# Applying Ateb–Gabor Filters to Biometric Imaging Problems

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## Abstract

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## 1. Introduction

- The symmetry of Ateb–Gabor functions enables us to create a great variety and amount of filters, that differ in form and size. All of them supplement Gabor function.
- The basis of wavelet-Ateb–Gabor functions allows us to create a large number of different filters that will effectively convert images into a skeleton and provide fast and reliable image identification, including fingerprints images.
- This filtering method can provide universal filtering, thus reducing the time spent on pre-processing images. This will reduce the pre-processing time of the images by applying the filter shape that will be most desirable.

## 2. Filtering of Biometric Images

#### 2.1. Fingerprint Filtering

#### 2.2. Wavelet Transformation of the Gabor Function

#### 2.3. Ateb-Functions as a New Tool to Develop Filtering

#### 2.4. Models of the Periodic Symmetrical Ateb-Functions

## 3. Wavelet Transformation of the Ateb–Gabor Function

#### 3.1. New Type of Filtering

#### 3.2. Mathematical Model of Wavelet Transform Ateb–Gabor Function

#### 3.3. Wavelet-Ateb–Gabor Function $ATEBG$($a,b,m,n,\theta ,\sigma ,t$) with Different Parameters $\sigma $

#### 3.4. Simulation of Wavelet-Ateb–Gabor Function with Parameters n, 0 < n < 1

#### 3.5. Simulation of Wavelet-Ateb–Gabor Function with Parameters m, 1 < m < 10

#### 3.6. Simulation of Wavelet-Ateb–Gabor Function with Parameters n = m = 3, 1 < σ < 4

## 4. Modeling, Results

#### 4.1. Dataset for Filtering

#### 4.2. Wavelet-Ateb–Gabor Fingerprint Image Filtering

#### 4.3. Comparison of the Efficiency of the Wavelet-Ateb–Gabor Filter with the Existing Ones

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Construction of Wavelet-Gabor filter $\psi \left(\zeta ,\upsilon ,{\zeta}_{0},{\upsilon}_{0},\theta ,\sigma ,\beta \right)$ in OX—$\zeta $, in OY—υ: (

**a**) $\sigma $ = 6; (

**b**) $\sigma $ = 7; (

**c**) $\sigma $ = 8; (

**d**) $\sigma $ = 9; (

**e**) $\sigma $ = 10.

**Figure A2.**Construction of wavelet-Ateb–Gabor filter $\psi \left(\zeta ,\upsilon ,{\zeta}_{0},{\upsilon}_{0},\theta ,\sigma ,\beta \right)$ with parameters m = 1, n = 1 in OX—$a$, in OY—b: (

**a**) $\sigma $ = 6; (

**b**) $\sigma $ = 7; (

**c**) $\sigma $ = 8; (

**d**) $\sigma $ = 9; (

**e**) $\sigma $ = 10.

**Figure A3.**Construction of wavelet-Ateb–Gabor filter $\psi \left(\zeta ,\upsilon ,{\zeta}_{0},{\upsilon}_{0},\theta ,\sigma ,\beta \right)$ with parameters m = 1, $\sigma $ = 1; in OX—$a$, in OY—b: (

**a**) n = 0.5; (

**b**) $n=1$.

**Figure A4.**Construction of Wavelet-Ateb–Gabor filter $\psi \left(\zeta ,\upsilon ,{\zeta}_{0},{\upsilon}_{0},\theta ,\sigma ,\beta \right)$ with parameters n = 1, $\sigma $ = 1; in OX—$a$, in OY—b: (

**a**) m = 5; (

**b**) $m=6$.

**Figure A5.**Construction of wavelet-Ateb–Gabor filter $\psi \left(\zeta ,\upsilon ,{\zeta}_{0},{\upsilon}_{0},\theta ,\sigma ,\beta \right)$ with parameters n = 3, m = 3; in OX—$a$, in OY—b: (

**a**) $\sigma $ = 4; (

**b**) $\sigma $ = 5; (

**c**) $\sigma $ = 6; (

**d**) = 7; (

**e**) = 8; (

**f**) = 9; (

**g**) $\sigma $ = 10.

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**Figure 1.**Construction of wavelet-Gabor filter $\psi \left(\zeta ,\upsilon ,{\zeta}_{0},{\upsilon}_{0},\theta ,\sigma ,\beta \right)$ in OX—$\zeta $, in OY—υ: (

