1 Introduction

With the explosive growth in computer technology and signal analysis tools, computer-aided analysis of bio-signals has become a common part of clinical. Bio-signal reconstruction is one of the important applications in bio-signal 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 one-dimensional 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 high-order 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 re-orthonormalization 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 high-order DOPs orthogonality property effectively. Thus, using the DOMs to analyze large-size 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 bio-signals 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 bio-signals.

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

$$\sigma (x)\Delta \nabla {p}_{n}(x)+\tau (x)\Delta {p}_{n}(x)+{\lambda }_{n}{p}_{n}(x)=0$$
(1.a)

where \(\Delta {p}_{n}\left(x\right)={p}_{n}\left(x+1\right)-{p}_{n}\left(x\right) and \nabla {p}_{n}(x)={p}_{n}(x)-{p}_{n}(x-1)\) indicates backward finite-difference 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

$$\sum\limits_{{x = 0}}^{s} {p_{n} } \left( x \right)p_{m} \left( x \right)w\left( x \right) = d_{n}^{2} \cdot \delta _{{mn}} ,\quad 0 \le m,n \le s$$
(1.b)

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

$${\widetilde{p}}_{n}(x)={p}_{n}(x)\sqrt{\frac{w(x)}{{d}_{n}^{2}}}, \quad n={0,1},\dots ,s$$
(1.c)

Therefore, the orthogonal property of normalized orthogonal polynomials in (1b) can be rewritten as

$$\sum_{x=0}^{s} {\widetilde{p}}_{m}(x){\widetilde{p}}_{n}(x)={\delta }_{mn}, \quad 0\le m,n\le s$$
(1.d)

A general formula for getting the normalized discrete orthogonal polynomials \({\widetilde{p}}_{n}(x)\) of order \(n\) is defined three-term recursive relation as follows [11]:

$${\widetilde{p}}_{n}\left(x\right)=\left(\frac{B*D}{A}\right){\widetilde{p}}_{n-1}\left(x\right)+\left(\frac{C*E}{A}\right){\widetilde{p}}_{n-2}\left(x\right)$$
(1.e)

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.\({\widetilde{p}}_{n-1}(x)\) and \({\widetilde{p}}_{n-2}(x)\) are the zero-order and first-order polynomials, respectively.

Table 1 Values of A, B, C, D, and E for each of the polynomial set 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 \({m}_{n}^{\left(a,b\right)}\left(x\right)\)
Full size table

2.1 Tchebichef polynomials

The nth Tchebichef polynomials \(t_{n} \left( x \right)\) are defined by hypergeometric function as the follows

$$t_{n} \left( x \right) = (1 - N)_{n3} F_{2} \left( { - n, - x,1 + n;1,1 - N;1} \right),\; n,x = 0,1,2, \ldots ,N - 1.$$
(2)

From Eq. (1.e) and Table 1, we obtain the recursive relation of discrete orthogonal Tchebichef polynomials as follows:

$${t}_{n}(x)=\left({\beta }_{1}x+{\beta }_{2}\right){t}_{n-1}(x)+{\beta }_{3}{t}_{n-2}(x)$$
(3)

with

$$\begin{aligned} t_{0} \left( x \right) & = \frac{1}{{\sqrt N }}, \\ t_{1} \left( x \right) & = \left( {2x + 1 - N} \right)\sqrt {\frac{3}{{N\left( {N^{2} - 1} \right)}}} . \\ \end{aligned}$$
(4)

where

$$\begin{aligned} \beta _{1} & = \frac{2}{n}\sqrt {\frac{{4n^{2} - 1}}{{N^{2} - n^{2} }}} \\ \beta _{2} & = \frac{{1 - N}}{n}\sqrt {\frac{{4n^{{2 - 1}} }}{{N^{2} - n^{2} }},} \\ \beta _{3} & = \frac{{1 - n}}{n}\sqrt {\frac{{2n + 1}}{{2n - 3}}} \sqrt {\frac{{N^{2} - (n - 1)^{2} }}{{N^{2} - n^{2} }}.} \\ \end{aligned}$$
(5)

2.2 Krawtchouk polynomials

Krawtchouk polynomials \(k_{n} \left( {x,p} \right)\) of order n are defined by hypergeometric function as the follows

$$k_{n} \left( {x,p} \right) =_{ 2} F_{1} \left( { - n, - x; - N;\frac{1}{p}} \right) ,\quad n,x = 0,1,2, \ldots ,N - 1.$$
(6)

