New Algorithm for Large-Sized 2D and 3D Image Reconstruction using Higher-Order Hahn Moments

Daoui, Achraf; Yamni, Mohamed; ogri, O. El; Karmouni, Hicham; Sayyouri, Mhamed; Qjidaa, Hassan

doi:10.1007/s00034-020-01384-z

Cited by 31 publications

(41 citation statements)

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“…When the order n of HPs is high, the computation of these polynomials becomes unstable and expansive in terms of computation time because of gamma and factorial functions that define HPs (Eq. ( 1 )) [ 9 , 27 ]. To improve the computation speed and the numerical stability of high-order HPs, researchers use the recurrence relations with respect to the order n or with respect to the variable x [ 46 , 50 ].…”

Section: Conventional and Proposed Computation Of Hpsmentioning

confidence: 99%

“…This problem is discussed by Camacho-bello et al in [ 5 ] when computing high-order Tchebichef polynomials. To overcome the encountered problem for Tchebichef, Meixner and Hahn polynomials, the modified Gram-Schmidt orthonormalization process (GSOP) is used [ 5 , 8 , 9 , 12 ]. However, the correction of numerical instability by GSOP is very time-consuming because GSOP involves a complex calculation process [ 5 , 12 ].…”

Section: Conventional and Proposed Computation Of Hpsmentioning

confidence: 99%

“…This problem limits the fast reconstruction of large-size signals and images by high-order HPs. Indeed, Table 3 shows a comparison in terms of execution time in second ( s ) by using algorithm 1, and GSOP method presented in [ 9 ]. The computation of HPs by these methods is performed 100 times, and then the average execution time is computed and illustrated in Table 3 .…”

Section: Conventional and Proposed Computation Of Hpsmentioning

confidence: 99%

“…This method succeeded to ensure the numerical stability of higher-order Tchebichef polynomials. More recently, GSOP method is used by Daoui et al to introduce a numerically stable computation of Hahn [ 9 ], Charlier [ 8 ] and Meixner [ 12 ] polynomials.…”

Section: Introductionmentioning

confidence: 99%

“…To overcome the problem ( ii ), a new numerical stability method is proposed. This method that is based on appropriate stability conditions, is very low in terms of computational cost compared to the GSOP method that is presented in [ 9 ]. As result, the computation of high-order Hahn moments (HMs) becomes stable and fast when reconstructing large-size signals and images.…”

Section: Introductionmentioning

confidence: 99%

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

Daoui

Karmouni

Sayyouri

et al. 2021

Multimed Tools Appl

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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.

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