Four-Term Recurrence for Fast Krawtchouk Moments Using Clenshaw Algorithm

Abstract

Krawtchouk polynomials (KPs) are discrete orthogonal polynomials associated with the Gauss hypergeometric functions. These polynomials and their generated moments in 1D or 2D formats play an important role in information and coding theories, signal and image processing tools, image watermarking, and pattern recognition. In this paper, we introduce a new four-term recurrence relation to compute KPs compared to their ordinary recursions (three-term) and analyse the proposed algorithm speed. Moreover, we use Clenshaw’s technique to accelerate the computation procedure of the Krawtchouk moments (KMs) using a fast digital filter structure to generate a lattice network for KPs calculation. The proposed method confirms the stability of KPs computation for higher orders and their signal reconstruction capabilities as well. The results show that the KMs calculation using the proposed combined method based on a four-term recursion and Clenshaw’s technique is reliable and fast compared to the existing recursions and fast KMs algorithms.

High-Performance Krawtchouk Polynomials of High Order Based on Multithreading

“…Therefore, it is not generally recommended to evaluate orthogonal geometric moments by expanding them into standard powers because this will lead to possible overflow and/or underflow and cause a loss of precision [5]. Moreover, most orthogonal polynomials concern hypergeometric functions; therefore, the polynomial computation and the corresponding moments are considered time-consuming [19]. Specifically, directly computing the Krawtchouk moment using factorial and gamma functions leads to two main issues: high numerical instability and significant computation time, especially when higher-order moments are required for an improved description of image contents.…”

“…Various methods have been presented to accelerate the calculation of moment kernels and the summation of these kernels for different signal dimensions. These methods are based on recursive relationships and symmetry properties, which make the computation easy, fast, and effective [19][20][21]. For example, two original methods are proposed using the outputs of cascaded digital filters in deriving the Krawtchouk moment in [5].…”

“…Moreover, recent developments in algorithms for computing Krawtchouk polynomial coefficients have been proposed to address instability issues associated with the computation of high-order polynomials [19,22,23]. One such approach involves the use of the three-term recurrence relation, which expresses each coefficient of the Krawtchouk polynomial in terms of the previous two coefficients.…”

Seismic Image Identification and Detection Based on Tchebichef Moment Invariant

“…Mukundan et al [22] and Yap et al [23] introduced a set of discrete orthogonal moments based on the discrete Tchebichef polynomials and Krawtchouk polynomials, respectively. The development of these discrete orthogonal moments has several applications in image detection, classification, and fast computational methods [24][25][26].…”