# Face Recognition Algorithm Based on Fast Computation of Orthogonal Moments

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## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review and Discussion

#### 1.2. Contributions

## 2. Preliminaries of Orthogonal Polynomials and Moments

#### 2.1. Squared Krawtchouk–Tchebichef Polynomials

#### 2.2. Squared Krawtchouk–Tchebichef Moments

## 3. Methodology

## 4. Experiments and Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | Two-dimensional |

FOBP | Fast Overlapped Block Processing |

HOP | Hybrid Orthogonal Polynomials |

KPs | Krawtchouk polynomials |

OP | Orthogonal Polynomials |

OM | Orthogonal Moments |

PSNR | Peak Signal-to-Noise Ratio |

SKTM | Squared Krawtchouk–Tchebichef Moment |

SKTP | Squared Krawtchouk–Tchebichef polynomials |

TPs | Tchebichef polynomials |

## Appendix A. Computation of the KP and TP Coefficients

#### Appendix A.1. Computation of the KP Coefficients

- 1.
- The initial values are computed as follows:
- 1.1.
- The value at $n=0$ and $x={x}_{0}$ is computed by$${K}_{0}({x}_{0};p)=exp\left(\frac{{k}_{0}}{2}\right),$$
- 1.2.
- The value at $n=0$ and $x={x}_{1}$ is computed by$${K}_{0}({x}_{1};p)=\sqrt{\left(\frac{{N}_{k}}{p{N}_{k}+1}-1\right)\xb7\left(\frac{p}{1-p}\right)}\phantom{\rule{1.em}{0ex}}{K}_{0}({x}_{0};p)$$
- 1.3.
- The value at $n=1$ and $x={x}_{0},\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{x}_{1}$ are computed by$$\begin{array}{cc}\hfill {K}_{1}({x}_{0};p)& =\frac{p}{\sqrt{p(1-p)({N}_{k}-1)}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{K}_{0}({x}_{0};p)\hfill \end{array}$$$$\begin{array}{cc}\hfill {K}_{1}({x}_{1};p)& =\frac{p+1}{\sqrt{p(1-p)({N}_{k}-1)}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{K}_{0}({x}_{1};p)\hfill \end{array}$$
- 1.4.
- The values in the range $n=2,3,\cdots ,x$ and $x={x}_{0},{x}_{1}$ are computed by$$\begin{array}{cc}\hfill {K}_{n}(x;p)=& \frac{p({N}_{k}-2n+1)+n-x-1}{\sqrt{pn(1-p)({N}_{k}-n)}}\phantom{\rule{0.166667em}{0ex}}{K}_{n-1}(x;p)-\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \sqrt{\frac{({N}_{k}-n+1)(n-1)}{n({N}_{k}-n)}}\phantom{\rule{0.166667em}{0ex}}{K}_{n-2}(x;p)\hfill \end{array}$$

- 2.
- The values in part P1 ($n=0,1,\cdots ,{x}_{0}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}x={x}_{0},{x}_{0}-1,\cdots ,n$) are computed as follows:$$\begin{array}{cc}\hfill {K}_{n}(x-1;p)& =-\frac{({N}_{k}-2x-1)p-n+x}{\sqrt{px(1-p)({N}_{k}-x)}}\phantom{\rule{0.166667em}{0ex}}{K}_{n}(x;p)-\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}\phantom{\rule{2.em}{0ex}}\sqrt{\frac{({N}_{k}-x-1)(x+1)}{x({N}_{k}-x)}}\phantom{\rule{0.166667em}{0ex}}{K}_{n}(x+1;p)\hfill \end{array}$$
- 3.
- The values in part P2 are computed as follows:
- 3.1.
- the values in the range ($n=0,1,\cdots ,{x}_{0}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}x={x}_{0},{x}_{0}+1,\cdots ,N-n-1$) are provided by$$\begin{array}{cc}\hfill {K}_{n}(x+1;p)=& \frac{p({N}_{k}-2x-1)-n+x}{\sqrt{p(1-p)(x+1)({N}_{k}-x-1)}}{K}_{n}(x;p)-\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \sqrt{\frac{x({N}_{k}-x)}{(x+1)({N}_{k}-x-1)}}{K}_{n}(x-1;p)\hfill \end{array}$$
- 3.2.
- The values in the range ($x={x}_{1},{x}_{1}+1,\cdots ,{N}_{k}/2-1;\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}n=x$) are provided by$$\begin{array}{cc}\hfill {K}_{n+1}& (x+1;p)=\frac{p({N}_{k}-2n-1)+n-x-1}{\sqrt{p(1-p)(n+1)({N}_{k}-n-1)}}{K}_{n}(x+1;p)-\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \sqrt{\frac{n({N}_{k}-n){(({N}_{k}-2x-1)p+x-n+1)}^{2}}{p(1-p)(n+1)(x+1)({N}_{k}-n-1)({N}_{k}-x-1)}}{K}_{n-1}(x;p)+\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \sqrt{\frac{nx({N}_{k}-n)({N}_{k}-x)}{(n+1)(x+1)({N}_{k}-n-1)({N}_{k}-x-1)}}{K}_{n-1}(x-1;p)\hfill \end{array}$$
- 3.3.
- The values in the range ($n={x}_{1},{x}_{1}+1,{N}_{k}/2-2$ and $n+2\le x\le {N}_{k}-n+1$) are provided by Equation (25).

