Efficient Analysis of Large-Size Bio-Signals Based on Orthogonal Generalized Laguerre Moments of Fractional Orders and Schwarz–Rutishauser Algorithm

Abstract

Orthogonal generalized Laguerre moments of fractional orders (FrGLMs) are signal and image descriptors. The utilization of the FrGLMs in the analysis of big-size signals encounters three challenges. First, calculating the high-order moments is a time-consuming process. Second, accumulating numerical errors leads to numerical instability and degrades the reconstructed signals’ quality. Third, the QR decomposition technique is needed to preserve the orthogonality of the higher-order moments. In this paper, the authors derived a new recurrence formula for calculating the FrGLMs, significantly reducing the computational CPU times. We used the Schwarz–Rutishauser algorithm as an alternative to the QR decomposition technique. The proposed method for computing FrGLMs for big-size signals is accurate, simple, and fast. The proposed algorithm has been tested using the MIT-BIH arrhythmia benchmark dataset. The results show the proposed method’s superiority over existing methods in terms of processing time and reconstruction capability. Concerning the reconstructed capability, it has achieved superiority with average values of 25.3233 and 15.6507 with the two metrics PSNR and MSE, respectively. Concerning the elapsed reconstruction time, it also achieved high superiority with an efficiency gain of 0.8. The proposed method is suitable for utilization in the Internet of Healthcare Things.

1. Introduction

The study of moments as a mathematical tool for describing 1-D, 2-D, and 3-D objects has become the focus of attention of many researchers since the middle of the previous century. Mathematically, the moments represent the projection of the object function onto certain functions called basis functions. This projection results in some statistical quantities representing significant features of the object [1]. One important type of moment is the Orthogonal Moments (OMs) set, where the basis functions are orthogonal. According to the object’s coordinate space type, OMs exist in Continuous Orthogonal Moments (COMs) and Discrete Orthogonal Moments (DOMs) moments.

The first set of DOMs was discrete Tchebichef moments, which were introduced in 2001 by Mukundan et al. [2]. Later, many sets of DOMs were introduced, such as Krawtchouk moments, Hahn moments, Dual Hahn moments, Charlier moments, and Racah moments [3,4,5,6,7]. The different sets of DOMs are widely used in many signal and image processing applications, such as image reconstruction [8], image retrieval [9,10], image compression [11,12], image steganography [13], and image watermarking [14]. DOMs reveal high-efficiency concerns representing different bio-medical signals: ECG, EEG, and EMG. Hosny et al. [15] suggested an efficient Algorithm for the compression of different types of bio-medical signals based on discrete Tchebichef moments and the Artificial Bee Colony (ABC). Fathi et al. [16,17,18] introduced different algorithms based on different types of DOMs for the efficient energy compression of bio-medical signals: ECG and FPCG. Such algorithms reveal high superiority concerning remote healthcare monitoring systems. In Cardiovascular Diseases (CVDs), continuous patient monitoring is very important for better diagnosis; healthcare systems that use ECG signals need long-term monitoring. As such, the size of the ECG signal collected by sensors to use in heart diagnosis is large. Concerning large-size bio-signals, Daoui et al. [19,20] suggested different approaches for reconstructing and compressing such signals. The presented approaches are based on Tchebichef moments and Meixner moments, respectively.

The successful utilization of DOMs with 2-D objects (i.e., images) motivates researchers to use these DOMs with 1-D objects (i.e., signals). Many studies used the DOMs as a feature descriptor for signals such as signal reconstruction and compression [15,16,17,21] and signal watermarking [22,23]. Other sets of moments are called complex moments. It is used basically in solving eigenvalue problems and has been used in image reconstruction applications. In the image reconstruction application, a formulation is shown to obtain an estimated original image from the degraded image moments and the blur parameter [24,25]. Complex Moments (CMs) were initially proposed as a straightforward and uncomplicated approach to reconstruction images. Still, the kernel functions of CMs lack orthogonality, which complicates reconstructing an image from its CMs.

Previous studies show that orthogonal moments of fractional orders outperformed their corresponding integer orthogonal moments of integer orders [26,27]. Orthogonal Moments of fractional orders are used to analyze medical signals, especially those characterized by their large size [19,20,28,29]. Despite the superiority of fractional versions of DOMs in many applications, some common problems occur when deriving the polynomial values: first, numerical instability results from the large increase (i.e., fluctuation) of the basis function values that occurs while increasing the moments’ order. Secondly, a high computational time is required when calculating the polynomial values with the classical form of the polynomial equation. Thirdly, there is the propagation and accumulation of numerical errors, especially at higher orders, with large signals. These drawbacks cause the loss of the polynomial’s orthogonality.

To overcome the previous problems, researchers used the weighted form of the fractional Chebychev polynomials, resulting in more numerical stability during calculations of polynomial values. Also, to overcome the highly time-consuming problem, they used a recurrence formula based on a normalization factor. The third problem vanished using the most common QR decomposition algorithms: the Gram–Schmidt Method (GSM), Householder Method (HM), and Given Rotations Method (GRM) [18].

In the current study, we introduced an efficient algorithm for large-size bio-signal signal reconstruction and analysis. The importance of this proposed algorithm lies in its high ability to compress the electrocardiogram (ECG) biomedical signals and transmit them over long distances in a short time. Also, due to this compression, the IoHT can transmit the ECG signals through a small bandwidth; hence, there is low energy consumption. The proposed algorithm depends on the set fractional-order generalized Laguerre moments (FrGLMs) and the Schwarz–Rutishauser approach. Through the current study, we derived three three-term second-order recurrence formulas for the normalized form of the FrGLMs. An important term when computing the normalized form of FrGLMs is the squared norm $h_k^{\left(\propto ,\lambda \right)}$. By using the simple recursive formula of the gamma function, we deduced a recursive formula for evaluating the squared norm $h_k^{\left(\propto ,\lambda \right)}$. The derived three-term recurrence formula with the recursive formula of the squared norm $h_k^{\left(\propto ,\lambda \right)}$ has great benefits in saving computation time.

