Numerical Stability Quality-Factor for Orthogonal Polynomials: Zernike Radial Polynomials Case Study

Abstract

In this work, we propose a Quality Factor (QF) that makes use of the orthogonal property in Orthogonal Polynomials (OPs). We have chosen Zernike Radial Polynomials (ZRPs), which were heavily investigated in fields related to digital image/signal processing, as a case study. We also implemented analysis of Zernike moments to investigate how the proposed QF relates to their computation error. Besides ZRPs direct computation, several fast algorithms have been proposed, e.g. the q-recursive, Kintner’s method, and Prata’s method. Our simulations using the proposed QF showed that the direct method and all forms of Prata’s method have numerical instability with orders ≥42 and ≥87 respectively. QF analyses using Kintner’s method and the q-recursive method have shown that these methods are numerically stable up to order 800 (the maximum targeted order in this work) and 180 respectively. We recommend using Prata’s method with orders less than 70, which contradict pervious findings suggesting that using Prata’s method(s) with orders up to 90 is legitimate. All the methods that we have investigated, except Kintner’s method, have numerical instability at high orders.