 Research
 Open access
 Published:
An efficient computation of discrete orthogonal moments for biosignals reconstruction
EURASIP Journal on Advances in Signal Processing volume 2022, Article number: 104 (2022)
Abstract
Biosignals are extensively used in diagnosing many diseases in wearable devices. In signal processing, signal reconstruction is one of the essential applications. Discrete orthogonal moments (DOMs) are effective analysis tools for signals that can extract digital information without redundancy. The propagation of numerical errors is a significant challenge for the computation of DOMs at high orders. This problem damages the orthogonality property of these moments, which restricts the ability to recover the signal's distinct and unique components with no redundant information. This paper proposes a stable computation of DOMs based on QR decomposition methods: the Gram–Schmidt, Householder, and Given Rotations methods. It also presents a comparative study on the performance of the types of moments: Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner moments. The proposed algorithm's evaluation is done using the MITBIH arrhythmia dataset in terms of mean square error and peak signal to noise ratio. The results demonstrate the superiority of the proposed method in computing DOMs, especially at high moment orders. Moreover, the results indicate that the Householder method outperforms Gram–Schmidt and Given Rotations methods in execution time and reconstruction quality. The comparative results show that Tchebichef, Krawtchouk, and Charlier moments have superior reconstruction quality than Hahn and Meixner moments, and Tchebichef generally has the highest performance in signal reconstruction.
1 Introduction
With the explosive growth in computer technology and signal analysis tools, computeraided analysis of biosignals has become a common part of clinical. Biosignal reconstruction is one of the important applications in biosignal processing. The study of the literature on image analysis techniques indicates that the method of orthogonal moments plays a significant role in each of its important fields. These fields include image reconstruction [1, 2], face recognition [3], image classification [4, 5], image watermarking [6], image encryption [7], image compression [8, 9], color stereo image analysis [10]. Orthogonal moments are classified as continuous or discrete depending on whether the kernel functions are orthogonal in the continuous or discrete domain. Continuous orthogonal can be utilized to characterize an image with minimal redundant information. But even so, computing these moments needs a coordinate transformation and an estimate of the continuous moment's integrals. This adds computational complexity and introduces approximation errors [11]. To this end, many researchers have started to use discrete orthogonal moments [12, 13]. Zhu et al. [14, 15] demonstrated that discrete orthogonal moments are more effective than continuous orthogonal moments at representing images. The types of discrete orthogonal moments (DOMs) according to their corresponding discrete orthogonal polynomials include Tchebichef [16, 17], Krawtchouk[18,19,20,21], Charlier [22,23,24], Hahn [25, 26] and Meixner [27, 28] moments. At the present time, Discrete Orthogonal Moments (DOMs) are gaining popularity in analyzing onedimensional signals due to their effectiveness in capturing digital information without redundancy. In order to compute DOMs, we have to compute Kernel discrete orthogonal polynomials (DOPs).
The computation of highorder DOPs faces a major problem which is the propagation of numerical errors. This problem destroys the orthogonality property of these polynomials, which affects the ability to extract the signal's distinct and unique components with no information redundancy. To address this problem, we propose using QR decomposition methods to maintain the orthogonality property by reorthonormalization DOPs. There are many ways for QR decomposition, like the Gram–Schmidt method, the Householder method, and the Given Rotations method [29]. These methods maintain the highorder DOPs orthogonality property effectively. Thus, using the DOMs to analyze largesize signals will become highly efficient Due to significant improvements in the computation of DOPs. Our paper presents several contributions that can be summarized as follows:

Testing Discrete Orthogonal Moments (DOMs) in biosignals analysis and reconstruction.

Proposing a new modified version of DOPs by the QR decomposition methods like the Gram–Schmidt, Householder, and Given Rotations methods. In addition to comparing methods of QR decomposition to estimate the best methods.