The utilization of the Schwarz–Rutishauser algorithm [30,31] is another contribution of this study, where the Schwarz–Rutishauser algorithm is used to preserve the orthogonality property when analyzing the large-sized bio-signals using high-order moments. The researchers are accustomed to using classical QR decomposition methods: the Gram–Schmidt Method (GSM), Householder Method (HM), and Given Rotations Method (GRM), but the empirical experiments reveal that the use of these leads to a significant increase in signal processing time, which leads to a lack of efficiency in both IoHT devices and real-time applications with the use of Schwarz–Rutishauser, which is an adaptation of the classical Gram–Schmidt. The complexity of the Schwarz–Rutishauser algorithm is (mn²), which is (n)-times less than the complexity of the classical QR decomposition methods: GSM, HM, and GRM, which motivated us to use it in the proposed algorithm, which obtains better results concerning complexity and numerical stability. Uwe et al. [32] proved the efficiency of the Schwarz–Rutishauser algorithm concerning real-time fetal ECG monitoring systems, and it helped in raising the efficiency of the proposed system in terms of improving the system performance and energy consumption rate. To confirm the robustness of the FrGLMs with Schwarz–Rutishauser, samples of ECG bio-medical signal are used. These samples have been obtained from a benchmark dataset called MIT-BIH arrhythmia. An empirical experiment was carried out with the ECG bio-medical signals and revealed high superiority.

The contributions of this study are:

• A three-term second-order recurrence formula for the normalized form of FrGLMs has been derived.
• A recursive formula for the squared norm $h_k^{\left(\propto ,\lambda \right)}$ has been derived.
• A novel QR-decomposition approach called Schwarz–Rutishauser gives more numerical stability and less processing time than the classical approaches.

The rest of this paper is: Section 2 briefly reviews the FrGLMs. Section 3 presents the proposed computations of the fractional generalized Laguerre polynomial (FrGLP). In Section 4, we introduce the proposed Schwarz–Rutishauser. The proposed algorithm of the FrGLMs based on the three-term second-order recurrence formula of the normalized FrGLP and the Schwarz–Rutishauser is discussed in detail in Section 5. The results are figured out and discussed in Section 6 and Section 7, Respectively. The conclusions of this study are presented in Section 8.

2. Fractional-Order Generalized Laguerre Orthogonal Moments

The one-dimensional (1D) fractional-order generalized Laguerre orthogonal moments. ${\widehat{{\mathrm{F}}_{\mathrm{r}}\mathrm{L}\mathrm{M}}}_{\mathrm{i}}$ with the order, $\mathrm{i}$ can be defined by the fractional-order generalized Laguerre polynomials ${\widetilde{{\mathrm{F}}_{\mathrm{r}}\mathrm{L}}}_{\mathrm{i}}^{\left(\propto ,\mathsf{\lambda }\right)}$ as follows:

$${\widehat{{\mathrm{F}}_{\mathrm{r}}\mathrm{L}\mathrm{M}}}_i=\sum _{x=0}^{N-1}{\widetilde{{\mathrm{F}}_{\mathrm{r}}\mathrm{L}}}_{\mathrm{i}}^{\left(\propto ,\mathsf{\lambda }\right)}s\left(x\right), i=\mathrm{0,1},2,\dots ,N-1.$$

(1)

where $s\left(x\right)$ is a $(1\times N$) signal.

From the inverse transformation of fractional Laguerre moments, the original signal S(x) is reconstructed as follows:

$$S\left(x\right)={\sum }_{i=0}^{i_{max}}{\widehat{{\mathrm{F}}_{\mathrm{r}}\mathrm{L}\mathrm{M}}}_i{\widetilde{{\mathrm{F}}_{\mathrm{r}}\mathrm{L}}}_{\mathrm{i}}^{\left(\propto ,\mathsf{\lambda }\right)}, x=\mathrm{0,1},2,\dots ,N-1.$$

(2)

where ${\widehat{F_r\mathrm{L}\mathrm{M}}}_{\mathrm{i}} and {\widetilde{F_{r}L}}_{\mathrm{i}}^{\left(\propto ,\lambda \right)}$ are the fractional-order Laguerre moments and polynomials, respectively.

3. Proposed Computation of Fractional Laguerre Orthogonal Polynomials

The fractional-order generalized Laguerre polynomials (FGLPs) can be generated rapidly according to the following three-term recurrence formula [31]:

$$F_r{L}_{i+1}^{\left(\propto ,\lambda \right)}\left(x\right)=\frac{1}{i+1}\left[\left(2i+\propto +1-x^{\lambda }\right)F_r{L}_i^{\left(\propto ,\lambda \right)}\left(x\right)-\left(i+\propto \right)F_r{L}_{i-1}^{\left(\propto ,\lambda \right)}\left(x\right)\right], i=1,2,\dots ,N$$

(3)

with the two initial conditions:

$$F_r{L}_0^{(\propto ,\lambda )}\left(x\right)=1 \mathrm{and} F_r{L}_1^{(\propto ,\lambda )}\left(x\right)=1+\propto -x^{\lambda }.$$

Also, the analytical form of $F_r{L}_i^{(\propto ,\lambda )}\left(x\right)$ of the fractional degree $i\lambda$ can be obtained as:

$$F_r{L}_i^{(\propto ,\lambda )}\left(x\right)=\sum _{k=0}^i{(-1)}^k\frac{\Gamma (i+\propto +1)}{\Gamma \left(k+\propto +1\right)\left(i-k\right)!k!}{x}^{\lambda k}, i=0,1,\dots$$