- 4.
- To compute the rest of the KP coefficients, the following relations are used:
- 4.1.
- The values in the range $x=0,1,\cdots ,N/2-1$ and $n=x+1,x+2,\cdots ,{N}_{k}-x-1$ are computed using$${K}_{n}(x;p)={K}_{x}(n;p)$$
- 4.2.
- The values in the range $x=0,1,\cdots ,{N}_{k}-1$ and $n={N}_{k}-x,{N}_{k}-x+1,\cdots ,{N}_{k}-1$ are computed using$${K}_{n}(x;p)={(-1)}^{{N}_{k}-n-x-1}{K}_{{N}_{k}-n}({N}_{k}-x;p)$$

#### Appendix A.2. Computation of the TP Coefficients

- 1.
- The initial set of values are computed as follows:
- 1.1.
- The initial value at ${T}_{0}\left(0\right)$ is computed by$${T}_{0}\left(0\right)=\frac{1}{\sqrt{{N}_{t}}}$$
- 1.2.
- The initial values at the range $x=0$ and $n=2,3,\cdots ,{N}_{t}-1$ are computed by$${T}_{n}\left(0\right)=-\sqrt{\frac{N-n}{N+n}}\sqrt{\frac{2n+1}{2n-1}}\phantom{\rule{1.em}{0ex}}{T}_{n-1}\left(0\right)$$
- 1.3.
- The initial values at the range $x=1$ and $n=1,2,\cdots ,{N}_{t}-1$ are computed by$${T}_{n}\left(1\right)=\left(1+\frac{n(1+n)}{1-N}\right)\phantom{\rule{1.em}{0ex}}{T}_{n}\left(0\right)$$

- 2.
- The values in the range $n=0,1,\cdots ,{N}_{t}-1;\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}x=2,3,\cdots ,\frac{{N}_{t}}{2}-1$ are computed by$$\begin{array}{cc}\hfill {T}_{n}\left(x\right)=& \frac{-n(n+1)-(2x-1)(x-{N}_{t}-1)-x}{x({N}_{t}-x)}{T}_{n}(x-1)+\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \frac{(x-1)(x-{N}_{t}-1)}{x({N}_{t}-x)}{T}_{n}(x-2)\hfill \end{array}$$
- 3.
- The values in the range $n=0,1,\cdots ,{N}_{t}-1$ and $x={N}_{t}/2,{N}_{t}/2+1,\cdots ,{N}_{t}-1$ are computed using the relation$${T}_{n}({N}_{t}-1-x)={(-1)}^{n}\phantom{\rule{4pt}{0ex}}{T}_{n}\left(x\right)$$

## Appendix B. Detailed Results of the Individual Runs for Different σ Values for the Used Smoothing Kernel

Run ID | $\mathit{\sigma}=0.5$ | $\mathit{\sigma}=1.0$ | $\mathit{\sigma}=1.5$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Overlap Size | Overlap Size | Overlap Size | |||||||

(0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | |

1 | 97.00 | 98.00 | 97.50 | 97.50 | 97.50 | 97.50 | 97.50 | 97.00 | 97.50 |

2 | 99.00 | 99.00 | 99.00 | 99.00 | 98.50 | 99.50 | 99.50 | 98.50 | 99.50 |

3 | 97.00 | 98.00 | 97.00 | 97.00 | 96.00 | 97.00 | 96.00 | 95.50 | 97.50 |

4 | 98.00 | 97.50 | 97.50 | 98.00 | 97.50 | 98.00 | 97.50 | 97.50 | 97.00 |

5 | 98.50 | 98.00 | 97.50 | 99.00 | 98.00 | 98.00 | 98.50 | 97.50 | 98.50 |

6 | 97.50 | 97.50 | 98.00 | 97.50 | 98.50 | 99.00 | 98.50 | 98.00 | 99.00 |

7 | 96.50 | 96.50 | 97.00 | 97.00 | 97.00 | 97.50 | 97.00 | 97.00 | 97.50 |

8 | 98.50 | 97.50 | 98.00 | 97.50 | 98.00 | 98.50 | 97.50 | 98.00 | 98.00 |

9 | 96.50 | 96.00 | 96.50 | 96.50 | 96.00 | 97.00 | 96.50 | 95.50 | 97.00 |

10 | 97.50 | 97.50 | 98.00 | 97.50 | 98.50 | 99.00 | 98.50 | 98.00 | 99.00 |

11 | 98.00 | 98.00 | 97.00 | 98.50 | 98.50 | 98.50 | 99.00 | 99.00 | 99.00 |

12 | 98.00 | 98.00 | 98.00 | 98.00 | 97.50 | 98.00 | 97.00 | 97.00 | 96.50 |

13 | 95.00 | 95.50 | 96.00 | 96.00 | 96.00 | 96.50 | 96.50 | 96.00 | 97.00 |

14 | 98.50 | 98.50 | 98.00 | 99.00 | 98.50 | 99.00 | 99.00 | 98.50 | 98.50 |

15 | 99.00 | 99.00 | 99.00 | 98.50 | 98.50 | 99.00 | 98.50 | 99.00 | 98.50 |

16 | 98.50 | 98.50 | 98.00 | 98.50 | 99.00 | 99.00 | 98.50 | 99.00 | 98.50 |

17 | 98.50 | 98.00 | 96.50 | 98.00 | 98.00 | 98.50 | 97.50 | 97.50 | 98.50 |

18 | 97.00 | 97.50 | 97.00 | 98.00 | 97.50 | 98.50 | 98.50 | 97.50 | 97.00 |

19 | 97.50 | 97.50 | 98.50 | 97.50 | 98.00 | 98.50 | 97.50 | 98.00 | 98.00 |

20 | 98.50 | 97.50 | 99.00 | 96.50 | 97.50 | 98.00 | 96.50 | 97.50 | 97.50 |

Average | 97.73 | 97.68 | 97.65 | 97.75 | 97.73 | 98.23 | 97.78 | 97.58 | 97.98 |

**Table A2.**The results of the runs for Gaussian noise with standard deviation of 0.01 using different values of $\sigma $.

