Fast and accurate computation of high-order Tchebichef polynomials

Discrete Tchebichef polynomials (DTPs) and their moments are effectively utilized in different fields such as video and image coding, pattern recognition, and computer vision due to their remarkable performance. However, when the moments order becomes large (high), DTPs prone to exhibit numerical instabilities. In this article, a computationally efficient and numerically stable recurrence algorithm is proposed for high order of moment. The proposed algorithm is based on combining two recurrence algorithms, which are the recurrence relations in the n$$ n $$ and x$$ x $$-directions. In addition, an adaptive threshold is used to stabilize the generation of the DTP coefficients. The designed algorithm can generate the DTP coefficients for high moment's order and large signal size. By large signal size, we mean the samples of the discrete signal are large. To evaluate the performance of the proposed algorithm, a comparison study is performed with state-of-the-art algorithms in terms of computational cost and capability of generating DTPs with large polynomial size and high moment order. The results show that the proposed algorithm has a remarkably low computation cost and is numerically stable, where the proposed algorithm is 27x$$ \times $$ times faster than the state-of-the-art algorithm.

Automated summary

This article introduces a computationally efficient and numerically stable algorithm for generating discrete Tchebichef polynomials (DTPs) coefficients, which can be applied to high order moments and large signal sizes.