The orthogonality property for FGLPs is also satisfied according to the following:

$${\int }_0^{\mathrm{\infty }}{F}_r{L}_j^{\left(\propto ,\lambda \right)}\left(x\right)F_r{L}_k^{\left(\propto ,\lambda \right)}\left(x\right)w^{\left(\propto ,\lambda \right)}\left(x\right)dx=h_k,$$

where $w^{\left(\propto ,\lambda \right)}\left(x\right)$ is the weight function, defined as follows:

$$w^{\left(\propto ,\lambda \right)}\left(x\right)=\lambda x^{\left(\propto +1\right)\lambda -1}{e}^{-x^{\lambda }},$$

Also, the squared norm $h_k$ is defined as follows:

$$h_k^{\left(\propto ,\lambda \right)}=\left\{\begin{array}{c}\frac{\Gamma (i+\propto +1)}{k!} j=k,\\ 0, j \ne k.\end{array}\right.$$

(4)

The normalized form of fractional-order generalized Laguerre polynomials (FGLPs) can obtained in terms of the weight and squared norm functions as follows:

$${\widetilde{F_{r}L}}_i^{(\propto ,\lambda )}\left(x\right)=\sqrt{\frac{w^{\left(\propto ,\lambda \right)}\left(x\right)}{h_k}}{F}_r{L}_i^{(\propto ,\lambda )}\left(x\right),$$

(5)

Figure 1a,b illustrate the values of polynomials for normalized fractional Laguerre and the values of polynomials for fractional Laguerre for N = 100 and n = 0, 1, 2, 3, 4, and 5.

α = 1, λ = 1.1.

The three-term recurrence relation of the normalized FGLPs (${\widetilde{F_{r}L}}_i^{(\propto ,\lambda )}\left(x\right)$) can be deduced as follows.

At first, the two initial conditions can be obtained as the follows.

From Equations (3) and (5), one can deduce the following:

$${\widetilde{F_{r}L}}_0^{(\propto ,\lambda )}\left(x\right)=\sqrt{\frac{w^{\left(\propto ,\lambda \right)}\left(x\right)}{h_0}}{F}_r{L}_0^{(\propto ,\lambda )}\left(x\right)=\sqrt{\frac{w^{\left(\propto ,\lambda \right)}\left(x\right)0!}{\Gamma (0+\propto +1)}}{F}_r{L}_0^{(\propto ,\lambda )}\left(x\right)=\sqrt{\frac{w^{\left(\propto ,\lambda \right)}\left(x\right)}{\Gamma (\propto +1)}},$$

(6)

and

$${\widetilde{F_{r}L}}_1^{(\propto ,\lambda )}\left(x\right)=\sqrt{\frac{w^{\left(\propto ,\lambda \right)}\left(x\right)}{h_1}}{F}_r{L}_1^{(\propto ,\lambda )}\left(x\right)=\sqrt{\frac{w^{\left(\propto ,\lambda \right)}\left(x\right)1!}{\Gamma (1+\propto +1)}}{F}_r{L}_1^{(\propto ,\lambda )}\left(x\right)=\left(1+\propto -x^{\lambda }\right)\sqrt{\frac{w^{\left(\propto ,\lambda \right)}\left(x\right)}{\Gamma \left(\propto +2\right)}}.$$

(7)

Secondly, the recurrence formula can be obtained as follows.

Based on the three-term recurrence relation of Equations (5) and (6), we can deduce the following:

$$\begin{array}{c}\sqrt{\frac{h_{i+1}}{w^{\left(\propto ,\lambda \right)}\left(x\right)}}{\widetilde{F_{r}L}}_{i+1}^{\left(\propto ,\lambda \right)}\left(x\right)=\frac{1}{i+1}\left[\left(2i+\propto +1-x^{\lambda }\right)\sqrt{\frac{h_1}{w^{\left(\propto ,\lambda \right)}\left(x\right)}}{\widetilde{F_{r}L}}_i^{\left(\propto ,\lambda \right)}\left(x\right)-\left(i+\propto \right)\sqrt{\frac{h_{i-1}}{w^{\left(\propto ,\lambda \right)}\left(x\right)}}{\widetilde{F_{r}L}}_{i-1}^{\left(\propto ,\lambda \right)}\left(x\right)\right],i=1,2,\dots \\ \sqrt{h_{i+1}}{\widetilde{F_{r}L}}_{i+1}^{\left(\propto ,\lambda \right)}\left(x\right)=\frac{1}{i+1}\left[\left(2i+\propto +1-x^{\lambda }\right)\sqrt{h_i}{\widetilde{F_{r}L}}_i^{\left(\propto ,\lambda \right)}\left(x\right)-\left(i+\propto \right)\sqrt{h_{i-1}}{\widetilde{F_{r}L}}_{i-1}^{\left(\propto ,\lambda \right)}\left(x\right)\right].\end{array}$$

(8)

By using the recursive formula of the gamma function, one can deduce the following:

$$\Gamma \left(k+\propto +1\right)=\left(k+\propto \right)\Gamma \left(k+\propto \right),$$

(9)

by substituting from Equation (9) into Equation (4):

$$\begin{array}{c}{h}_k^{\left(\propto ,\lambda \right)}=\frac{\Gamma (\propto +k+1)}{k!}=\frac{\left(k+\propto \right)\Gamma \left(k+\propto \right)}{k(k-1)!}=\frac{\left(k+\propto \right)}{k}\frac{\Gamma \left(k+\propto \right)}{(k-1)!}=\frac{\left(k+\propto \right)}{k}{h}_{k-1}^{\left(\propto ,\lambda \right)},\\ h_k^{\left(\propto ,\lambda \right)}=\frac{\left(k+\propto \right)}{k}{h}_{k-1}^{\left(\propto ,\lambda \right)}.\end{array}$$

(10)

Equation (10) represents a recursive formula for evaluating the squared norm $h_k^{\left(\propto ,\lambda \right)}$. This formula has great benefits in saving computation time.

