Accelerated and Improved Stabilization for High Order Moments of Racah Polynomials

TL;DR

This paper introduces an improved algorithm for computing discrete Racah polynomials that enhances numerical stability and allows for higher degree moments, benefiting applications in image processing and computer vision.

Contribution

The paper presents a novel stabilization algorithm (ImSt) that partitions the polynomial plane, uses the gamma function for initial values, and optimizes stability conditions for high-order moments.

Findings

ImSt extends the range of stable high-order moments.

It outperforms existing methods in stability and accuracy.

Applicable to large polynomial sizes and diverse parameters.

Abstract

One of the most effective orthogonal moments, discrete Racah polynomials (DRPs) and their moments are used in many disciplines of sciences, including image processing, and computer vision. Moments are the projections of a signal on the polynomial basis functions. Racah polynomials were introduced by Wilson and modified by Zhu for image processing and they are orthogonal on a discrete set of samples. However, when the moment order is high, they experience the issue of numerical instability. In this paper, we propose a new algorithm for the computation of DRPs coefficients called Improved Stabilization (ImSt). In the proposed algorithm, {the DRP plane is partitioned into four parts, which are asymmetric because they rely on the values of the polynomial size and the DRP parameters.} The logarithmic gamma function is utilized to compute the initial values, which empower the computation of the initial value for a wide range of DRP parameter values as well as large size of the polynomials. In addition, a new formula is used to compute the values of the initial sets based on the initial value. Moreover, we optimized the use of the stabilizing condition in specific parts of the algorithm. ImSt works for wider range of parameters until higher degree than the current algorithms. We compare it with the other methods in a number of experiments.