A Constant Space Representation of Digital Cubic Parabolas

Abstract

The concept of ``noisy'' straight line introduced by Melter and Rosenfeld is generalized and applied to digital cubic parabolas. It is proved that digital cubic parabola segments and their least square cubic parabola fits are in one-to-one correspondence. This leads to a constant space representation of a digital cubic parabola segment. One such representation is $(x_1,n,a,b,c,d)$, where $x_1$ and $n$ are the left endpoint and the number of digital points, respectively, while $a$, $b$, $c$ and $d$ are the coefficients of the least square cubic parabola fit $Y = aX^3+bX^2+cX+d$ for the given cubic parabola segment.