From Equation (10), we can deduce the following:

$$h_k^{\left(\propto ,\lambda \right)}=\frac{\left(k+\propto \right)}{k}{h}_{k-1}^{\left(\propto ,\lambda \right)}=\frac{\left(k+\propto \right)}{k}\frac{\left(k+\propto -1\right)}{k-1}{h}_{k-2}^{\left(\propto ,\lambda \right)},$$

(11)

$$h_{k+1}^{\left(\propto ,\lambda \right)}=\frac{\left(k+\propto +1\right)}{k+1}{h}_k^{\left(\propto ,\lambda \right)}=\frac{\left(k+\propto +1\right)}{k+1}\frac{\left(k+\propto \right)}{k}\frac{\left(k+\propto -1\right)}{k-1}{h}_{k-2}^{\left(\propto ,\lambda \right)},$$

(12)

$$h_{k-1}^{\left(\propto ,\lambda \right)}=\frac{\left(k+\propto -1\right)}{k-1}{h}_{k-2}^{\left(\propto ,\lambda \right)}.$$

(13)

Equation (8) can be rewritten as follows:

$${\widetilde{F_{r}L}}_{i+1}^{\left(\propto ,\lambda \right)}\left(x\right)=\frac{1}{i+1}\left[\left(2i+\propto +1-x^{\lambda }\right)\frac{\sqrt{h_i}}{\sqrt{h_{i+1}}}{\widetilde{F_{r}L}}_i^{\left(\propto ,\lambda \right)}\left(x\right)-\left(i+\propto \right)\frac{\sqrt{h_{i-1}}}{\sqrt{h_{i+1}}}{\widetilde{F_{r}L}}_{i-1}^{\left(\propto ,\lambda \right)}\left(x\right)\right],$$

(14)

from Equations (11)–(13):

$$\frac{\sqrt{h_k}}{\sqrt{h_{k+1}}}=\sqrt{\frac{\left(k+1\right)}{\left(\propto +k+1\right)}},$$

(15)

$$\frac{\sqrt{h_{k-1}}}{\sqrt{h_{k+1}}}=\sqrt{\frac{k\left(k+1\right)}{\left(\propto +k+1\right)(\propto +k)}}.$$

(16)

By substituting from Equations (15) and (16) into Equation (14):

$${\widetilde{F_{r}L}}_{i+1}^{\left(\propto ,\lambda \right)}\left(x\right)=\frac{\left(2i+\propto +1-x^{\lambda }\right)}{\left(i+1\right)}\sqrt{\frac{\left(i+1\right)}{\left(\propto +i+1\right)}}{\widetilde{F_{r}L}}_i^{\left(\propto ,\lambda \right)}\left(x\right)-\frac{\left(i+\propto \right)}{\left(i+1\right)}\sqrt{\frac{i\left(i+1\right)}{\left(\propto +i+1\right)(\propto +i)}}{\widetilde{F_{r}L}}_{i-1}^{\left(\propto ,\lambda \right)}\left(x\right),$$

(17)

4. Schwarz–Rutishauser Algorithm

The Schwarz–Rutishauser algorithm adapts the classical Gram–Schmidt methods [30,33,34,35]. The Gram–Schmidt is a decomposition of matrix A as:

$$A=QR$$

(18)

where $Q$ is an orthogonal matrix and $R$ is an upper triangular matrix. The orthogonal matrix $Q$ is produced based on the orthogonal projection. An orthogonal projection of $a$ onto $q$ is:

$${proj}_{\stackrel{⃑}{q}}\stackrel{⃑}{a}=\frac{〈\stackrel{⃑}{q},\stackrel{⃑}{a}〉}{〈\stackrel{⃑}{q},\stackrel{⃑}{q}〉}\times \stackrel{⃑}{q}=\frac{〈\stackrel{⃑}{q},\stackrel{⃑}{a}〉}{{‖\stackrel{⃑}{q}‖}^2}\times \stackrel{⃑}{q}$$

(19)

Each orthogonal vector q ∈ Q is determined by calculating the sum of projections and subtracting it from the corresponding vector a ∈ A. The entire Gram–Schmidt process yields an orthogonal matrix $Q$ and can be expressed as:

$${\stackrel{⃑}{q}}_k={\stackrel{⃑}{a}}_k-{\sum }_{i=1}^{k-1}{proj}_{\stackrel{⃑}{q_i}}\stackrel{⃑}{a_k},$$

(20)

$${\stackrel{⃑}{q}}_k=\frac{{\stackrel{⃑}{q}}_k}{‖{\stackrel{⃑}{q}}_k‖}.$$

(21)

The Gram–Schmidt method exhibits several drawbacks, including numerical instability and a significant increase in computational complexity for orthogonalizing matrices of a considerable size. The motivation behind the development of the Schwarz–Rutishauser algorithm stemmed from the objective of minimizing the computational complexity associated with the prevailing Gram–Schmidt projection-based approaches and improving their numerical stability.

