Efficient computation of high-order Meixner moments for large-size signals and images analysis
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Efficient computation of high-order Meixner moments for large-size signals and images analysis Achraf Daoui 1
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& Mhamed Sayyouri & Hassan Qjidaa
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Received: 2 May 2020 / Revised: 19 August 2020 / Accepted: 26 August 2020 # Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract
In this paper, we present an in-depth study on the computational aspects of high-order discrete orthogonal Meixner polynomials (MPs) and Meixner moments (MMs). This study highlights two major problems related to the computation of MPs. The first problem is the overflow and the underflow of MPs values (“Nan” and “infinity”). To overcome this problem, we propose two new recursive Algorithms for MPs computation with respect to the polynomial order n and with respect to the variable x. These Algorithms are independent of all functions that are the origin the numerical overflow and underflow problem. The second problem is the propagation of rounding errors that lead to the loss of the orthogonality property of high-order MPs. To fix this problem, we implement MPs based on the following orthogonalization methods: modified Gram-Schmidt process (MGS), Householder method, and Givens rotation method. The proposed Algorithms for the stable computation of MPs are efficiently applied for the reconstruction and localization of the region of interest (ROI) of large-sized 1D signals and 2D/3D images. We also propose a new fast method for the reconstruction of large-size 1D signal. This method involves the conversion of 1D signal into 2D matrix, then the reconstruction is performed in the 2D domain, and a 2D to 1D conversion is performed to recover the reconstructed 1D signal. The results of the performed simulations and comparisons clearly justify the efficiency of the proposed Algorithms for the stable analysis of largesize signals and 2D/3D images. Keywords Discrete orthogonal Meixner moments . Discrete orthogonal Meixner polynomials . Signal and image reconstruction . Orthogonalization methods . Numerical stability . High-order polynomials
* Achraf Daoui [email protected] Extended author information available on the last page of the article
Multimedia Tools and Applications
1 Introduction Orthogonal moments (OM) are widely applied in different fields of signal and image analysis. There are two categories of orthogonal moments, namely continuous orthogonal moments (COM) and discrete orthogonal moments (DOM). COM are defined as the projection of the data space (signal or image) onto a continuous orthogonal polynomial kernel, such as Zernike [8], Legendre [2], Fourier-Mellin [27], pseudo-Zernike [37] and Gaussian-Hermite [15] polynomial. The applicability of COM for digital image and signal analysis is limited by the problem of the continuous polynomials space discretization, thus leading to discretization and approximation errors. In order to solve this problem, researchers proposed to substitute COM by DOM, because the latter are defined on a discrete domain and therefore there is no need to make approximations or dis
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