Fast and stable computation of higher-order Hahn polynomials and Hahn moment invariants for signal and image analysis

Abstract

This article presents, on the one hand, new algorithms for the fast and stable computation of discrete orthogonal Hahn polynomials of high order (HPs) based on the elimination of all gamma and factorial functions that cause the numerical fluctuations of HPs, and based on the use of appropriate stability conditions. On the other hand, a new method for the fast and numerically stable computation of Hahn moment invariants (HMIs) is also proposed. This method is mainly based on the use of new recursive relations of HPs and of matrix multiplications when calculating HMIs. To validate the efficiency of the algorithms proposed for the calculation of HPs, several signals and large images (≥4000 × 4000) are reconstructed by Hahn moments (HMs) up to the last order with a reconstruction error tending towards zero (MSE ≃ 10−10). The efficiency of the proposed method for calculating HMIs is demonstrated on large medical images (2048 × 2048) with a very low relative error (RE ≃ 10−10). Finally, comparisons with some recent work in the literature are provided.