Equation (21) can be simplified by removing the division by the squared norm from the sum of projections as follows:

$${\stackrel{⃑}{q}}_k={\stackrel{⃑}{a}}_k-{\sum }_{i=1}^{k-1}〈{\stackrel{⃑}{q}}_i^T,{\stackrel{⃑}{a}}_k〉\times {\stackrel{⃑}{q}}_i$$

(22)

We can easily find the i-th element of each column vector r by using the formula:

$${\stackrel{⃑}{\mathrm{e}}}_{\mathrm{k}}{\mathrm{r}}_{\mathrm{i},\mathrm{k}}=〈{\stackrel{⃑}{\mathrm{q}}}_{\mathrm{i}}^{\mathrm{T}},{\stackrel{⃑}{\mathrm{a}}}_{\mathrm{k}}〉{\stackrel{⃑}{e}}_k$$

(23)

By substituting the Equation (23) in (22):

$${\stackrel{⃑}{q}}_k={\stackrel{⃑}{a}}_k-{\sum }_{i=1}^{k-1}{\mathrm{r}}_{\mathrm{k}-1,\mathrm{k}}\times {\stackrel{⃑}{\mathrm{q}}}_{\mathrm{k}-1}$$

(24)

Based on Equations (23) and (24) above, this can be done recursively:

$$r_{i,k}=〈{\stackrel{⃑}{q}}_i^T,{\stackrel{⃑}{q}}_k〉$$

(25)

$${\stackrel{⃑}{\mathrm{q}}}_{\mathrm{k}}={\stackrel{⃑}{\mathrm{a}}}_{\mathrm{k}} - {\mathrm{r}}_{\mathrm{k}-1,\mathrm{k}} \times {\stackrel{⃑}{\mathrm{q}}}_{\mathrm{k}-1}$$

(26)

We subtract the product of the vector $q_{\mathrm{i}}\in \left(k\right)$ and r_i from the vector $q$, which is the vector rejection of $a$. That is why the sum operator can be removed from Equation (24).

Besides the $q$, we compute the $r k$th diagonal element in $R$, as the norm $\left|q\right|:$

$$r_{k,k=}‖{\stackrel{⃑}{q}}_k‖$$

(27)

5. Proposed Computation of Fractional Laguerre Orthogonal Moments Based on the Schwarz–Rutishauser Algorithm

Generally, the proposed signal analysis algorithm has two main methods: FrGLMs and Schwarz–Rutishauser. FrGLMs extract features from the signals and Schwarz–Rutishauser to preserve numerical stability with less complexity. The proposed algorithm is described as six steps. In step 1, set the maximum value (L) and (${\mathrm{L}}_{\mathrm{m}\mathrm{a}\mathrm{x}})$ as the order of polynomials. Step 2 illustrates calculating the two initial conditions, ${\widetilde{F_{r}L}}_0^{(\propto ,\lambda )}\left(x\right)$ and ${\widetilde{F_{r}L}}_1^{(\propto ,\lambda )}\left(x\right)$, using the following equations:

$$\begin{gathered} {\widetilde{F_{r}L}}_0^{(\propto ,\lambda )}\left(x\right)=\sqrt{\frac{w^{\left(\propto ,\lambda \right)}\left(x\right)}{\Gamma (\propto +1)}}, \\ {\widetilde{F_{r}L}}_1^{(\propto ,\lambda )}\left(x\right)=\left(1+\propto -x^{\lambda }\right)\sqrt{\frac{w^{\left(\propto ,\lambda \right)}\left(x\right)}{\Gamma \left(\propto +2\right)}} \end{gathered}$$

In step 3, calculate the polynomials of order $i$ depending on the initial conditions by the below equation:

$${\widetilde{F_{r}L}}_{i+1}^{\left(\propto ,\lambda \right)}\left(x\right)=\frac{\left(2i+\propto +1-x^{\lambda }\right)}{\left(i+1\right)}\sqrt{\frac{\left(i+1\right)}{\left(\propto +i+1\right)}}{\widetilde{F_{r}L}}_i^{\left(\propto ,\lambda \right)}\left(x\right)-\frac{\left(i+\propto \right)}{\left(i+1\right)}\sqrt{\frac{i\left(i+1\right)}{\left(\propto +i+1\right)(\propto +i)}}{\widetilde{F_{r}L}}_{i-1}^{\left(\propto ,\lambda \right)}\left(x\right)$$

After computing Laguerre polynomials of orders i, in step 4, we use the Schwarz–Rutishauser to obtain the orthogonal matrix $Q$ and normalize $Q$ by dividing it by its norm using the following equations:

$$\begin{array}{c}{r}_{n,m}=〈{\stackrel{⃑}{Q}}_{1:L,n}^T,{\stackrel{⃑}{Q}}_{1:L,m}〉,\\ Q_{1:L,m}=Q_{1:L,m}-r_{n,m}\times Q_{1:L,n},\\ Q_{1:L,m}=Q_{1:L,m}/‖Q_{1:L,m}‖.\end{array}$$

In step 5, the fractional-order generalized Laguerre moments are applied on matrix $Q$ to get the features from the signal using the following equation:

$${\widehat{{\mathrm{F}}_{\mathrm{r}}\mathrm{L}\mathrm{M}}}_{\mathrm{i}}=\sum _{\mathrm{x}=0}^{\mathrm{N}-1}{\widetilde{{\mathrm{F}}_{\mathrm{r}}\mathrm{L}}}_{\mathrm{i}}^{\left(\propto ,\mathsf{\lambda }\right)}\mathrm{s}\left(\mathrm{x}\right), \mathrm{i}=\mathrm{0,1},2,\dots ,\mathrm{N}-1$$

Then, apply the inverse of the fractional Laguerre moments to get the reconstructed signal $\mathrm{S}\left(\mathrm{x}\right)$ based on the below equation:

$$\mathrm{S}\left(\mathrm{x}\right)=\sum _{\mathrm{i}=0}^{{\mathrm{i}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\widehat{{\mathrm{F}}_{\mathrm{r}}\mathrm{L}\mathrm{M}}}_{\mathrm{i}}{\widetilde{{\mathrm{F}}_{\mathrm{r}}\mathrm{L}}}_{\mathrm{i}}^{\left(\propto ,\mathsf{\lambda }\right)}, \mathrm{x}=\mathrm{0,1},2,\dots ,\mathrm{N}-1.$$

An Algorithm of the Proposed Method

The algorithm’s pseudo code is presented in Algorithm 1.

