# Applying Ateb–Gabor Filters to Biometric Imaging Problems

^{1}

^{2}

^{*}

## Abstract

**:**

## 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.

## References

- Savchenko, V.V.; Savchenko, A.V. Method for Measuring Distortions in Speech Signals during Transmission over a Communication Channel to a Biometric Identification System. Meas. Tech.
**2021**, 63, 917–925. [Google Scholar] [CrossRef] - Al Solami, E.; Kamran, M.; Alkatheiri, M.S.; Rafiq, F.; Alghamdi, A.S. Fingerprinting of Relational Databases for Stopping the Data Theft. Electronics
**2020**, 9, 1093. [Google Scholar] [CrossRef] - Hrytsyk, V.; Grondzal, A.; Bilenkyj, A. Augmented reality for people with disabilities. In Proceedings of the 2015 Xth International Scientific and Technical Conference Computer Sciences and Information Technologies (CSIT), Lviv, Ukraine, 14–17 September 2015; pp. 188–191. [Google Scholar]
- Valechha, L.; Valecha, H.; Ahuja, V.; Chawla, T.; Sengupta, S. Orisyncrasy—An Ear Biometrics on the Fly Using Gabor Filter. In Advances in Data Sciences, Security and Applications; Springer: Singapore, 2020; pp. 457–466. [Google Scholar]
- Le, N.T.; Wang, J.-W.; Le, D.H.; Wang, C.-C.; Nguyen, T.N. Fingerprint Enhancement Based on Tensor of Wavelet Subbands for Classification. IEEE Access
**2020**, 8, 6602–6615. [Google Scholar] [CrossRef] - Chanukya, P.S.V.V.N.; Thivakaran, T.K. Multimodal biometric cryptosystem for human authentication using fingerprint and ear. Multimed. Tools Appl.
**2019**, 79, 659–673. [Google Scholar] [CrossRef] - Wagh, D.P.; Fadewar, H.S.; Shinde, G.N. Biometric Finger Vein Recognition Methods for Authentication. In Computing in Engineering and Technology; Springer: Singapore, 2020; pp. 45–53. [Google Scholar]
- Lee, C.-J.; Wang, S.-D. Fingerprint feature extraction using Gabor filters. Electron. Lett.
**1999**, 35, 288. [Google Scholar] [CrossRef][Green Version] - Yang, J.; Liu, L.; Jiang, T.; Fan, Y. A modified Gabor filter design method for fingerprint image enhancement. Pattern Recognit. Lett.
**2003**, 24, 1805–1817. [Google Scholar] [CrossRef] - Wang, W.; Li, J.; Huang, F.; Feng, H. Design and implementation of Log-Gabor filter in fingerprint image enhancement. Pattern Recognit. Lett.
**2008**, 29, 301–308. [Google Scholar] [CrossRef] - Areekul, V.; Watchareeruetai, U.; Suppasriwasuseth, K.; Tantaratana, S. Separable Gabor filter realization for fast fingerprint enhancement. In Proceedings of the IEEE International Conference on Image Processing 2005, Genoa, Italy, 11–14 September 2005; IEEE: Piscataway, NJ, USA, 2005; Volume 3, p. III-253. [Google Scholar]
- Usmonov, B.; Evsutin, O.; Iskhakov, A.; Shelupanov, A.; Iskhakova, A.; Meshcheryakov, R. The cybersecurity in the development of IoT embedded technologies. In Proceedings of the International Conference on Information Science and Communications Technologies (ICISCT) 2017, Tashkent, Uzbekistan, 2–4 November 2017; pp. 1–4. [Google Scholar]
- Lytvynenko, V.; Bidyuk, P.; Myrgorod, V. Application of the method and combined algorithm on the basis of immune network and negative selection for identification of turbine engine surging. In Proceedings of the 2nd International Conference on Inductive Modelling, Kyiv, Ukraine, 15–19 September 2008; pp. 116–123. [Google Scholar]
- Hryshchuk, R.; Molodetska, K. Synergetic control of social networking services actors’interactions. In Proceedings of the International Conference on Systems, Control and Information Technologies 2016, Warsaw, Poland, 20–21 May 2016; pp. 34–42. [Google Scholar]