Run ID | $\mathit{\sigma}=0.5$ | $\mathit{\sigma}=1.0$ | $\mathit{\sigma}=1.5$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Overlap Size | Overlap Size | Overlap Size | |||||||

(0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | |

1 | 97.00 | 97.50 | 97.50 | 97.50 | 97.50 | 97.50 | 97.50 | 97.50 | 97.50 |

2 | 99.00 | 99.00 | 99.00 | 99.00 | 98.50 | 99.50 | 99.50 | 98.50 | 99.50 |

3 | 97.00 | 97.50 | 96.50 | 97.00 | 96.00 | 97.00 | 96.00 | 96.00 | 97.50 |

4 | 98.50 | 98.00 | 97.50 | 98.00 | 98.00 | 98.00 | 97.50 | 97.50 | 97.00 |

5 | 98.50 | 98.50 | 98.00 | 99.00 | 98.50 | 98.00 | 98.50 | 98.00 | 98.50 |

6 | 97.00 | 97.50 | 98.00 | 97.50 | 99.00 | 99.00 | 98.50 | 98.00 | 99.00 |

7 | 96.50 | 96.50 | 96.50 | 97.00 | 97.00 | 97.50 | 97.00 | 97.00 | 97.50 |

8 | 98.00 | 97.50 | 98.00 | 98.00 | 98.00 | 98.50 | 97.50 | 99.00 | 98.00 |

9 | 97.00 | 96.50 | 96.50 | 96.50 | 96.00 | 97.00 | 96.50 | 96.00 | 97.00 |

10 | 97.00 | 97.50 | 98.00 | 97.50 | 99.00 | 99.00 | 98.50 | 98.00 | 99.00 |

11 | 98.00 | 98.00 | 97.50 | 98.50 | 98.50 | 98.50 | 99.00 | 99.00 | 99.00 |

12 | 98.50 | 97.50 | 97.50 | 97.50 | 98.00 | 98.50 | 97.00 | 96.50 | 96.50 |

13 | 94.50 | 95.50 | 95.50 | 95.50 | 96.00 | 96.50 | 96.50 | 96.00 | 96.50 |

14 | 98.50 | 98.50 | 98.00 | 99.00 | 98.50 | 99.00 | 98.50 | 98.50 | 98.50 |

15 | 99.50 | 99.00 | 99.00 | 98.00 | 99.00 | 99.00 | 98.50 | 99.00 | 98.50 |

16 | 98.50 | 98.50 | 98.00 | 98.50 | 99.00 | 99.00 | 98.50 | 99.00 | 98.50 |

17 | 98.00 | 98.00 | 96.50 | 98.00 | 98.00 | 98.50 | 97.50 | 97.50 | 98.50 |

18 | 97.50 | 97.50 | 96.50 | 97.00 | 97.50 | 98.00 | 98.50 | 97.50 | 97.00 |

19 | 97.50 | 98.00 | 98.50 | 97.50 | 97.50 | 98.50 | 97.50 | 98.00 | 98.50 |

20 | 98.50 | 97.50 | 98.50 | 96.50 | 97.50 | 98.00 | 97.00 | 97.50 | 97.50 |

Average | 97.73 | 97.70 | 97.55 | 97.65 | 97.85 | 98.23 | 97.78 | 97.70 | 97.98 |

**Table A3.**The results of the runs for Gaussian noise with standard deviation of 0.05 using different values of $\sigma $.

Run ID | $\mathit{\sigma}=0.5$ | $\mathit{\sigma}=1.0$ | $\mathit{\sigma}=1.5$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Overlap Size | Overlap Size | Overlap Size | |||||||