Presenting comparative study between the different types of moments to estimate the best moment in analyzing and reconstructing biosignals.
The rest of the paper is structured into five sections: Sect. 2 outlines the Recursive relation of Discrete Orthogonal Polynomials (DOPs). Discrete Orthogonal Moments (DOMs) will be discussed in Sect. 3. Section 4 shows the proposed procedure for ensuring the orthogonality property of discrete polynomials. Experimental results and discussion are presented in Sect. 5. Discussions are shown in Sect. 6. In Sect. 7, we conclude our work.
2 Recursive relation of discrete orthogonal polynomials (DOPs)
The discrete orthogonal polynomials are the polynomial solutions of the given difference equation
where \(\Delta {p}_{n}\left(x\right)={p}_{n}\left(x+1\right){p}_{n}\left(x\right) and \nabla {p}_{n}(x)={p}_{n}(x){p}_{n}(x1)\) indicates backward finitedifference operator and forward finite difference operator, respectively. \(\sigma (x)\text{ and }\tau (x)\) denote first and second degree functions. \({\lambda }_{n}\) indicate a suitable constant.
The polynomials \({p}_{n}(x)\) satisfy an orthogonality relation of the form
where \(w\left(x\right)\) is the weight function, \({d}_{n}^{2}\) denotes the square of the norm of the corresponding orthogonal polynomials and \({\delta }_{mn}\) denotes the Dirac function. The normalized orthogonal polynomials can be obtained by utilizing the square norm and weighted function
Therefore, the orthogonal property of normalized orthogonal polynomials in (1b) can be rewritten as
A general formula for getting the normalized discrete orthogonal polynomials \({\widetilde{p}}_{n}(x)\) of order \(n\) is defined threeterm recursive relation as follows [11]:
where A, B, C, D, and E are coefficients independent of each polynomial set shown in Table 1. These discrete orthogonal polynomials include Tchebichef\({{t}}_{{n}}\left({x};{ N}\right)\), Krawtchouk \({k}_{n}\left(x;P,N\right)\), Charlier \({c}_{n}(x)\), Hahn \({h}_{n}^{(\alpha ,\beta )}(x)\) and Meixner \({m}_{n}^{(a,b)}(x)\) Polynomials.\({\widetilde{p}}_{n1}(x)\) and \({\widetilde{p}}_{n2}(x)\) are the zeroorder and firstorder polynomials, respectively.
2.1 Tchebichef polynomials
The nth Tchebichef polynomials \(t_{n} \left( x \right)\) are defined by hypergeometric function as the follows
From Eq. (1.e) and Table 1, we obtain the recursive relation of discrete orthogonal Tchebichef polynomials as follows:
with
where
2.2 Krawtchouk polynomials
Krawtchouk polynomials \(k_{n} \left( {x,p} \right)\) of order n are defined by hypergeometric function as the follows
The recursive relation of discrete orthogonal Krawtchouk polynomials can be calculated using Eq. (1.e) and Table 1 as:
with
where
2.3 Charlier polynomials
Charlier polynomials \({C}_{n}^{{a}_{1}}(x)\) of order n are defined by hypergeometric function as the follows
By substituting coefficients of A, B, C, D, and E from Table 1 in Eq. (1.e), we conclude the recursive relation of discrete orthogonal Charlier polynomials as follows:
with
where
2.4 Hahn polynomials
The nth Hahn polynomials \({h}_{n}^{\left(\alpha ,\beta \right)}(x)\) are defined by hypergeometric function as the follows.
By substituting coefficients of A, B, C, D, and E from Table 1 in Eq. (1.e), we conclude the recursive relation of discrete orthogonal Charlier polynomials as follows:
with
where
\(\text{where }\,\alpha ,\beta >0\text{.}\)
2.5 Meixner polynomials
Meixner polynomials \({m}_{n}^{\left(a,b\right)}(x)\) of order n are defined by hypergeometric function as the follows
From Eq. (1.e) and Table 1, we obtain the recursive relation of discrete orthogonal Meixner polynomials as follows:
with
where
\(\text{where }\,0<b<1\text{ and }a>0\text{.}\)
3 Discrete orthogonal moments (DOMs)
The discrete orthogonal moments are a set of moments calculated by discrete orthogonal polynomials. The set of discrete orthogonal onedimensional (1D) moments are defined as follows [11]:
where \(s\left(x\right)\) is a onedimensional signal of size \(1\times N,\) \({M}_{n}\) is a set of moment coefficients of the signal \(s\left(x\right)\) and \(p(x)\) is orthogonal polynomials of order n (Tchebichef \({{t}}_{{n}}\left({x};{ N}\right)\), Krawtchouk \({k}_{n}\left(x;P,N\right)\), Charlier \({c}_{n}\left(x\right)\), Hahn \({h}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)\) and Meixner \({m}_{n}^{\left(a,b\right)}\left(x\right)\)).
The reconstructed signal \(S(x)\) is calculated from the inverse transformation of the orthogonal moment as follows:
Using the following matrix form decreases the time and complexity of 1D orthogonal moment computations significantly:
\(\text{where }\,{ M}_{n}\) indicates orthogonal polynomials of order n \(,s\text{ denotes } 1\times N \text{signal vector.}\)
4 Ensuring the orthogonality property of discrete polynomials
In this section, we propose a procedure for ensuring the orthogonality property of discrete polynomials. According to the orthogonality property, polynomials matrix (\({p}_{n}\left(x\right)\)) satisfies the following relation:
where \({I}_{n}\) denotes the identity matrix.
To avoid numerical errors propagation and preserve the orthogonality property of DOPs, we present an efficient method for reorthonormalizing \({p}_{n}(x)\) matrix columns using QR decomposition methods. In these methods, a matrix \(A=[ {u}_{1}\),\({u}_{2},\dots ,{u}_{n1},{u}_{n}]\) of size \(n\times m\) factored as\(A=QR\), where Q is an \(n\times m\) matrix with orthogonal columns (\({Q}^{T}Q=I\)) and \(R\) is an \(m\times m\) upper triangular matrix [29]. In our situation, \(R\) matrix contains just recursive computation errors. The primary purpose of these ways is to generate the orthogonal \(Q(n\times m)\) matrix from \({p}_{n}(x)\) that contains roundoff errors. Many ways are used in \(QR\) decomposition, such as the Gram–Schmidt method, the Householder method, and the Given Rotations method [30].
4.1 Computation DOPs with modified Gram–Schmidt method (MGSM)
One of the most common algorithms for applying QR decomposition is the Gram–Schmidt (GS) method. It is a simple procedure for generating an orthogonal or orthonormal basis for any nonzero \({R}^{n}\) subspace [31]. Although the modified Gram–Schmidt method is always preferred because it avoids potentially costly cancellation errors, it is not as good numerically as the Givens or Householder approaches [29]. Algorithm 1 summarizes the proposed implementation of DOPs using MGSM.
4.2 Computation DOPs with Householder method (HM)
The main way to apply QR decomposition is with the Householder method. [29]. This approach is regarded to be more numerically stable than the Gram–Schmidt orthogonalization method for QR matrix decomposition. The proposed computation of DOPs with HM is illustrated in Algorithm 2.
4.3 Computation DOPs with Given Rotations method (GRM)
The Given Rotations method is an alternative to the Modified Gram–Schmidt method and Householder method for calculating QR decomposition [29]. The Proposed Algorithm for computing DOPs using GRM is reported in Algorithm 3.
5 Results
The experiments of this study are performed on a personal computer using Matlab Software (version R2014a) on Microsoft Windows 7, 32bit Edition, Intel Core i3 processor, and 4 GB RAM machine. Performance evaluation has been done by ECG signals from MITBIH arrhythmia dataset [32], which contain cardiac information from large numbers of patients. These recordings were obtained at a sampling frequency of 360 Hz (360 samples per second) with 11bit resolution. Our numerical simulations are presented in three sections: the first is to evaluate the performance of the proposed reorthonormalization methods in the quality of reconstruction signals. The second compares the three proposed reorthogonalization methods (Gram–Schmidt, Householder, and Given Rotations) in signal reconstruction quality. The third is a comparative study on the performance of Discrete Orthogonal Moments in signal reconstruction. The quality of the reconstructed signal is evaluated based on the following criteria:

Peak signal to noise ratio (\(\mathrm{PSNR}\)):
\(\mathrm{PSNR}\) is the highest possible signal power ratio to the corrupting noise power. It is presented as follows:
$$\mathrm{PSNR}=20\times {{log}}_{10}\frac{max\lefts(x)\right}{\sqrt{MSE}}$$ 
MeanSquare Error (MSE): the reconstruction error between the original and reconstructed signals.
$$MSE=\frac{1}{N}\sum_{x=0}^{N1} (s\left(x\right)S\left(x\right){)}^{2}$$where \(s\left(x\right)\) and \(S\left(x\right)\) are the original signal and reconstructed signal, respectively.
In this experiment, the parameters of polynomials are set as \(p=0.5\) for Krawtchouk, \({a}_{1}=140\) for Charlier,\(\alpha ,\beta =100\) for Hahn, and \(a=512 ,b= 0.5\) for Meixner. The signal size is N = 3600, and the order of the DOMs used is 200.
5.1 Reconstruction quality of DOMs computed using the proposed reorthonormalization method
We started by investigating the superiority of the proposed reorthogonalization methods with the discrete orthogonal moments in reconstruction quality signals. As shown in Table 2, we test the Tchebichef moments with and without the Householder method as one of the proposed methods for reconstructing the signals. The results obtained in Table 2 show that using the Householder method significantly improves the reconstruction quality of all records used. Tchebichef moments with Householder provide a high Peak signal to noise ratio (PSNR) with very low MeanSquare Error (MSE) values compared to Tchebichef moments. The average reconstruction errors PSNR and MSE of the proposed methods are 109.147 and 0.0564, respectively, as reported in Table 2. Figure 1 presents the reconstructed signal's reconstruction errors (PSNR, MSE) using Tchebichef moments with and without Householder method. It confirms the superiority of the proposed methods in reconstructed signals. Figure 2 shows the reconstructed “Rec. 107” signal by Tchebichef moments with and without the Householder method.
5.2 Comparison of reconstruction quality for the proposed reorthonormalization methods
In the previous section, we investigated the ability of the proposed procedure to maintain the orthogonality property of the discrete polynomials in the reconstruction of the signal. There are three methods in the proposed procedure mentioned, and they are the Gram–Schmidt method (MGSM), the Householder method (HM), and the Given Rotations method (GRM). This section will investigate which of the three methods is preferable in signal reconstruction quality and execution time. We have used Tchebichef moments with the three proposed methods (MGSM, HM, and GRM) to reconstruct the signals and summarized the results in Table 3. Figure 3 also illustrates the reconstruction errors (PSNR, MSE) of the three proposed methods using Tchebichef moments. The results displayed in Table 3 and Fig. 3 demonstrate outperforming HM on MGSM and GRM in PSNR and MSE on all records used. The reconstructed “Rec. 115” signal by using Tchebichef moments with Gram–Schmidt, Householder, and Given Rotations methods are shown in Fig. 4.
We compare the execution time of HM on MGSM and GRM to discover which of the three methods is best in terms of execution time, as shown in Fig. 5. The visual inspection from Fig. 5 indicates that HM is faster than MGSM and GRM.
5.3 Comparison of reconstruction quality for DOMs
This section determines which of the different types of moments is the best in the quality of the reconstructed signals. The compression performance of the discrete orthogonal moments (Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner) in signal reconstruction is presented in Table 4. In these experiments, the Householder method is used to preserve the orthogonality property methods in discrete orthogonal moments. Table 4 illustrates the resulting PSNR and MSE as reconstruction error metrics for 30 records from MITBIH arrhythmia dataset. The obtained results generally indicate that Tchebichef, Krawtchouk, and Charlier are superior to Hahn and Meixner in terms of PSNR and MSE. As for the three methods, Tchebichef, Krawtchouk, and Charlier, Tchebichef is relatively superior to Krawtchouk and Charlier. The average performance of the Tchebichef in terms of PSNR and MSE is 108.924 and 0.0580, respectively. Figures 6 and 7 depict the compression of the average PSNR and MSE of discrete orthogonal moments (Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner) in signal reconstruction. The reconstructed “Rec. 234” signal by using Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner moment with Householder method is depicted in Fig. 8.
To further validate the efficiency of DOMs, reconstruction is conducted using Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner with orders ranging between 50 and 200. Table 5 compares the quality of the signals reconstructed for the five moments in terms of PSNR and MSE in different moment orders. Figures 9 and 10 depict the curves of PSNR and MSE values corresponding to the reconstructed MITBIH Rec. 101 in different moments, respectively. As can be seen from the results in Table 5 and Fig. 9, The PSNR values improve appropriately with moment order increases, indicating an improvement in the reconstructed signal quality. The best quality of the reconstructed signal (lower MSE) is likewise obtained at the last moment order, as shown in Table 5 and Fig. 10.
6 Discussion
This paper contributes to the ongoing discussions about using DOMs in analyzing onedimensional biosignals. In addition, it also introduces an algorithm to overcome the propagation of numerical errors problem faces highorder computation of DOPs. The comparative experiments shown in the above tabular and graphical results assure the superiority of DOMs in reconstruction biosignals. It also demonstrates the advantages of the proposed reorthonormalization methods (Gram–Schmidt, Householder, and Given Rotations).
Generally, the increase in polynomial order, the increase in error propagation. Therefore, many researchers used QR decomposition methods to overcome these errors. In this work, we present Discrete Orthogonal Moments (DOMs) in biosignals analysis and reconstruction, which are gaining popularity in analyzing onedimensional signals due to their effectiveness in capturing digital information without redundancy. The works addressed by others faced propagation errors at high order polynomials which destroy the orthogonality property of these polynomials. While in our work, the problem of error propagation at high order polynomials is solved using QR decomposition methods. Hence, the OP reconstructs the biosignals efficiently. Moreover, to highlight the efficiency of the different forms of QR decomposition like the Gram–Schmidt method, the Householder method, and the Given Rotations method in different situations, we compare the different methods to each other to show the differences between them. Consequently, we introduce a road map to the interested researchers. Additionally, we compare five common types of moments (Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner) to estimate the best moment in analyzing and reconstructing biosignals which gives a clear guide to researchers in this area.
The results discussion obtained can be divided into three sections. Section one is the performance of the DOMs in biosignal reconstruction and the effect of using reorthonormalization methods in maintaining the orthogonality property at the highorder computation of DOPs. In general, the superiority of DOMs in the reconstruction of biosignals can be attributed to the following worthwhile factors:

DOMs are orthogonal moments with orthogonal basis functions. Each moment coefficient can capture the signal's distinct and unique components with no information redundancy.

According to the order value, orthogonal moments' basis functions can extract various distinct types of information from the signals.

Moments generated from discrete orthogonal polynomials are effective at compressing signals. This is because they have a higher efficiency of energy compression for common signals. If the discrete orthogonal moment is chosen correctly, the energy in the signal is concentrated on a small fraction of the moment coefficients; these coefficients are then stored and used to generate the reconstructed signal.

The ability of DOMs on local and global feature extraction.

Using recursive formulas to compute polynomial values by using lower polynomial orders instead of directly computing them causes computational efficiency in the computation of the moments.
Section two determines which of the three reorthogonalization methods (Gram–Schmidt, Householder, and Given Rotations) best preserves the orthogonality property. The comparative results indicate that the Householder method is the best in signal reconstruction in terms of reconstruction errors (PSNR, MSE) and execution time. The most likely explanation of the result has explained the fact that using Gram–Schmidt after computation of each nth Polynomial order minimizes the numerical error propagation considerably. Therefore, the Gram–Schmidt method is not stable when used in a reorthogonalization matrix with large size. To this end, the Householder method outperforms the Gram–Schmidt and Givens rotation methods in numerical stability in the \(QR\) decomposition of a matrix with large size. In addition to Householder method is faster compared to the Gram–Schmidt and Givens rotation methods. Because of this, the Householder method is better for realtime applications. The last section investigates which type of discrete orthogonal moments (Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner) provides betterreconstructed signals. Besides that, tracking the reconstructed signal quality for DOMs at various orders of moments. The obtained results demonstrate that all of the used moment types are stable since they enabled the reconstruction of the signals until the high moment order. It reflects the effectiveness and numerical stability of the orthogonal moment for largesize signal reconstruction. This numerical stability is ensured by reorthogonalization methods (Gram–Schmidt, Householder, and Given Rotations), especially the Householder method.
Despite the development of several reconstruction methods, substantial limitations must still be addressed. In the design of reconstruction methods, computational complexity and memory management play a crucial role, particularly in realtime applications like Remote Monitoring Systems. Reconstruction techniques increase the complexity of memory management. When the memory required to conduct the compression technique exceeds the available device memory, efficient reconstruction cannot be accomplished. Even though some reconstruction techniques achieve higher reconstruction quality, they do not manage memory effectively. Consequently, memory management and computational complexity in reconstruction techniques are interesting future research directions.
7 Conclusion
This article presents a method for biosignal reconstruction based on Discrete Orthogonal Moments (DOMs). It also proposes a modified version of DOPs using the QR decomposition methods such as the Gram–Schmidt, Householder, and Given Rotations methods. The purpose of the proposed modification is to preserve the orthogonality property in the computation of high polynomials order. Based on the results, it can be concluded that the research into DOPs has been very successful. DOMs of various types: Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner moments provide good results in reconstruction quality (PSNR, MSE). The comparative experiments demonstrate the superiority of the proposed modification of DOMs in reconstruction quality. This improvement in DOMs performance is due to QR decomposition methods to preserve the orthogonality property and then overcome the propagation of numerical errors. We also conclude that Tchebichef, Krawtchouk, and Charlier moments are better than Hahn and Meixner moments in reconstruction quality, and generally, Tchebichef has the best performance in signal reconstruction. The experiments of performance DOMs in reconstruction quality at a high order of moments are performed. We have noticed that the reconstruction quality improvement (PSNR highest, MSE lower) with moment orders increases. It means that the DOMs used in the proposed modification are efficient in largesize signal reconstruction. We could use the proposed method for largesize signal compression and classification in our future work and research direction. In addition, other applications will be used instead of biosignal, such as volumetric medical images, Galaxies images, and Retrieval systems for Biomedical Images. The proposed method's ability in reconstruction could be improved by using a new version of DOMs as fractional DOMs, Radial DOMs.
Availability of data and materials
The data that used this study belong to Physionet database; fPCG signals from the fPCG Dataset are used.
Abbreviations
 DOMs:

Discrete orthogonal moments
 MSE:

Mean square error
 PSNR:

Peak signal to noise ratio
 DOPs:

Discrete orthogonal polynomials
 MGSM:

Gram–Schmidt method
 HM:

Householder method
 GRM:

Given Rotations method
References
A. Daoui et al., New algorithm for largesized 2D and 3D image reconstruction using higherorder Hahn moments. Circuits Syst. Signal Process. 39(9), 4552–4577 (2020)
G. Hassan et al., An efficient retrieval system for biomedical images based on Radial Associated Laguerre Moments. IEEE Access 8, 175669–175687 (2020)
S.M.M. Rahman, T. Howlader, D. Hatzinakos, On the selection of 2D Krawtchouk moments for face recognition. Pattern Recognit. 54, 83–93 (2016)
R. Benouini et al., Fast and accurate computation of Racah moment invariants for image classification. Pattern Recognit. 91, 100–110 (2019)
M. Abd Elaziz, K.M. Hosny, I.M. Selim, Galaxies image classification using artificial bee colony based on orthogonal Gegenbauer moments. Soft. Comput. 23(19), 9573–9583 (2019)
M. Yamni et al., Fractional Charlier moments for image reconstruction and image watermarking. Signal Process. 171, 107509 (2020)
B. Xiao et al., Fractional discrete Tchebyshev moments and their applications in image encryption and watermarking. Inf. Sci. 516, 545–559 (2020)
F. Ernawan, N. Kabir, K.Z. Zamli, An efficient image compression technique using Tchebichef bit allocation. Optik 148, 106–119 (2017)
K.M. Hosny, A.M. Khalid, E.R. Mohamed, Efficient compression of volumetric medical images using Legendre moments and differential evolution. Soft. Comput. 24(1), 409–427 (2020)
M. Yamni et al., Novel Octonion moments for color stereo image analysis. Digit. Signal Process. 108, 102878 (2021)
H. Zhu et al., General form for obtaining discrete orthogonal moments. IET Image Process. 4(5), 335–352 (2010)
M. Yamni et al., Robust audio watermarking scheme based on fractional Charlier moment transform and dual tree complex wavelet transform. Expert Syst. Appl. 203, 117325 (2022)
M. Yamni et al., Efficient watermarking algorithm for digital audio/speech signal. Digit. Signal Process. 120, 103251 (2022)
K.M. Hosny, A.M. Khalid, E.R. Mohamed, Efficient compression of biosignals by using Tchebichef moments and Artificial Bee Colony. Biocybern. Biomed. Eng. 38(2), 385–398 (2018)
I.S. Fathi, M.A. Ahmed, M.A. Makhlouf, E.A. Osman, Compression techniques of biomedical signals in remote healthcare monitoring systems: a comparative study. Int. J. Hybrid Inf. Technol. 1(1), 33–50 (2021)
S.H. Abdulhussain et al., On computational aspects of Tchebichef polynomials for higher polynomial order. IEEE Access 5, 2470–2478 (2017)
C. Camachobello, J.S. Riveralopez, Some computational aspects of Tchebichef moments for higher orders. Pattern Recognit. Lett. 112, 332–339 (2018)
E.J. Huertas, A. Lastra, A. Sorialorente, watermarking applications of Krawtchouk–Sobolev type orthogonal moments. Electronics 11(3), 500 (2022)
S.H. Abdulhussain et al., Fast recursive computation of Krawtchouk polynomials. J. Math. Imaging Vis. 60(3), 285–303 (2018)
I.S. Fathi et al., An energyefficient compression algorithm of ECG signals in remote healthcare monitoring systems. IEEE Access 10, 39129–39144 (2022)
G. Hassan et al., New set of invariant quaternion Krawtchouk moments for color image representation and recognition. Int. J. Image. Graph 22(04), 2250037 (2021)
H. Karmouni et al., Fast and stable computation of the Charlier moments and their inverses using digital filters and image block representation. Circuits Syst. Signal Process. 37(9), 4015–4033 (2018)
A. Daoui et al., Stable computation of higher order Charlier moments for signal and image reconstruction. Inf. Sci. 521, 251–276 (2020)