The recursive relation of discrete orthogonal Krawtchouk polynomials can be calculated using Eq. (1.e) and Table 1 as:

$$k_{n} \left( {x,p} \right) = \left( {x + \beta_{1} } \right)\beta_{2} k_{n - 1} \left( {x,p} \right) - \beta_{3} k_{n - 2} \left( {x,p} \right)$$
(7)

with

$$\begin{aligned} k_{0} \left( {x,p} \right) & = \sqrt {\frac{{N!p^{x} (1 - p)^{{N - x}} }}{{x!\left( {N - x} \right)!}}} , \\ k_{1} \left( {x,p} \right) & = ( - p\left( {N - x} \right) + x(1 - p)) \times \sqrt {\frac{{(N - 1)!p^{{x - 1}} (1 - p)^{{N - x - 1}} }}{{x!\left( {N - x} \right)!}}} \\ \end{aligned}$$
(8)

where

$$\begin{aligned} \beta _{1} & = \left( {1 - n - p\,\left( {N - 2n + 2} \right)} \right), \\ \beta _{2} & = \sqrt {\frac{1}{{p(1 - p)(N - n + 1)n}}} \\ \beta _{3} & = \sqrt {\frac{{\left( {N - n + 2} \right)\left( {n - 1} \right)}}{{\left( {N - n + 1} \right)n}}} ,\quad 0 < p < 1. \\ \end{aligned}$$
(9)

2.3 Charlier polynomials

Charlier polynomials \({C}_{n}^{{a}_{1}}(x)\) of order n are defined by hypergeometric function as the follows

$$c_{n}^{{a_{1} }} \left( x \right) =_{2} F_{0} \left( { - n, - x;; - 1/a_{1} } \right),\quad n,x = 0,1,2, \ldots ,N - 1 \;{\text{and}}\;a_{1} > 0$$
(10)

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:

$$C_{n}^{{a_{1} }} \left( x \right) = \left( {\beta_{1} - x} \right) \beta_{2} C_{n - 1}^{{a_{1} }} \left( x \right) - \beta_{3} C_{n - 2}^{{a_{1} }} \left( x \right)$$
(11)

with

$$\begin{aligned} c_{0}^{{a_{1} }} \left( x \right) & = \sqrt {\frac{{e^{{ - a_{1} }} a_{1}^{x} }}{{x!}},} \\ C_{1}^{{a_{1} }} (x) & = \frac{{a_{1} - x}}{{a_{1} }}\sqrt {\frac{{e^{{ - a_{1} }} a_{1}^{{x + 1}} }}{{x!}}} \\ \end{aligned}$$
(12)

where

$$\begin{aligned} \beta _{1} & = \left( {a_{1} + n - 1} \right), \\ \beta _{2} & = \sqrt {\frac{1}{{na_{1} }}} , \\ \beta _{3} & = \sqrt {\frac{{n - 1}}{n}} . \\ \end{aligned}$$
(13)

2.4 Hahn polynomials

The nth Hahn polynomials \({h}_{n}^{\left(\alpha ,\beta \right)}(x)\) are defined by hypergeometric function as the follows.

$$h_{n}^{{\left( {\alpha ,\beta } \right)}} \left( x \right) = \frac{{( - 1)^{n} (\beta + 1)_{n} (N - n)_{n} }}{{n!}} \times _{3} F_{2} \left( { - n, - x,n + 1 + \alpha + \beta ;\beta + {\text{1}},{\text{1}} - N;1} \right),\quad n,x = {\text{0}},{\text{1}},2, \ldots ,N - 1.$$
(14)

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:

$$h_{n}^{{\left( {\alpha ,\beta } \right)}} \left( x \right) = \left( {x - \beta_{1} } \right)\beta_{2} h_{n - 1}^{{\left( {\alpha ,\beta } \right)}} \left( x \right) - \beta_{3} h_{n - 2}^{{\left( {\alpha ,\beta } \right)}} \left( x \right)$$
(15)

with

$$\begin{aligned} h_{0}^{{\left( {\alpha ,\beta } \right)}} \left( x \right) & = \sqrt {\frac{{(\alpha + 1)_{\beta } (\alpha + \beta + 1)}}{{(N - \alpha )_{{\beta + 1}} }}} \\ h_{1}^{{\left( {\alpha ,\beta } \right)}} \left( x \right) & = \left( {\alpha + \beta + 2} \right)x - \left( {\beta + 1} \right)\left( {N - 1} \right) \times \sqrt {\frac{{\alpha + \beta + 3}}{{\left( {\alpha + 1} \right)\left( {\beta + 1} \right)\left( {n - 1} \right)\left( {N + 1 + \beta + 1} \right)}}} . \\ \end{aligned}$$
(16)