Algorithm 1. The algorithm’s pseudo-code

{—— Step 1: Determine the value L and ${\mathbf{L}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$——} Input the original signal $\mathit{s}\left(\mathit{x}\right)$ Set the highest value (L) of variable x. Set a polynomial’s order (${\mathit{L}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$). {——Step 2: Calculate the initial conditions of the polynomials——} for x←0 to L-1 do Calculate ${\widetilde{{\mathit{F}}_{\mathit{r}}\mathit{L}}}_{\mathbf{0}}^{(\propto ,\mathit{\lambda })}\left(\mathit{x}\right)$ using Equation (6). Calculate ${\widetilde{{\mathit{F}}_{\mathit{r}}\mathit{L}}}_{\mathbf{1}}^{(\propto ,\mathit{\lambda })}\left(\mathit{x}\right)$ using Equation (7). {—— Step 3: Calculate the polynomials of order i ——} for $\mathit{i}\leftarrow \mathbf{2}$ to ${\mathit{L}}_{\mathbf{m}\mathbf{a}\mathbf{x}}-\mathbf{1}$ do Calculate ${\widetilde{{\mathit{F}}_{\mathit{r}}\mathit{L}}}_{\mathit{i}}^{(\propto ,\mathit{\lambda })}\left(\mathit{x}\right)$ using Equation (17). end for {—— Step 4: Obtain the orthogonal matrix Q using the Schwarz-Rutishauser ——} $\mathit{F}\mathit{L}={\widetilde{{\mathit{F}}_{\mathit{r}}\mathit{L}}}_{\mathit{i}}^{(\propto ,\mathit{\lambda })}\left(\mathit{x}\right)$ For n$\leftarrow \mathbf{1}$ to L do ${\mathit{Q}}_{\mathbf{1}:\mathit{N},\mathit{n}}={\mathit{F}\mathit{L}}_{\mathbf{1}:\mathit{N},\mathit{n}}$ for $\mathit{m}\leftarrow \mathbf{0}$ to $\mathbf{n}-\mathbf{1}$ do ${\mathit{r}}_{\mathit{n},\mathit{m}}=〈{\stackrel{⃑}{\mathit{Q}}}_{\mathbf{1}:\mathit{L},\mathit{n}}^{\mathit{T}},{\stackrel{⃑}{\mathit{Q}}}_{\mathbf{1}:\mathit{L},\mathit{m}}〉$ ${\mathit{Q}}_{\mathbf{1}:\mathit{L},\mathit{m}}={\mathit{Q}}_{\mathbf{1}:\mathit{L},\mathit{m}}-{\mathit{r}}_{\mathit{n},\mathit{m}}\times {\mathit{Q}}_{\mathbf{1}:\mathit{L},\mathit{n}}$ end for ${\mathit{Q}}_{\mathbf{1}:\mathit{L},\mathit{m}}={\mathit{Q}}_{\mathbf{1}:\mathit{L},\mathit{m}}/‖{\mathit{Q}}_{\mathbf{1}:\mathit{L},\mathit{m}}‖$ end for end for {— —Step 5: Get the features of the input signal using Fractional Laguerre moments ——} Apply fractional Laguerre moments ${\widehat{({\mathbf{F}}_{\mathbf{r}}\mathbf{L}\mathbf{M}}}_{\mathit{i}}$) using Equation (1). {—— Step 6: return the reconstructed signal ——} Apply the inverse of fractional Laguerre moments to get the reconstructed signal $\mathit{S}\left(\mathit{x}\right)$ using Equation (2).

6. Experiments and Discussion

We utilized some ECG signals from the MIT-BIH Arrhythmia Dataset in the experiments. The Mean Squared Error (MSE) and Peak Signal-to-Noise Ratio (PSNR) are utilized to evaluate the efficacy of the introduced algorithm [36].

• Relative error (RelErr (%))

The relative error is calculated by dividing the absolute error of the measurement by the value of the measurement itself:

$$\mathrm{RelErr}=\left(\frac{stdev(f\left(x\right)-F\left(x\right))}{\left|f\left(x\right)\right|}\right)\times 100\%$$

(28)

• Mean Squared Error (MSE)

The Mean Squared Error (MSE) measures the average of the squares of the errors, that is, the average squared difference between the reconstructed signal $\mathrm{F}\left(\mathrm{x}\right)$ and the original signal $\mathrm{f}\left(\mathrm{x}\right)$:

$$\mathrm{M}\mathrm{S}\mathrm{E}=\frac{1}{\mathrm{N}}\sum _{\mathrm{x}=0}^{\mathrm{N}-1} (\mathrm{f}\left(\mathrm{x}\right)-\mathrm{F}\left(\mathrm{x}\right){)}^2$$

(29)

• Peak Signal-to-Noise Ratio ($\mathbf{P}\mathbf{S}\mathbf{N}\mathbf{R}$)

PSNR is a metric that quantifies the relationship between the maximum achievable signal power and the power of the noise that distorts it:

$$\mathrm{S}\mathrm{N}\mathrm{R}=20\times {\log }_{10}\frac{max\left|f\left(x\right)\right|}{\sqrt{MSE}}$$

(30)

where $f\left(x\right)$ is the original signal, and $F\left(x\right)$ is a reconstructed signal.

Results

In this section, an experimental examination of the proposed algorithm is carried out to investigate and assess its performance in reconstructing the signals utilizing various scenarios.

Table 1. Empirical results of the introduced algorithm compared with FrGLMs.