(0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | |

1 | 97.00 | 97.50 | 98.00 | 97.50 | 97.50 | 97.50 | 97.50 | 97.50 | 97.50 |

2 | 99.00 | 99.00 | 98.50 | 99.00 | 98.50 | 99.50 | 99.50 | 98.00 | 99.50 |

3 | 97.00 | 98.00 | 97.00 | 96.00 | 96.00 | 97.00 | 96.00 | 96.00 | 97.00 |

4 | 98.50 | 97.50 | 98.00 | 98.00 | 98.00 | 97.00 | 96.50 | 97.50 | 97.00 |

5 | 98.50 | 98.00 | 97.50 | 99.00 | 98.50 | 98.50 | 98.50 | 98.00 | 99.00 |

6 | 97.50 | 97.50 | 98.00 | 97.50 | 98.50 | 99.00 | 98.00 | 98.00 | 99.00 |

7 | 96.50 | 96.50 | 97.00 | 97.00 | 97.00 | 97.50 | 97.00 | 97.00 | 97.50 |

8 | 97.50 | 98.00 | 98.50 | 97.50 | 98.00 | 98.50 | 97.50 | 98.00 | 98.00 |

9 | 96.50 | 96.00 | 96.00 | 96.00 | 96.00 | 97.00 | 96.50 | 95.50 | 97.50 |

10 | 97.50 | 97.50 | 98.00 | 97.50 | 98.50 | 99.00 | 98.00 | 98.00 | 99.00 |

11 | 98.00 | 97.50 | 97.00 | 98.50 | 98.50 | 98.50 | 99.00 | 99.50 | 98.50 |

12 | 98.50 | 98.50 | 98.00 | 97.50 | 98.00 | 98.00 | 96.50 | 97.00 | 96.50 |

13 | 95.50 | 95.50 | 95.50 | 96.00 | 96.00 | 97.00 | 96.00 | 96.00 | 97.00 |

14 | 98.50 | 98.50 | 98.50 | 99.00 | 98.50 | 99.00 | 99.00 | 98.50 | 98.50 |

15 | 99.00 | 98.50 | 99.00 | 98.00 | 98.50 | 99.00 | 98.50 | 99.00 | 98.50 |

16 | 98.50 | 98.50 | 98.00 | 98.50 | 99.00 | 99.00 | 98.50 | 99.00 | 99.00 |

17 | 98.00 | 98.00 | 97.50 | 98.00 | 98.00 | 98.50 | 97.50 | 97.50 | 98.50 |

18 | 97.00 | 97.50 | 97.00 | 98.00 | 97.50 | 98.50 | 98.50 | 97.00 | 97.00 |

19 | 97.50 | 98.00 | 98.50 | 97.50 | 98.00 | 98.00 | 97.50 | 98.00 | 98.00 |

20 | 98.00 | 96.50 | 98.00 | 96.00 | 97.50 | 97.50 | 96.00 | 97.00 | 97.50 |

Average | 97.70 | 97.63 | 97.68 | 97.60 | 97.80 | 98.18 | 97.60 | 97.60 | 98.00 |

**Table A4.**The results of the runs for Salt and Pepper noise with density of 0.05 using different values of $\sigma $.

Run ID | $\mathit{\sigma}=0.5$ | $\mathit{\sigma}=1.0$ | $\mathit{\sigma}=1.5$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Overlap Size | Overlap Size | Overlap Size | |||||||

(0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | |

1 | 97.50 | 97.50 | 98.50 | 97.50 | 98.00 | 97.50 | 97.50 | 98.00 | 97.50 |

2 | 99.50 | 98.50 | 99.50 | 99.00 | 99.00 | 99.50 | 99.00 | 99.00 | 99.50 |

3 | 96.50 | 97.50 | 97.50 | 96.50 | 97.00 | 97.00 | 96.00 | 96.00 | 97.00 |

4 | 97.50 | 97.00 | 97.00 | 97.50 | 96.50 | 96.50 | 97.00 | 96.50 | 96.50 |

5 | 98.50 | 98.50 | 98.00 | 99.00 | 98.50 | 99.00 | 98.50 | 98.50 | 99.00 |

6 | 97.50 | 97.50 | 97.50 | 97.50 | 99.00 | 98.50 | 98.00 | 98.50 | 98.50 |

7 | 97.00 | 96.00 | 97.00 | 97.00 | 97.00 | 97.50 | 97.00 | 97.00 | 97.50 |

8 | 98.50 | 97.50 | 98.50 | 97.50 | 98.00 | 98.50 | 97.50 | 98.50 | 98.00 |

9 | 96.50 | 96.00 | 96.00 | 96.00 | 96.50 | 96.50 | 96.00 | 96.00 | 96.50 |

10 | 97.50 | 97.50 | 97.50 | 97.50 | 99.00 | 98.50 | 98.00 | 98.50 | 98.50 |

11 | 97.50 | 98.00 | 97.50 | 98.00 | 98.50 | 98.50 | 99.00 | 99.50 | 98.50 |

12 | 98.00 | 97.00 | 97.00 | 97.00 | 96.50 | 97.00 | 96.50 | 96.00 | 95.50 |

13 | 96.00 | 95.00 | 96.50 | 95.50 | 97.00 | 96.50 | 96.50 | 96.50 | 97.00 |

14 | 99.00 | 98.50 | 98.50 | 99.00 | 98.50 | 99.00 | 98.50 | 98.50 | 98.50 |

15 | 98.00 | 98.50 | 98.50 | 98.00 | 98.50 | 98.50 | 98.50 | 98.50 | 98.50 |

16 | 98.50 | 98.50 | 98.00 | 98.50 | 99.00 | 99.00 | 98.50 | 98.50 | 99.00 |

17 | 97.50 | 98.00 | 97.50 | 97.50 | 98.50 | 98.00 | 97.50 | 97.50 | 98.00 |

18 | 97.50 | 97.50 | 97.50 | 97.50 | 98.00 | 98.50 | 98.50 | 98.00 | 97.50 |

19 | 97.50 | 97.50 | 98.50 | 98.00 | 97.00 | 98.00 | 98.00 | 97.50 | 98.00 |

20 | 96.50 | 96.00 | 97.50 | 96.00 | 97.00 | 97.00 | 96.50 | 97.00 | 96.50 |

Average | 97.63 | 97.40 | 97.70 | 97.50 | 97.85 | 97.95 | 97.63 | 97.70 | 97.78 |

**Table A5.**The results of the runs for Salt and Pepper noise with density of 0.10 using different values of $\sigma $.

Run ID | $\mathit{\sigma}=0.5$ | $\mathit{\sigma}=1.0$ | $\mathit{\sigma}=1.5$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Overlap Size | Overlap Size | Overlap Size | |||||||

