Accelerated and Improved Stabilization for High Order Moments of Racah Polynomials
This work addresses a specific numerical issue in image processing and computer vision, offering an incremental improvement for researchers using Racah polynomials.
The paper tackles the numerical instability of high-order discrete Racah polynomial moments by proposing an Improved Stabilization (ImSt) algorithm, which achieves stable computation for a wider range of parameters and higher degrees than existing methods.
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.