M. Yamni et al., Robust zerowatermarking scheme based on novel quaternion radial fractional Charlier moments. Multimedia Tools Appl. 80(14), 21679–21708 (2021)
J.S. Riveralópez et al., Color image reconstruction by discrete orthogonal moment. J. Data Anal. Inf. Process. 5, 156–166 (2017)
A. Daoui et al., On computational aspects of highorder dual Hahn moments. Pattern Recognit. 127, 108596 (2022)
M. Sayyouri, A. Hmimid, H. Qjidaa, A fast computation of novel set of Meixner invariant moments for image analysis. Circuits Syst. Signal Process. 34(3), 875–900 (2015)
T. Jahid et al., Image analysis by Meixner moments and a digital filter. Multimedia Tools Appl. 77(15), 19811–19831 (2018)
W. Ford, Numerical Linear Algebra with Applications: Using MATLAB (Academic Press, Cambridge, 2014)
A. Daoui, M. Sayyouri, H. Qjidaa, Efficient computation of highorder Meixner moments for largesize signals and images analysis. Multimedia Tools Appl. 80(2), 1641–1670 (2021)
L.N. Trefethen, D. Bau III., Numerical Linear Algebra (Siam, Philadelphia, 1977)
Acknowledgements
Not applicable.
Funding
Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).
Author information
Authors and Affiliations
Contributions
I.S. conceived of the problem, gathered data, conducted the analysis, and produced the first draft of the paper. M.A. reorganized the paper and provided the related work. M.M. evaluated the arguments critically, reviewed the versions, and provided helpful comments. All authors provided general feedback and helped to revisions. All authors read and approved the final manuscript.
Authors’ Information
Islam S. Fathi received the B.Sc. and M.Sc. degrees in math and computer sciences from the Faculty of Science, Zagazig University, Egypt, in 2013 and 2019. He is currently working as a Teaching Assistant with the Department of Information Systems, Al Alson Academy, Cairo, Egypt. His research interests include signal processing, metaheuristic optimization, and bioinformatics.
Mohamed Ali Ahmed is currently an assistant professor with the Mathematics Department in the Faculty of Science Suez Canal University. His research interests include Computer Networks, Performance Evaluation, Queuing Systems, metaheuristic optimization and bioinformatics.
Mohamed Abd Allah Makhlouf is currently an associate professor in the Faculty of Computer Science and informatics at Suez Canal University and Faculty of Computer Science, Nahda University. He received his first degree in Computer Science and Operation Research, Faculty of Science, Master’s degree in Expert Systems, Faculty of Science Cairo University. He received his Ph.D. degree in computer science from the Faculty of Science, Zagazig University. He got PostDoctoral studies in Computer Science from Granada University, Spain, in 2016. His research interests include Machine learning, data mining, Intelligent Bioinformatics, metaheuristic optimization, Decision support systems and predictive models, Bioinformatics, metaheuristic optimization, Decision support systems and predictive models.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
We declare that the studies in this work do not involve any human participants, human data, human tissue and animal.
Consent for publication
We declare that the manuscript does not contain any individual person’s data in any form (including individual details, images or videos).
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Fathi, I.S., Ahmed, M.A. & Makhlouf, M.A. An efficient computation of discrete orthogonal moments for biosignals reconstruction. EURASIP J. Adv. Signal Process. 2022, 104 (2022). https://doi.org/10.1186/s13634022009384
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13634022009384