(0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | |

1 | 98.00 | 97.50 | 98.00 | 97.50 | 97.00 | 98.00 | 98.00 | 97.00 | 97.00 |

2 | 98.00 | 98.50 | 97.50 | 98.00 | 98.50 | 99.50 | 99.00 | 98.50 | 99.50 |

3 | 98.00 | 97.50 | 97.50 | 96.00 | 97.00 | 96.50 | 96.00 | 95.00 | 96.50 |

4 | 97.50 | 95.00 | 97.00 | 96.50 | 96.50 | 96.00 | 96.50 | 96.00 | 96.50 |

5 | 98.50 | 97.50 | 97.50 | 98.50 | 97.50 | 98.50 | 98.00 | 98.00 | 98.00 |

6 | 97.50 | 98.00 | 97.50 | 97.50 | 97.50 | 98.50 | 97.00 | 97.50 | 98.50 |

7 | 97.00 | 96.00 | 97.00 | 97.00 | 97.00 | 97.50 | 97.00 | 97.00 | 97.50 |

8 | 99.00 | 97.50 | 98.50 | 97.50 | 97.00 | 98.50 | 97.50 | 98.00 | 98.50 |

9 | 96.50 | 95.00 | 96.00 | 95.50 | 95.50 | 96.00 | 95.50 | 95.00 | 95.50 |

10 | 97.50 | 98.00 | 97.50 | 97.50 | 97.50 | 98.50 | 97.00 | 97.50 | 98.50 |

11 | 97.00 | 96.50 | 96.50 | 97.00 | 98.00 | 98.00 | 97.00 | 98.00 | 97.50 |

12 | 96.50 | 95.50 | 96.00 | 95.00 | 94.50 | 95.50 | 95.50 | 94.50 | 94.00 |

13 | 95.00 | 95.00 | 94.50 | 95.00 | 94.00 | 95.00 | 94.50 | 94.50 | 94.50 |

14 | 99.00 | 98.50 | 98.50 | 98.50 | 98.50 | 98.50 | 98.50 | 98.50 | 98.50 |

15 | 98.50 | 98.50 | 98.50 | 98.00 | 98.50 | 98.50 | 98.50 | 98.50 | 98.00 |

16 | 98.50 | 98.00 | 98.00 | 98.50 | 98.00 | 98.50 | 98.50 | 98.00 | 98.00 |

17 | 98.00 | 98.50 | 98.00 | 97.00 | 98.00 | 98.00 | 97.50 | 97.50 | 98.00 |

18 | 97.50 | 97.50 | 97.00 | 97.00 | 97.00 | 98.00 | 97.50 | 97.00 | 96.50 |

19 | 97.50 | 96.50 | 98.50 | 97.50 | 96.50 | 97.50 | 98.00 | 96.50 | 98.00 |

20 | 95.50 | 95.00 | 97.50 | 96.00 | 95.50 | 96.50 | 96.50 | 95.50 | 96.50 |

Average | 97.53 | 97.00 | 97.35 | 97.05 | 96.98 | 97.58 | 97.18 | 96.90 | 97.28 |

## Appendix C. The Detailed Results of the Individual Runs with and without Smoothing Kernel

Run ID | Without Smoothing Kernel | With Smoothing Kernel | ||||
---|---|---|---|---|---|---|

Overlap Size | Overlap Size | |||||

(0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | |

1 | 97.50 | 98.00 | 97.00 | 97.50 | 97.50 | 97.50 |

2 | 98.50 | 99.00 | 99.00 | 99.00 | 98.50 | 99.50 |

3 | 96.00 | 98.00 | 97.50 | 97.00 | 96.00 | 97.00 |

4 | 97.50 | 97.00 | 98.50 | 98.00 | 97.50 | 98.00 |

5 | 97.50 | 98.50 | 98.50 | 99.00 | 98.00 | 98.00 |

6 | 97.50 | 97.00 | 97.50 | 97.50 | 98.50 | 99.00 |

7 | 97.00 | 96.50 | 96.50 | 97.00 | 97.00 | 97.50 |

8 | 98.00 | 97.00 | 97.50 | 97.50 | 98.00 | 98.50 |

9 | 96.00 | 96.50 | 97.00 | 96.50 | 96.00 | 97.00 |

10 | 97.50 | 97.00 | 97.50 | 97.50 | 98.50 | 99.00 |

11 | 97.00 | 97.00 | 98.00 | 98.50 | 98.50 | 98.50 |

12 | 97.00 | 97.50 | 97.50 | 98.00 | 97.50 | 98.00 |

13 | 95.00 | 95.00 | 94.50 | 96.00 | 96.00 | 96.50 |

14 | 98.00 | 98.50 | 98.50 | 99.00 | 98.50 | 99.00 |

15 | 99.00 | 99.00 | 99.50 | 98.50 | 98.50 | 99.00 |

16 | 98.00 | 98.50 | 98.50 | 98.50 | 99.00 | 99.00 |

17 | 96.50 | 98.00 | 97.00 | 98.00 | 98.00 | 98.50 |

18 | 97.00 | 97.50 | 97.00 | 98.00 | 97.50 | 98.50 |

19 | 98.50 | 97.00 | 97.50 | 97.50 | 98.00 | 98.50 |

20 | 98.00 | 97.50 | 97.50 | 96.50 | 97.50 | 98.00 |

Average | 97.35 | 97.50 | 97.60 | 97.75 | 97.73 | 98.23 |

Run ID | Without Smoothing Kernel | With Smoothing Kernel | ||||
---|---|---|---|---|---|---|