where

$$\begin{aligned} \beta _{1} & = \frac{{\alpha - \beta + 2N - 2}}{4} + \frac{{\left( {\beta ^{2} - \alpha ^{2} } \right)(\alpha + \beta + 2N)}}{{4(\alpha + \beta + 2n - 2)(\alpha + \beta + 2n)}}, \\ \beta _{2} & = \sqrt {\frac{{\left( {\alpha + \beta + 2n} \right)^{4} - \left( {\alpha + \beta + 2n} \right)^{2} }}{{n(N - n)\left( {\alpha + n} \right)\left( {\beta + n} \right)\left( {\alpha + \beta + n + N} \right)(\alpha + \beta + n)}}} , \\ \beta _{3} & = \frac{{\alpha + \beta + 2n}}{{\alpha + \beta + 2n - 2}} \times \sqrt {\frac{{(n - 1)(\alpha + n - 1)(\beta + n - 1)(N - n + 1)}}{{n(\alpha + n)(\beta + n)(N - n)(\alpha + \beta + n)}}} \quad \times \sqrt {\frac{{(\alpha + \beta + n - 1)(\alpha + \beta + 2n + 1)(\alpha + \beta + N + n - 1)}}{{(\alpha + \beta + 2n - 3)(\alpha + \beta + n + N)}}} . \\ \end{aligned}$$
(17)

\(\text{where }\,\alpha ,\beta >0\text{.}\)

2.5 Meixner polynomials

Meixner polynomials \({m}_{n}^{\left(a,b\right)}(x)\) of order n are defined by hypergeometric function as the follows

$$m_{n}^{{\left( {a,b} \right)}} \left( x \right) = (a)_{n2} F_{1} \left( { - n, - x;a;1 - 1/b} \right),\;n,x = 0,1,2, \ldots ,N - 1.$$
(18)

From Eq. (1.e) and Table 1, we obtain the recursive relation of discrete orthogonal Meixner polynomials as follows:

$$m_{n}^{{\left( {a,b} \right)}} \left( x \right) = \left( {x\beta_{1} + \beta_{2} } \right)m_{n - 1}^{{\left( {a,b} \right)}} \left( x \right) - \beta_{3} m_{n - 2}^{{\left( {a,b} \right)}} \left( x \right)$$
(19)

with

$$\begin{aligned} m_{0}^{{\left( {a,b} \right)}} \left( x \right) & = \sqrt {\frac{{b^{x} \left( {a + x - 1} \right)!}}{{x!\left( {a - 1} \right)!}}(1 - b)^{a} } , \\ m_{1}^{{\left( {a,b} \right)}} \left( x \right) & = \left( {a + x - \frac{x}{b}} \right) \times \sqrt {\frac{{b^{x} \left( {a + x - 1} \right)!}}{{x!\left( {a - 1} \right)!}}\frac{{b(1 - b)^{a} }}{a}} . \\ \end{aligned}$$
(20)

where

$$\begin{aligned} \beta _{1} & = \left( {b - 1} \right)\sqrt {\frac{1}{{n\left( {a + n - 1} \right)b}}} , \\ \beta _{2} & = \left( {n - 1 + bn - b + ab} \right)\sqrt {\frac{1}{{n\left( {a + n - 1} \right)b}}} , \\ \beta _{3} & = \sqrt {\frac{{\left( {n - 1} \right)\left( {n - 2 + a} \right)}}{{n\left( {a + n - 1} \right)}}} , \\ \end{aligned}$$
(21)

\(\text{where }\,0<b<1\text{ and }a>0\text{.}\)

3 Discrete orthogonal moments (DOMs)

The discrete orthogonal moments are a set of moments calculated by discrete orthogonal polynomials. The set of discrete orthogonal one-dimensional (1D) moments are defined as follows [11]:

$${M}_{n}=\sum_{x=0}^{N-1} {p}_{n}\left(x\right)s\left(x\right), \quad n={0,1}\dots ,N-1.$$
(22)

where \(s\left(x\right)\) is a one-dimensional 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:

$$S\left( x \right) = \mathop \sum \limits_{n = 0}^{N - 1} { }M_{n} p_{n} \left( x \right), \quad x = 0,1,2, \ldots ,N - 1$$
(23)

Using the following matrix form decreases the time and complexity of 1D orthogonal moment computations significantly:

$${M}_{n}=\left[\begin{array}{cccc}{p}_{0}(0)& {p}_{0}(1)& \dots & {p}_{0}(N-1)\\ {p}_{1}(0)& {p}_{1}(1)& \dots & {p}_{1}(N-1)\\ \vdots & \vdots & \vdots & \vdots \\ {p}_{n}(0)& {p}_{n}(1)& \dots & {p}_{n}\left(N-1\right)\end{array}\right]\times \left[\begin{array}{c}s(0)\\ s(1)\\ \vdots \\ s(N-1)\end{array}\right]$$
(24)