Signal	FrGLMs			Proposed Algorithm
	PSNR	MSE	RelErr (%)	PSNR	MSE	RelErr (%)
Rec. 101	84.61	0.23005	0.57	141.05	0.00992	0.025
Rec. 108	70.06	0.967	2.398	119.92	0.01496	0.037
Rec. 115	110.88	0.037	0.092	155.46	0.00328	0.008
Rec. 209	66.65	0.83865	2.079	135.82	0.00728	0.018
Rec. 214	80.83	0.60155	1.492	139.8	0.0304	0.075
Rec. 219	112.49	0.05465	0.136	163.2	0.00808	0.02
Rec. 230	78.05	0.722	1.79	151.41	0.00456	0.011
Rec. 234	74.11	0.49475	1.227	139.05	0.00936	0.023
Average	89.21	0.4932	1.223	143.21	0.01096	0.027

presents the proposed algorithm’s and FrGLMs’ comparative results on some signals from the MIT-BIH Arrhythmia Dataset. The results illustrate the superiority of the proposed algorithm in reconstruction metrics (PSNR = 143.21, MSE = 0.01096, and RelErr (%) = 0.027) over using FrGLMs; this can also be seen in Figure 2.

Table 2. Comparative results of the introduced algorithm with FrGLMs—GSM, FrGLMs—HM, and FrGLMs—GRM.

Signal	FrGLMs—GSM			FrGLMs—HM			FrGLMs—GRM			Proposed Algorithm
	PSNR	MSE	RelErr (%)	PSNR	MSE	RelErr (%)	PSNR	MSE	RelErr (%)	PSNR	MSE	RelErr (%)
Rec. 101	109.60	0.0359	0.089	128.22	0.01743	0.043	111.91	0.02548	0.063	141.05	0.00992	0.025
Rec. 108	93.022	0.2066	0.512	108.39	0.1064	0.264	88.38	0.2723	0.675	119.92	0.01496	0.037
Rec. 115	124.20	0.0203	0.050	139.72	0.01358	0.034	118.55	0.029792	0.074	155.46	0.00328	0.008
Rec. 209	109.06	0.0210	0.052	120.78	0.01967	0.049	104.73	0.029792	0.074	135.82	0.00728	0.018
Rec. 214	112.64	0.0508	0.126	124.86	0.04214	0.104	109.09	0.05985	0.148	139.80	0.0304	0.075
Rec. 219	129.54	0.0195	0.048	144.22	0.01491	0.037	128.10	0.02044	0.051	163.20	0.0080	0.020
Rec. 230	123.72	0.0142	0.035	135.56	0.01351	0.033	120.46	0.01631	0.040	151.41	0.00456	0.011
Rec. 234	112.09	0.0219	0.054	125.26	0.01652	0.041	107.287	0.02947	0.073	139.05	0.00936	0.023
Average	114.23	0.0487	0.121	128.37	0.03052	0.076	111.06	0.06041	0.150	143.21	0.01096	0.027

shows the numerical results of the proposed algorithm against the algorithms that use GSM, HM, and GRM. The results in Table 1 demonstrate the advantage of the proposed algorithm over the other methods (GSM, HM, and GRM). The proposed algorithm achieves the best PSNR (143.21), MSE (0.01096), and RelErr (0.027%). Figure 3 depicts the average PSNR and RelErr of the proposed algorithm and other signal reconstruction algorithms employing GSM, HM, and GRM. It demonstrates that the introduced algorithm is superior in the reconstructed signal quality.

To confirm the superiority of the proposed algorithm in reconstruction quality, Figure 3 presents a set of reconstructed “Rec. 101” signal in terms of PSNR and RelErr for the introduced algorithm and FrGLMs—GSM, FrGLMs—HM, and FrGLMs—GRM. The results in Table 2 and Figure 3 and Figure 4 demonstrate that the proposed algorithm has the best PSNR, MSE, and RelErr corresponding to all the used signals and the total average of PSNR, MSE, and RelErr.

A comparison is conducted between the proposed algorithm and FrGLMs—HM in reconstruction with orders from 50 and 500 to make the results more accurate.

Table 3. Performance of introduced algorithm compared to FrGLMs—HM in various orders.

Signal	Order	FrGLMs—HM		Proposed Algorithm
		PSNR	MSE	PSNR	MSE
Rec. 101	50	85.104	0.8541	88.104	0.3479
	100	90.658	0.3140	93.487	0.2011
	200	110.847	0.0220	119.639	0.0194
	300	116.014	0.0200	132.541	0.0148
	400	121.583	0.1984	137.965	0.0117
	500	128.22	0.01743	141.05	0.0099
Rec. 219	50	89.417	0.3851	105.487	0.1961
	100	101.541	0.2604	116.541	0.1358
	200	114.981	0.0833	129.574	0.0504
	300	125.635	0.0293	145.654	0.0117
	400	136.992	0.0199	157.541	0.0098
	500	144.22	0.0149	163.20	0.0080

presents the comparative results of the proposed algorithm and FrGLMs—HM in PSNR, MSE, and RelErr (%). The PSNR and MSE curves for the reconstructed signals (Rec. 101 and Rec. 219) at various orders from 50 to 500 are shown in Figure 5 and Figure 6, respectively. As shown in Table 3 and Figure 5, the PSNR values increase proportionally as the number of moment orders increases, indicating an enhancement in the reconstructed signal quality. As shown in Table 3 and Figure 6, the best value of MSE is likewise obtained as the number of moment orders increases. The results also outperform the introduced algorithm compared to FrGLMs—HM at each order in terms of PSNR and MSE.

A comparative investigation of other existing algorithms [8,18,20,36] in reconstruction quality is summarized in

Table 4. Comparative results of the introduced algorithm with other existing algorithms.

