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

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## Abstract

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## 1. Introduction

**mn**

^{2}), 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.

- 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.

## 2. Fractional-Order Generalized Laguerre Orthogonal Moments

## 3. Proposed Computation of Fractional Laguerre Orthogonal Polynomials

## 4. Schwarz–Rutishauser Algorithm

**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).

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

#### An Algorithm of the Proposed Method

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 ${\stackrel{~}{{\mathit{F}}_{\mathit{r}}\mathit{L}}}_{\mathbf{0}}^{(\propto ,\mathit{\lambda})}\left(\mathit{x}\right)$ using Equation (6). Calculate ${\stackrel{~}{{\mathit{F}}_{\mathit{r}}\mathit{L}}}_{\mathbf{1}}^{(\propto ,\mathit{\lambda})}\left(\mathit{x}\right)$ 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 ${\stackrel{~}{{\mathit{F}}_{\mathit{r}}\mathit{L}}}_{\mathit{i}}^{(\propto ,\mathit{\lambda})}\left(\mathit{x}\right)$ using Equation (17). end for {—— Step 4: Obtain the orthogonal matrix Q using the Schwarz-Rutishauser ——} $\mathit{F}\mathit{L}={\stackrel{~}{{\mathit{F}}_{\mathit{r}}\mathit{L}}}_{\mathit{i}}^{(\propto ,\mathit{\lambda})}\left(\mathit{x}\right)$ 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 ${\mathit{r}}_{\mathit{n},\mathit{m}}=\u2329{\stackrel{\u20d1}{\mathit{Q}}}_{\mathbf{1}:\mathit{L},\mathit{n}}^{\mathit{T}},{\stackrel{\u20d1}{\mathit{Q}}}_{\mathbf{1}:\mathit{L},\mathit{m}}\u232a$ ${\mathit{Q}}_{\mathbf{1}:\mathit{L},\mathit{m}}={\mathit{Q}}_{\mathbf{1}:\mathit{L},\mathit{m}}-{\mathit{r}}_{\mathit{n},\mathit{m}}\times {\mathit{Q}}_{\mathbf{1}:\mathit{L},\mathit{n}}$ end for ${\mathit{Q}}_{\mathbf{1}:\mathit{L},\mathit{m}}={\mathit{Q}}_{\mathbf{1}:\mathit{L},\mathit{m}}/\Vert {\mathit{Q}}_{\mathbf{1}:\mathit{L},\mathit{m}}\Vert $ 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

**Relative error (RelErr (%))**

**Mean Squared Error (MSE)**

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

#### Results

## 7. Discussion

**mn**

^{2}), which is (

**n**)-times less than GSM.

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Plots of (

**a**) ${\stackrel{~}{{\mathrm{F}}_{\mathrm{r}}\mathrm{L}}}_{\mathrm{i}}^{(\propto ,\mathsf{\lambda})}\left(\mathrm{x}\right)$ and (

**b**) ${\mathrm{F}}_{\mathrm{r}}{\mathrm{L}}_{\mathrm{i}+1}^{\left(\propto ,\mathsf{\lambda}\right)}\left(\mathrm{x}\right)$ for the first even orders with parameter values.

**Figure 2.**The average values of quality metrics (PSNR and RelErr (%)) for the introduced algorithm and FrGLMs.

**Figure 3.**The average values of quality metrics (PSNR and RelErr (%)) for the introduced algorithm and FrGLMs—GSM, FrGLMs—HM, and FrGLMs—GRM.

**Figure 4.**Set of reconstructed “Rec. 101” signal employing the introduced algorithm and FrGLMs—GSM, FrGLMs—HM, and FrGLMs—GRM.

**Figure 5.**Values of PSNR for the reconstructed signal “Rec. 219” based on the proposed algorithm and FrGLMs—HM at different orders.

**Figure 6.**Values of MSE for the reconstructed signal “Rec. 219” based on the proposed algorithm and FrGLMs—HM at different orders.

**Figure 7.**The average values of PSNR for the introduced algorithm compared with other existing algorithms.

**Figure 8.**The average values of MSE for the introduced algorithm compared with other existing algorithms.

**Figure 9.**Values of PSNR of the reconstructed signal “Rec. 101” for the proposed algorithm and other existing algorithms at different orders.

**Figure 10.**Values of MSE of the reconstructed signal “Rec. 101” for the proposed algorithm and other existing algorithms at different orders.

**Figure 11.**Elapsed reconstruction time for the proposed algorithm, FrGLMs—GSM, FrGLMs—HM, and FrGLMs—GRM at different orders.

**Figure 12.**Comparison of elapsed reconstruction time between the proposed algorithm and FrGLMs—HM for “Rec. 101” and “Rec. 219” signals.

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 |

**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 |

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 |

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 |

**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 |

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 |

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## Share and Cite

**MDPI and ACS Style**

Aldakheel, E.A.; Khafaga, D.S.; Fathi, I.S.; Hosny, K.M.; Hassan, G.
Efficient Analysis of Large-Size Bio-Signals Based on Orthogonal Generalized Laguerre Moments of Fractional Orders and Schwarz–Rutishauser Algorithm. *Fractal Fract.* **2023**, *7*, 826.
https://doi.org/10.3390/fractalfract7110826

**AMA Style**

Aldakheel EA, Khafaga DS, Fathi IS, Hosny KM, Hassan G.
Efficient Analysis of Large-Size Bio-Signals Based on Orthogonal Generalized Laguerre Moments of Fractional Orders and Schwarz–Rutishauser Algorithm. *Fractal and Fractional*. 2023; 7(11):826.
https://doi.org/10.3390/fractalfract7110826

**Chicago/Turabian Style**

Aldakheel, Eman Abdullah, Doaa Sami Khafaga, Islam S. Fathi, Khalid M. Hosny, and Gaber Hassan.
2023. "Efficient Analysis of Large-Size Bio-Signals Based on Orthogonal Generalized Laguerre Moments of Fractional Orders and Schwarz–Rutishauser Algorithm" *Fractal and Fractional* 7, no. 11: 826.
https://doi.org/10.3390/fractalfract7110826