**a**) $\sigma $ = 1; (

**b**) $\sigma $ = 2; (

**c**) $\sigma $ = 3; (

**d**) $\sigma $ = 4; (

**e**) $\sigma $ = 5.

**Figure 3.**Construction of wavelet-Ateb–Gabor filter $ATEBG\left(a,b,m,n,\theta ,\sigma ,t\right)$ in OX—$a$, in OY—b: (

**a**) $\sigma $ = 1; (

**b**) $\sigma $ = 2; (

**c**) $\sigma $ = 3; (

**d**) $\sigma $ = 4; (

**e**) $\sigma $ = 5.

**Figure 4.**Construction of wavelet-Ateb–Gabor filter $\psi \left(\zeta ,\upsilon ,{\zeta}_{0},{\upsilon}_{0},\theta ,\sigma ,\beta \right)$ with parameters m = 1, $\sigma $ = 1; in OX—$a$, in OY—b: (

**a**) n = 0.1; (

**b**) $n=0.2$; (

**c**) $n=0.3$; (

**d**) $n=0.4$.

**Figure 5.**Construction of wavelet-Ateb–Gabor filter $\psi \left(\zeta ,\upsilon ,{\zeta}_{0},{\upsilon}_{0},\theta ,\sigma ,\beta \right)$ with parameters n = 1; $\sigma $ = 1; in OX—$a$, in OY—b: (

**a**) m = 1; (

**b**) $m=2$; (

**c**) $m=3$; (

**d**) $m=4$.

**Figure 6.**Construction of wavelet-Ateb–Gabor filter $\psi \left(\zeta ,\upsilon ,{\zeta}_{0},{\upsilon}_{0},\theta ,\sigma ,\beta \right)$ with parameters n = 3, m = 3; in OX—$a$, in OY—b: (

**a**) $\sigma $ = 1; (

**b**) $\sigma $ = 2; (

**c**) $\sigma $ = 3; (

**d**) $\sigma $ = 4.

**Figure 7.**The examples of images (

**a**) Sample1; (

**b**) Sample 2; (

**c**) Sample 3; (

**d**) Sample 4; (

**e**) Sample 5; (

**f**) Sample 6.

**Figure 8.**Dependence of the change of distortion level of wavelet-Ateb–Gabor filter with Gabor filter according to PSNR depending on the change of the parameters m and n.

**Figure 9.**Dependence of the change of distortion level of wavelet-Ateb–Gabor filter with Gabor filter according to MSE depending on the change of the parameters m and n.

**Figure 10.**Dependence of PSNR between wavelet-Ateb–Gabor filter with Gabor filter on change of parameter σ.

**Figure 11.**Dependence of MSE between wavelet-Ateb–Gabor filter with Gabor filter on change of parameter σ.

**Table 1.**The period of the wavelet function Ateb–Gabor, which is calculated by (6) when n is changing from 0.1 to 1, and m = 1.

$\mathit{m}$ | $\mathit{n}$ | The Period of the Wavelet Function Ateb–Gabor |
---|---|---|

1 | 0.1 | 2.12142061299 |

1 | 0.2 | 2.24050260067 |

1 | 0.3 | 2.35762298776 |

1 | 0.4 | 2.47307918393 |

1 | 0.5 | 2.58710955923 |

1 | 0.6 | 2.6999077953 |

1 | 0.7 | 2.81163314784 |

1 | 0.8 | 2.92241794389 |

1 | 0.9 | 3.03237316197 |

1 | 1 | 3.14159265359 |

$\mathit{m}$ | $\mathit{n}$ | The Period of the Wavelet Function Ateb–Gabor |
---|---|---|

1 | 1 | 3.14159265359 |

2 | 1 | 4.20654631598 |

3 | 1 | 5.24411510858 |

4 | 1 | 6.26865312409 |

5 | 1 | 7.28595194366 |

6 | 1 | 8.29880821421 |

7 | 1 | 9.30874056975 |

8 | 1 | 10.3166455868 |

9 | 1 | 11.3230869752 |

10 | 1 | 12.3284370431 |

$\mathit{m}$ | $\mathit{n}$ | The Period of the Wavelet Function Ateb–Gabor |
---|---|---|

3 | 3 | 7.41629870921 |

Ateb Filtering, m | Comparison, m | Filtration Time | Sample 1 | Sample 2 | Sample 3 | Sample 4 | Sample 5 | Sample 6 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PSNR | MSE | PSNR | MSE | PSNR | MSE | PSNR | MSE | PSNR | MSE | PSNR | MSE | |||