Overlap Size | Overlap Size | |||||

(0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | |

1 | 97.00 | 97.50 | 97.00 | 97.50 | 97.50 | 97.50 |

2 | 98.50 | 99.00 | 98.50 | 99.00 | 98.50 | 99.50 |

3 | 96.50 | 98.00 | 97.50 | 97.00 | 96.00 | 97.00 |

4 | 97.50 | 97.50 | 98.00 | 98.00 | 98.00 | 98.00 |

5 | 97.50 | 98.50 | 98.50 | 99.00 | 98.50 | 98.00 |

6 | 97.50 | 97.00 | 97.00 | 97.50 | 99.00 | 99.00 |

7 | 96.50 | 96.50 | 96.50 | 97.00 | 97.00 | 97.50 |

8 | 98.00 | 97.00 | 98.50 | 98.00 | 98.00 | 98.50 |

9 | 96.00 | 96.50 | 96.50 | 96.50 | 96.00 | 97.00 |

10 | 97.50 | 97.00 | 97.00 | 97.50 | 99.00 | 99.00 |

11 | 97.00 | 97.00 | 98.00 | 98.50 | 98.50 | 98.50 |

12 | 96.50 | 97.00 | 97.50 | 97.50 | 98.00 | 98.50 |

13 | 95.00 | 95.00 | 94.50 | 95.50 | 96.00 | 96.50 |

14 | 98.00 | 98.50 | 98.50 | 99.00 | 98.50 | 99.00 |

15 | 99.00 | 99.00 | 99.50 | 98.00 | 99.00 | 99.00 |

16 | 98.00 | 98.50 | 98.50 | 98.50 | 99.00 | 99.00 |

17 | 96.50 | 98.00 | 97.50 | 98.00 | 98.00 | 98.50 |

18 | 97.00 | 97.50 | 98.00 | 97.00 | 97.50 | 98.00 |

19 | 98.50 | 97.50 | 97.50 | 97.50 | 97.50 | 98.50 |

20 | 98.50 | 98.00 | 97.50 | 96.50 | 97.50 | 98.00 |

Average | 97.33 | 97.53 | 97.60 | 97.65 | 97.85 | 98.23 |

Run ID | Without Smoothing Kernel | With Smoothing Kernel | ||||
---|---|---|---|---|---|---|

Overlap Size | Overlap Size | |||||

(0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | |

1 | 98.00 | 98.00 | 97.00 | 97.50 | 97.50 | 97.50 |

2 | 98.00 | 99.00 | 98.50 | 99.00 | 98.50 | 99.50 |

3 | 97.00 | 98.00 | 97.50 | 96.00 | 96.00 | 97.00 |

4 | 97.50 | 96.50 | 98.50 | 98.00 | 98.00 | 97.00 |

5 | 97.50 | 98.50 | 98.00 | 99.00 | 98.50 | 98.50 |

6 | 98.00 | 97.50 | 97.50 | 97.50 | 98.50 | 99.00 |

7 | 97.00 | 96.50 | 96.50 | 97.00 | 97.00 | 97.50 |

8 | 98.00 | 98.00 | 97.50 | 97.50 | 98.00 | 98.50 |

9 | 96.00 | 96.00 | 96.00 | 96.00 | 96.00 | 97.00 |

10 | 98.00 | 97.50 | 97.50 | 97.50 | 98.50 | 99.00 |

11 | 96.00 | 97.00 | 97.50 | 98.50 | 98.50 | 98.50 |

12 | 98.00 | 98.50 | 98.00 | 97.50 | 98.00 | 98.00 |

13 | 95.50 | 95.00 | 95.00 | 96.00 | 96.00 | 97.00 |

14 | 98.00 | 98.50 | 98.50 | 99.00 | 98.50 | 99.00 |

15 | 98.50 | 99.00 | 99.00 | 98.00 | 98.50 | 99.00 |

16 | 98.00 | 98.50 | 98.50 | 98.50 | 99.00 | 99.00 |

17 | 97.00 | 98.00 | 98.00 | 98.00 | 98.00 | 98.50 |

18 | 96.50 | 97.50 | 97.00 | 98.00 | 97.50 | 98.50 |

19 | 98.50 | 97.50 | 97.50 | 97.50 | 98.00 | 98.00 |

20 | 97.00 | 96.50 | 97.50 | 96.00 | 97.50 | 97.50 |

Average | 97.40 | 97.58 | 97.55 | 97.60 | 97.80 | 98.18 |

Run ID | Without Smoothing Kernel | With Smoothing Kernel | ||||
---|---|---|---|---|---|---|

