Abstract
Krawtchouk polynomials (KPs) and their moments are promising techniques for applications of information theory, coding theory, and signal processing. This is due to the special capabilities of KPs in feature extraction and classification processes. The main challenge in existing KPs recurrence algorithms is that of numerical errors, which occur during the computation of the coefficients in large polynomial sizes, particularly when the KP parameter (p) values deviate away from 0.5 to 0 and 1. To this end, this paper proposes a new recurrence relation in order to compute the coefficients of KPs in high orders. In particular, this paper discusses the development of a new algorithm and presents a new mathematical model for computing the initial value of the KP parameter. In addition, a new diagonal recurrence relation is introduced and used in the proposed algorithm. The diagonal recurrence algorithm was derived from the existing n direction and x direction recurrence algorithms. The diagonal and existing recurrence algorithms were subsequently exploited to compute the KP coefficients. First, the KP coefficients were computed for one partition after dividing the KP plane into four. To compute the KP coefficients in the other partitions, the symmetry relations were exploited. The performance evaluation of the proposed recurrence algorithm was determined through different comparisons which were carried out in state-of-the-art works in terms of reconstruction error, polynomial size, and computation cost. The obtained results indicate that the proposed algorithm is reliable and computes lesser coefficients when compared to the existing algorithms across wide ranges of parameter values of p and polynomial sizes N. The results also show that the improvement ratio of the computed coefficients ranges from 18.64% to 81.55% in comparison to the existing algorithms. Besides this, the proposed algorithm can generate polynomials of an order ∼8.5 times larger than those generated using state-of-the-art algorithms.
Highlights
Digital image processing plays an essential role in several aspects of our daily lives.Image signals are subject to several processes such as transmission [1], enhancement [2], transformation [3], hiding [4], and compression [5,6]
The results show that the proposed initial values are more computable for wide ranges of parameter p and large polynomial sizes N, as desired
63.11% is achieved by the proposed algorithm when it compares with the FRK algorithm while it is 81.55% when it compares with the recurrence algorithm in the n direction (RAN) and recurrence algorithm in the x direction (RAX) algorithms
Summary
Digital image processing plays an essential role in several aspects of our daily lives. There is an essential need to reduce the number of recurrences, especially when the polynomial size is increased Such a reduction could lead to a reduction in the propagation error, thereby leading to a more stable computation of polynomial coefficients, as desired. The work in [25] proposed a recurrence relation algorithm that has the ability to compute KPCs with very large sizes. The existing algorithms suffer from the following limitations: (1) no initial value is provided; (2) the propagation error is high; and (3) the implementation of these algorithms is limited to a specific value of the parameter p They suffer from numerical instabilities, especially when the polynomials orders and sizes become high. Notation: In this paper, the operator transpose is denoted by (·)T and (ba) denotes the binomial coefficients
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