1 | 1 | 1 min 54 s | 39.20 | 10.5 | 40.05 | 12.03 | 37.33 | 10.34 | 40.38 | 11.18 | 31.49 | 8.72 | 37.93 | 10.50 |

0.9 | 1 | 1 min 55 s | 38.77 | 10 | 39.59 | 10.96 | 31.71 | 8.78 | 34.43 | 9.53 | 31.49 | 8.72 | 31.86 | 8.82 |

0.8 | 1 | 1 min 54 s | 33.06 | 9.15 | 34.03 | 9.42 | 28.75 | 7.96 | 30.18 | 8.36 | 28.43 | 7.87 | 28.43 | 7.87 |

0.7 | 1 | 1 min 54 s | 29.66 | 8.21 | 30.82 | 8.53 | 26.60 | 7.36 | 26.06 | 7.21 | 26.37 | 7.87 | 25.37 | 7.02 |

0.6 | 1 | 2 min 3 s | 27.08 | 7.49 | 28.59 | 7.91 | 24.98 | 6.92 | 22.39 | 6.20 | 24.86 | 6.88 | 23.02 | 6.37 |

0.5 | 1 | 1 min 57 s | 24.95 | 6.91 | 26.91 | 7.45 | 23.84 | 6.60 | 20.34 | 5.63 | 12.51 | 3.46 | 21.23 | 5.88 |

0.4 | 1 | 2 min 1 s | 23.35 | 6.46 | 25.47 | 7.05 | 15.0 | 7.75 | 19.96 | 5.52 | 12.52 | 3.47 | 19.70 | 5.45 |

0.3 | 1 | 1 min 53 s | 22.20 | 6.14 | 24.79 | 6.86 | 2.65 | 9.58 | 19.75 | 5.47 | 12.70 | 3.52 | 19.09 | 5.28 |

0.2 | 1 | 1 min 55 s | 20.83 | 5.77 | 24.07 | 6.66 | 2.73 | 9.86 | 19.41 | 5.37 | 22.28 | 6.17 | 18.51 | 5.13 |

0.1 | 1 | 2 min 11 s | 19.35 | 5.35 | 23.10 | 6.39 | 3.27 | 11.83 | 2.65 | 9.58 | 19.54 | 5.41 | 2.65 | 9.58 |

Ateb Filtering m = 1, n = 1, σ | Comparison m = 1, n = 1, σ | Filtration Time | Sample 1 | Sample 2 | Sample 3 | Sample 4 | ||||
---|---|---|---|---|---|---|---|---|---|---|

PSNR | MSE | PSNR | MSE | PSNR | MSE | PSNR | MSE | |||

π/4 | π | 2 min 18 s | 12.77 | 3.14 | 4.56 | 17.01 | 4.76 | 17.21 | 17.59 | 4.87 |

π/3 | π | 2 min 3 s | 3.61 | 3.54 | 4.18 | 15.09 | 4.28 | 15.49 | 17.40 | 4.82 |

π/2 | π | 2 min 15 s | 4.03 | 14.55 | 4.00 | 16.06 | 4.50 | 16.26 | 17.22 | 4.76 |

2 × π | π | 2 min 15 s | 4.724 | 17.07 | 4.61 | 16.67 | 4.61 | 16.67 | 17.04 | 4.72 |

3 × π | π | 1 min 58 s | 4.25 | 15.35 | 4.63 | 16.72 | 4.63 | 16.72 | 17.00 | 4.71 |

4 × π | π | 2 min 3 s | 3.94 | 14.25 | 4.55 | 16.70 | 4.55 | 16.70 | 17.00 | 4.71 |

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**MDPI and ACS Style**

Nazarkevych, M.; Kryvinska, N.; Voznyi, Y.
Applying Ateb–Gabor Filters to Biometric Imaging Problems. *Symmetry* **2021**, *13*, 717.
https://doi.org/10.3390/sym13040717

**AMA Style**

Nazarkevych M, Kryvinska N, Voznyi Y.
Applying Ateb–Gabor Filters to Biometric Imaging Problems. *Symmetry*. 2021; 13(4):717.
https://doi.org/10.3390/sym13040717

**Chicago/Turabian Style**

Nazarkevych, Mariia, Natalia Kryvinska, and Yaroslav Voznyi.
2021. "Applying Ateb–Gabor Filters to Biometric Imaging Problems" *Symmetry* 13, no. 4: 717.
https://doi.org/10.3390/sym13040717