Overlap Size | Overlap Size | |||||

(0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | |

1 | 98.00 | 98.50 | 97.50 | 97.50 | 98.00 | 97.50 |

2 | 98.50 | 99.00 | 99.00 | 99.00 | 99.00 | 99.50 |

3 | 97.50 | 99.00 | 96.50 | 96.50 | 97.00 | 97.00 |

4 | 96.50 | 97.00 | 97.50 | 97.50 | 96.50 | 96.50 |

5 | 97.50 | 98.50 | 98.00 | 99.00 | 98.50 | 99.00 |

6 | 97.50 | 97.50 | 97.50 | 97.50 | 99.00 | 98.50 |

7 | 97.00 | 96.00 | 97.00 | 97.00 | 97.00 | 97.50 |

8 | 98.00 | 97.50 | 98.50 | 97.50 | 98.00 | 98.50 |

9 | 96.00 | 96.50 | 97.00 | 96.00 | 96.50 | 96.50 |

10 | 97.50 | 97.50 | 97.50 | 97.50 | 99.00 | 98.50 |

11 | 97.00 | 97.50 | 97.50 | 98.00 | 98.50 | 98.50 |

12 | 97.00 | 97.00 | 98.00 | 97.00 | 96.50 | 97.00 |

13 | 97.00 | 95.50 | 96.00 | 95.50 | 97.00 | 96.50 |

14 | 98.50 | 98.50 | 99.00 | 99.00 | 98.50 | 99.00 |

15 | 99.00 | 98.50 | 98.00 | 98.00 | 98.50 | 98.50 |

16 | 98.00 | 98.50 | 98.00 | 98.50 | 99.00 | 99.00 |

17 | 97.50 | 98.00 | 97.50 | 97.50 | 98.50 | 98.00 |

18 | 97.50 | 97.50 | 97.50 | 97.50 | 98.00 | 98.50 |

19 | 98.50 | 97.00 | 97.50 | 98.00 | 97.00 | 98.00 |

20 | 98.00 | 97.00 | 96.50 | 96.00 | 97.00 | 97.00 |

Average | 97.60 | 97.60 | 97.58 | 97.50 | 97.85 | 97.95 |

Run ID | Without Smoothing Kernel | With Smoothing Kernel | ||||
---|---|---|---|---|---|---|

Overlap Size | Overlap Size | |||||

(0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | |

1 | 97.00 | 98.00 | 97.50 | 97.50 | 97.00 | 98.00 |

2 | 98.00 | 98.50 | 98.00 | 98.00 | 98.50 | 99.50 |

3 | 97.00 | 98.50 | 95.50 | 96.00 | 97.00 | 96.50 |

4 | 96.00 | 96.00 | 97.00 | 96.50 | 96.50 | 96.00 |

5 | 97.50 | 98.50 | 98.00 | 98.50 | 97.50 | 98.50 |

6 | 97.50 | 98.00 | 97.50 | 97.50 | 97.50 | 98.50 |

7 | 96.50 | 96.00 | 97.00 | 97.00 | 97.00 | 97.50 |

8 | 98.50 | 97.50 | 99.00 | 97.50 | 97.00 | 98.50 |

9 | 96.00 | 95.00 | 96.50 | 95.50 | 95.50 | 96.00 |

10 | 97.50 | 98.00 | 97.50 | 97.50 | 97.50 | 98.50 |

11 | 96.00 | 96.50 | 97.00 | 97.00 | 98.00 | 98.00 |

12 | 96.00 | 95.50 | 96.50 | 95.00 | 94.50 | 95.50 |

13 | 94.50 | 94.50 | 95.50 | 95.00 | 94.00 | 95.00 |

14 | 98.50 | 98.00 | 99.00 | 98.50 | 98.50 | 98.50 |

15 | 97.50 | 98.50 | 98.50 | 98.00 | 98.50 | 98.50 |

16 | 98.00 | 98.00 | 98.00 | 98.50 | 98.00 | 98.50 |

17 | 98.00 | 98.50 | 97.00 | 97.00 | 98.00 | 98.00 |

18 | 96.50 | 97.00 | 96.50 | 97.00 | 97.00 | 98.00 |

19 | 98.00 | 96.50 | 97.50 | 97.50 | 96.50 | 97.50 |

20 | 97.00 | 95.00 | 95.50 | 96.00 | 95.50 | 96.50 |

Average | 97.08 | 97.10 | 97.23 | 97.05 | 96.98 | 97.58 |

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**Figure 4.**PSNR values for smoothing kernel test using Gaussian noise with standard deviation of (

**a**) 0.01 and (

**b**) 0.05.

Environment | ${\mathit{\sigma}}_{\mathit{smoothing}}$ | Overlap Size | ||
---|---|---|---|---|

(0,0) | (2,2) | (4,4) | ||

Noise-free | 0.5 | 97.73 | 97.68 | 97.65 |

1.0 | 97.75 | 97.73 | 98.23 | |

1.5 | 97.78 | 97.58 | 97.98 | |

Gaussian 0.005 | 0.5 | 97.73 | 97.70 | 97.55 |

1.0 | 97.65 | 97.85 | 98.23 | |

1.5 | 97.78 | 97.70 | 97.98 | |

Gaussian 0.010 | 0.5 | 97.70 | 97.63 | 97.68 |

1.0 | 97.60 | 97.80 | 98.18 | |

1.5 | 97.60 | 97.60 | 98.00 | |

Salt&Pepper 0.05 | 0.5 | 97.63 | 97.40 | 97.70 |

1.0 | 97.50 | 97.85 | 97.95 | |

1.5 | 97.63 | 97.70 | 97.78 | |

Salt&Pepper 0.10 | 0.5 | 97.53 | 97.00 | 97.35 |

1.0 | 97.05 | 96.98 | 97.58 | |

1.5 | 97.18 | 96.90 | 97.28 |

Environment | Without Smoothing Kernel | With Smoothing Kernel | ||||
---|---|---|---|---|---|---|

Overlap Size | Overlap Size | |||||

(0,0) | (2,2) | (4,4) | (0,0) | (2,2) | (4,4) | |

Noise-free | 97.35 | 97.50 | 97.60 | 97.75 | 97.73 | 98.23 |

Gaussian 0.005 | 97.33 | 97.53 | 97.60 | 98.65 | 97.85 | 98.23 |

Gaussian 0.010 | 97.40 | 97.58 | 97.55 | 97.60 | 97.80 | 98.18 |

Salt&Pepper 0.05 | 97.60 | 97.60 | 97.58 | 97.50 | 97.85 | 97.95 |

Salt&Pepper 0.10 | 97.08 | 97.10 | 97.23 | 97.05 | 96.98 | 97.58 |

Average | 97.35 | 97.46 | 97.51 | 97.51 | 97.64 | 98.03 |

**Table 3.**ORL time (milliseconds) with different parameters for the proposed and traditional algorithms.

Kernel Size | Overlap Size | Traditional Algorithm | Proposed Algorithm | Speedup Ratio |
---|---|---|---|---|

