Least squares fitting of digital polynomial segments

Abstract

It is proved that digital polynomial segments and their least squares polynomial fits are in one-to-one correspondence. This enables an efficient representation of digital polynomial segments by n+3 parameters, under the condition that an upper bound, say n, for the degrees of the digitized polynomials is assumed. One of such representations is (x 1, m, an, an−1,…, a 0), where x 1 and m are the x-coordinate of the left endpoint and the number of digital points, respectively, while a n, a n−1,..., a 0 are the coefficients of the least squares polynomial fit Y=a nXn+an− 1Xn−1+ ...+a0, for a given digital polynomial segment.