Techniques	MSE	PSNR
Charlier Moment—GSOP [8]	0.883	95.741
Krawtchouk—Householder [18]	0.0771	105.015
Meixner- MGS [20]	0.0948	85.654
Tchebichef–Householder [18]	0.0436	107.085
Hahn Moment Invariants (HMIs) [36]	0.0805	97.548
Proposed algorithm	0.0109	143.21

. Figure 7 and Figure 8 display a graphic comparison of the introduced algorithm and other existing algorithms concerning PSNR and MSE, respectively. As shown in Table 4 and Figure 7 and Figure 8, the proposed algorithm outperforms the existing algorithms, which have very high PSNR and the lowest MSE values.

A comparative analysis is performed between the proposed algorithm and other existing algorithms in the context of reconstruction to ensure the accuracy of the proposed method. The analysis includes orders ranging from 50 to 500.

Table 5. Performance of introduced algorithm compared to other existing algorithms in various orders.

Signal	Order	Tchebichef–Householder [18]		Krawtchouk—Householder [18]		Proposed Algorithm
		PSNR	MSE	PSNR	MSE	PSNR	MSE
Rec. 101	50	75.232	0.949	73.048	1.2201	88.104	0.3479
	100	80.335	0.5273	77.213	0.7554	93.487	0.2011
	200	106.854	0.0249	104.067	0.0343	119.639	0.0194
	300	122.548	0.0201	118.474	0.03	132.541	0.0148
	400	129.198	0.0197	125.985	0.0284	137.965	0.0117
	500	132.017	0.0158	128.811	0.0203	141.05	0.0099

illustrates the comparison outcomes of the introduced and existing algorithms in terms of PSNR and MSE. Figure 9 and Figure 10 display the PSNR and MSE curves corresponding to the reconstructed signal “Rec. 101” over different orders ranging from 50 to 500. The obtained results demonstrate the superior performance of the proposed method over the existing algorithms at every order, determined by PSNR and MSE.

To investigate the complexity of the introduced algorithm, we compare the elapsed reconstruction time of the proposed algorithm and FrGLMs—GSM, FrGLMs—HM, and FrGLMs—GRM. As shown in Figure 11, the proposed algorithm outperforms other algorithms in the elapsed reconstruction time at different orders. The visual inspection in Figure 11 shows that the introduced algorithm has less reconstruction time than FrGLMs—HM, FrGLMs—GRM, and FrGLMs—GSM.

Table 6. Elapsed reconstruction time of the proposed algorithm and FrGLMs—HM.

	Elapsed Reconstruction Time (s)
	Rec. 101	Rec. 219
FrGLMs—HM	1.5	1.7
Proposed algorithm	0.7	0.9
Efficiency Gain	0.8	0.8

and Figure 12 compare the introduced algorithm and the FrGLMs—HM algorithm in the elapsed reconstruction time and efficiency gain. The numerical results in Table 6 show that the elapsed reconstruction time of the proposed algorithm for the Rec. 101 signal is 0.7 s in contrast to 1.5 s for FrGLMs—HM. Also, concerning the Rec. 219 signal, the proposed algorithm elapses 0.9 s compared to 1.7 for FrGLMs—HM.

7. Discussion

This paper introduces a reconstruction algorithm based on FrGLMs and the Schwarz–Rutishauser algorithm. As seen previously and from the results of the comparative experiments, the proposed algorithm outperforms the algorithm that uses FrGLMs with GSM, HM, and GRM methods. The proposed algorithm was evaluated by assessing the quality of the reconstructed signal in terms of the elapsed time of reconstruction. The proposed method excelled in both aspects. The good results in quality reconstruction and elapsed reconstruction time go back to two main reasons. The first uses set fractional-order generalized Laguerre moments (FrGLMs), in which the term second-order recurrence formula is derived for the normalized form of the FrGLMs. Also, using the simple recursive formula of the gamma function, we deduced a recursive formula for evaluating the squared norm $h_k^{\left(\propto ,\lambda \right)}$. It has great benefits in saving computation time. The second reason is using the Schwarz–Rutishauser algorithm to preserve the orthogonality property. The empirical experiments reveal that using the Schwarz–Rutishauser algorithm with FrGLMs takes less elapse time in reconstruction. In contrast to using the GSM, HM, and GRM methods with FrGLMs, due to its complexity, the Schwarz–Rutishauser algorithm is (mn²), which is (n)-times less than GSM.

8. Conclusions

In conclusion, we would like to point out that we have provided an ideal approach for large-size bio-signals. The proposed approach is based on an ultramodern set of DOMs called FrGLMs. For more fast and accurate calculations, we derived a term second-order recurrence formula for the normalized form of FrGLMs. Also, we used a novel QR decomposition method called Schwarz–Rutishauser, which gives more accurate and stable calculations, especially at higher orders of FrGLMs. The MIT-BIH Arrhythmia Dataset analyzed the proposed algorithm’s performance concerning MSE, PSNR, and elapsed reconstruction time. The empirical results reveal superiority in favor of the proposed algorithm (i.e., FrGLMs with the Schwarz–Rutishauser algorithm) when compared with FrGLMs with the classical QR decomposition techniques (i.e., GSM, HM, and GRM). Regarding the PSNR metric, the proposed algorithm gives average efficiency gain with values of 28.98, 14.84, and 32.15 when compared with FrGLMs—MGSM, FrGLMs—HM, and FrGLMs—GRM, respectively. The MSE metric gives average efficiency gain with values of 0.03784, 14.84, and 32.15 when compared with FrGLMs—MGSM, FrGLMs—HM, and FrGLMs—GRM, respectively. The proposed approach also achieves high superiority concerning the elapsed reconstruction time.