3 | 0 | 9.461 | 1.172 | 8.07 |

3 | 2 | 14.385 | 1.198 | 12.01 |

3 | 4 | 29.601 | 1.212 | 24.42 |

5 | 0 | 10.207 | 1.168 | 8.74 |

5 | 2 | 15.590 | 1.172 | 13.30 |

5 | 4 | 31.965 | 1.193 | 26.79 |

7 | 0 | 12.974 | 1.177 | 11.02 |

7 | 2 | 19.855 | 1.185 | 16.75 |

7 | 4 | 41.193 | 1.214 | 33.93 |

Algorithm | Reference | Accuracy % |
---|---|---|

DWT–PCA | [76] | 96.75 |

DCT–PCA | [77] | 96.00 |

OLPP | [78] | 93.50 |

Wavelet + PCA | [79] | 94.20 |

Wavelet + LDA | [79] | 97.10 |

GELM | [80] | 96.30 |

NPE | [81] | 94.33 |

ENPE | [81] | 95.78 |

TSLDA | [82] | 93.75 |

Improved TSLDA | [82] | 94.58 |

DIWT-LBP | [51] | 97.00 |

MLDV | [83] | 85.36 |

RLD | [42] | 97.49 |

DLCDRC | [84] | 96.39 |

DLGWT | [7] | 96.00 |

RMDL | [85] | 95.00 |

Proposed Algorithm (smoothing kernel size = 5, overlap size = (4,4)) | 98.23 |

**Table 5.**The reported face recognition accuracy (%) for the FEI dataset using the proposed algorithm.

Image Size | Block Size | Environment | Overlap Size | ||||
---|---|---|---|---|---|---|---|

$480\times 640$ | $48\times 48$ | (0,0) | (2,2) | (4,4) | (8,8) | (12,12) | |

Noise-free | 95.50 | 97.00 | 96.75 | 97.50 | 96.50 | ||

Gaussian 0.01 | 94.50 | 95.50 | 95.50 | 95.75 | 95.25 | ||

Gaussian 0.05 | 94.00 | 95.00 | 95.00 | 96.00 | 95.50 | ||

Salt&Pepper 0.05 | 95.50 | 96.75 | 97.00 | 97.00 | 96.25 | ||

Salt&Pepper 0.10 | 95.25 | 95.50 | 96.50 | 96.75 | 96.75 | ||

Average | 94.64 | 95.89 | 96.11 | 96.46 | 96.04 | ||

$240\times 320$ | $24\times 24$ | (0,0) | (2,2) | (4,4) | (6,6) | (8,8) | |

Noise-free | 95.50 | 96.50 | 97.50 | 96.50 | 96.75 | ||

Gaussian 0.01 | 94.50 | 95.50 | 95.75 | 95.25 | 95.25 | ||

Gaussian 0.05 | 94.00 | 94.50 | 95.75 | 95.25 | 95.25 | ||

Salt&Pepper 0.05 | 95.50 | 96.50 | 97.50 | 96.50 | 97.00 | ||

Salt&Pepper 0.10 | 95.25 | 96.50 | 97.25 | 96.75 | 96.50 | ||

Average | 94.64 | 95.82 | 96.61 | 96.00 | 95.93 | ||

$120\times 160$ | $12\times 12$ | (0,0) | (1,1) | (2,2) | (3,3) | (4,4) | |

Noise-free | 95.75 | 96.25 | 97.25 | 96.50 | 96.50 | ||

Gaussian 0.01 | 95.00 | 94.50 | 95.25 | 95.50 | 94.50 | ||

Gaussian 0.05 | 93.75 | 95.00 | 95.00 | 94.50 | 94.50 | ||

Salt&Pepper 0.05 | 95.50 | 96.25 | 97.00 | 96.00 | 96.50 | ||

Salt&Pepper 0.10 | 94.25 | 95.75 | 96.50 | 96.00 | 96.50 | ||

Average | 94.54 | 95.29 | 96.00 | 95.54 | 95.64 |

Overlap Size | Traditional | Proposed | Improvement |
---|---|---|---|

(0,0) | 55.278 | 1.806 | 30.60 |

(2,2) | 65.585 | 1.967 | 33.35 |

(4,4) | 76.684 | 2.055 | 37.31 |

(8,8) | 115.391 | 2.293 | 50.32 |

(16,16) | 447.883 | 4.915 | 91.12 |

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## Share and Cite

**MDPI and ACS Style**

Abdulhussain, S.H.; Mahmmod, B.M.; AlGhadhban, A.; Flusser, J.
Face Recognition Algorithm Based on Fast Computation of Orthogonal Moments. *Mathematics* **2022**, *10*, 2721.
https://doi.org/10.3390/math10152721

**AMA Style**

Abdulhussain SH, Mahmmod BM, AlGhadhban A, Flusser J.
Face Recognition Algorithm Based on Fast Computation of Orthogonal Moments. *Mathematics*. 2022; 10(15):2721.
https://doi.org/10.3390/math10152721

**Chicago/Turabian Style**

Abdulhussain, Sadiq H., Basheera M. Mahmmod, Amer AlGhadhban, and Jan Flusser.
2022. "Face Recognition Algorithm Based on Fast Computation of Orthogonal Moments" *Mathematics* 10, no. 15: 2721.
https://doi.org/10.3390/math10152721