-coordinate of the left endpoint and the number of digital points, respectively, while a, b, and c are the coefficients of the least square parabola fit Y = aX2 + bX + c for the given parabola segment.