- Bogachuk, I.; Sokolov, V.; Buriachok, V. Monitoring Subsystem for Wireless Systems Based on Miniature Spectrum Analyzers. In Proceedings of the 2018 International Scientific-Practical Conference Problems of Infocommunications. Science and Technology (PIC S&T), Kharkiv, Ukraine, 9–12 October 2018; Institute of Electrical and Electronics Engineers (IEEE): Piscataway, NJ, USA, 2018; pp. 581–585. [Google Scholar]
- Korchenko, O.; Vasiliu, Y.; Gnatyuk, S. Modern quantum technologies of information security against cyber-terrorist attacks. Aviation
**2010**, 14, 58–69. [Google Scholar] [CrossRef][Green Version] - Vacca, J.R. Computer and Information Security Handbook; Elsevier: Amsterdam, The Netherlands, 2012. [Google Scholar]
- Nixon, M.; Aguado, A. Feature Extraction and Image Processing for Computer Vision; Academic Press: Cambridge, MA, USA, 2019. [Google Scholar]
- Nanavati, S.; Steinhardt, B. Face-off: Is the use of biometrics an invasion of privacy. Netw. World
**2000**, 17, 143. [Google Scholar] - Bihl, T.J.; Paciencia, T.J.; Bauer, K.W.; Temple, M.A. Cyber-Physical Security with RF Fingerprint Classification through Distance Measure Extensions of Generalized Relevance Learning Vector Quantization. Secur. Commun. Netw.
**2020**, 2020, 1–12. [Google Scholar] [CrossRef] - Okokpujie, K.; Etinosa, N.O.; John, S.; Joy, E. Comparative analysis of fingerprint preprocessing algorithms for electronic voting processes. In IT Convergence and Security 2017; Springer: Singapore, 2017; pp. 212–219. [Google Scholar]
- Lv, X.; Ding, L.; Zhang, G. Research on fingerprint feature recognition of access control based on deep learning. Int. J. Biomed.
**2021**, 13, 80–95. [Google Scholar] - Nazarkevych, M.; Logoyda, M.; Troyan, O.; Vozniy, Y.; Shpak, Z. The Ateb-Gabor Filter for Fingerprinting. In Proceedings of the Conference on Computer Science and Information Technologies 2019, Lviv, Ukraine, 17–20 September 2019; pp. 247–255. [Google Scholar]
- Barina, D. Gabor wavelets in image processing. arXiv
**2016**, arXiv:1602.03308. [Google Scholar] - Rosenberg, R.M. The Ateb (h)-functions and their properties. Q. Appl. Math.
**1963**, 21, 37–47. [Google Scholar] [CrossRef][Green Version] - Senyk, P.M. About Ateb-functions. Rep. AS URSR Ser. A
**1963**, 1, 23–27. [Google Scholar] - Vozny, A.M. Application of Ateb-functions for construction of solution of one class of essentially nonlinear differential equations. Rep. USSR Acad. Sci. Ser. A
**1970**, 9, 971–974. [Google Scholar] - Senyk, P.M. Inversion of the incomplete beta function. Ukr. Math. J.
**1969**, 21, 271–278. [Google Scholar] [CrossRef] - Mouley, J.; Panja, M.M.; Mandal, B.N. Numerical solution of an integral equation arising in the problem of cruciform crack using Daubechies scale function. Math. Sci.
**2019**, 14, 21–27. [Google Scholar] [CrossRef][Green Version] - Shensa, M.J. The discrete wavelet transform: wedding the a trous and Mallat algorithms. IEEE Trans. Signal Process.
**1992**, 40, 2464–2482. [Google Scholar] [CrossRef][Green Version] - Hore, A.; Ziou, D. Image quality metrics: PSNR vs. SSIM. In Proceedings of the 20th International Conference on Pattern Recognition, Istanbul, Turkey, 23–26 August 2010; pp. 2366–2369. [Google Scholar]
- Wang, J.; Chen, P.; Zheng, N.; Chen, B.; Principe, J.C.; Wang, F.-Y. Associations between MSE and SSIM as cost functions in linear decomposition with application to bit allocation for sparse coding. Neurocomputing
**2021**, 422, 139–149. [Google Scholar] [CrossRef]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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