\(\text{where }\,{ M}_{n}\) 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:

$$p_{n} \left( x \right)^{T} p_{n} \left( x \right) = I_{n}$$
(25)

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 re-orthonormalizing \({p}_{n}(x)\) matrix columns using QR decomposition methods. In these methods, a matrix \(A=[ {u}_{1}\),\({u}_{2},\dots ,{u}_{n-1},{u}_{n}]\) 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 round-off 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.

figure a

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.

figure b

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.

figure c

5 Results

The experiments of this study are performed on a personal computer using Matlab Software (version R2014a) on Microsoft Windows 7, 32-bit Edition, Intel Core i3 processor, and 4 GB RAM machine. Performance evaluation has been done by ECG signals from MIT-BIH 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 11-bit resolution. Our numerical simulations are presented in three sections: the first is to evaluate the performance of the proposed re-orthonormalization methods in the quality of reconstruction signals. The second compares the three proposed re-orthogonalization 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:

    $$\mathrm{PSNR}=20\times {{log}}_{10}\frac{max\left|s(x)\right|}{\sqrt{MSE}}$$

  • Mean-Square Error (MSE): the reconstruction error between the original and reconstructed signals.

    $$MSE=\frac{1}{N}\sum_{x=0}^{N-1} (s\left(x\right)-S\left(x\right){)}^{2}$$

    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 re-orthonormalization method

We started by investigating the superiority of the proposed re-orthogonalization 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 Mean-Square 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.

Table 2 Comparison of reconstruction errors (PSNR and MSE) using Tchebichef moments with and without Householder method
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Fig. 1
figure 1

The average values a PSNR and b MSE of the reconstructed signals using Tchebichef moments with and without Householder method

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Fig. 2
figure 2

Set of reconstructed “Rec. 107” signal using Tchebichef moments with and without the Householder method

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5.2 Comparison of reconstruction quality for the proposed re-orthonormalization 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.

Table 3 Comparison of reconstruction errors (PSNR and MSE) for re-orthogonalization methods (MGSM, HM, and GRM) using Tchebichef moments
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Fig. 3
figure 3

The average values a PSNR and b MSE of the reconstructed signals using Tchebichef moments with Gram–Schmidt, Householder, and Given Rotations methods

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Fig. 4
figure 4

Set of reconstructed “Rec. 115” signal using Tchebichef moments with Gram–Schmidt, Householder, and Given Rotations methods

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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.

Fig. 5
figure 5

Average execution time using Tchebichef moments with re-orthonormalization methods (Gram–Schmidt, Householder, and Given Rotations)

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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 MIT-BIH 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.

Table 4 Reconstructions errors (PSNR and MSE) of biosignal by Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner moments with the Householder method
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Fig. 6
figure 6

The average PSNR of the reconstructed signals using Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner moment with Householder method

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Fig. 7
figure 7

The average MSE of the reconstructed signals using Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner moment with Householder method

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Fig. 8
figure 8

Set of reconstructed “Rec. 234” signal using Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner moment with Householder method

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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 MIT-BIH 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.

Table 5 Performance of Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner moments with Householder method at different orders
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Fig. 9
figure 9

PSNR of the reconstructed signal "MIT-BIH Rec. 101" using Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner moment with Householder methods

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Fig. 10
figure 10

MSE of the reconstructed signal "MIT-BIH Rec. 101" using Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner moment with Householder methods

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6 Discussion

This paper contributes to the ongoing discussions about using DOMs in analyzing one-dimensional bio-signals. In addition, it also introduces an algorithm to overcome the propagation of numerical errors problem faces high-order 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 re-orthonormalization 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 bio-signals analysis and reconstruction, which are gaining popularity in analyzing one-dimensional 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 bio-signals 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 bio-signals 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 re-orthonormalization methods in maintaining the orthogonality property at the high-order computation of DOPs. In general, the superiority of DOMs in the reconstruction of bio-signals 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 re-orthogonalization 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 re-orthogonalization 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 real-time applications. The last section investigates which type of discrete orthogonal moments (Tchebichef, Krawtchouk, Charlier, Hahn, and Meixner) provides better-reconstructed 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 large-size signal reconstruction. This numerical stability is ensured by re-orthogonalization 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 real-time 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 bio-signal 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 large-size signal reconstruction. We could use the proposed method for large-size signal compression and classification in our future work and research direction. In addition, other applications will be used instead of bio-signal, 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.