Applying Ateb–Gabor Filters to Biometric Imaging Problems

S S symmetry

Article Applying Ateb–Gabor Filters to Biometric Imaging Problems

Mariia Nazarkevych 1 , Natalia Kryvinska 2,* and Yaroslav Voznyi 1

1 Department of Publishing Information Technologies, Lviv Polytechnic National University, 12 Bandera Str., 79000 Lviv, Ukraine; [email protected] (M.N.); [email protected] (Y.V.) 2 Faculty of Management, Comenius University in Bratislava, Odbojárov 10, 820 05 Bratislava, Slovakia * Correspondence: [email protected]

Abstract: This article presents a new method of image filtering based on a new kind of image processing transformation, particularly the wavelet-Ateb–Gabor transformation, that is a wider basis for Gabor functions. Ateb functions are symmetric functions. The developed type of filtering makes it possible to perform image transformation and to obtain better biometric image recognition results than traditional filters allow. These results are possible due to the construction of various forms and sizes of the curves of the developed functions. Further, the wavelet transformation of Gabor filtering is investigated, and the time spent by the system on the operation is substantiated. The filtration is based on the images taken from NIST Special Database 302, that is publicly available. The reliability of the proposed method of wavelet-Ateb–Gabor filtering is proved by calculating and comparing the values of peak signal-to-noise ratio (PSNR) and mean square error (MSE) between two biometric images, one of which is filtered by the developed filtration method, and the other by the Gabor filter. The time characteristics of this filtering process are studied as well.

Keywords: wavelet-Ateb–Gabor transformation; ﬁltration; biometric images

Citation: Nazarkevych, M.; Kryvinska, N.; Voznyi, Y. Applying Ateb–Gabor Filters to Biometric 1. Introduction Imaging Problems. Symmetry 2021, 13, The field of biometric identification systems (BISs) is developing quickly [1]. These sys- 717. https://doi.org/10.3390/ tems attract special attention due to their high reliability and recent significant price reduc- sym13040717 tions, and are often used for real-time recognition tasks [2]. Biometric security systems are gradually displacing traditional security systems Academic Editor: Dumitru Baleanu based on the entry of passwords, PIN codes, which are personal identification numbers, and printed IDs. Systems of this kind are an identification technology with various Received: 21 March 2021 means of protection and are used to compare the obtained physiological or behavioral Accepted: 13 April 2021 characteristics of a particular person. Published: 19 April 2021 The development of biometrics is the result of the wide usage of embedded miniature sensors in small personal devices such as smartphones, deployment of biometric data in Publisher’s Note: MDPI stays neutral various data systems and large-scale identification, and study of usage of biometric features with regard to jurisdictional claims in in wearable devices [3]. published maps and institutional affil- It is a common knowledge, that almost every security biometric system implicitly iations. imposes a certain type of restriction on its users. Such systems require the creation of a certain type of environment during data collection. It is also important to allocate enough memory to store the collected information. However, cheaper biometric sensors and the rapid development of smartphones, smart watches, and various other devices provide Copyright: © 2021 by the authors. wider opportunities for researchers. Thus, today, we observe a revolution in devices with Licensee MDPI, Basel, Switzerland. biometric logins. This article is an open access article Such devices store the personal information of the owner of the device or person who distributed under the terms and has the right to use this device. These devices that enable biometric authentication are conditions of the Creative Commons equipped with biometric sensors and are able to constantly check the owner of the certain Attribution (CC BY) license (https:// biometric data. Therefore, these attributes of the individuals can be used as a biometric creativecommons.org/licenses/by/ 4.0/). signature of the person.

Symmetry 2021, 13, 717. https://doi.org/10.3390/sym13040717 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 717 2 of 17

In [4], biometric images are viewed from different angles and different orientations. They are filtered by the Gabor function. The system is invariant for the rotation of the face of the profile in one or different lanes. Based on that fact, we can claim that Gabor filtration is promising. In [5], it is established a secure and reliable algorithm for distinguishing fingerprints from counterfeits, using circular Gabor Wavelet to segment the textures of captured images. The samples are subjected to extraction functions using a circular Gabor wavelet algorithm designed for segmentation. The enhancement of the biometric images is extremely essential during the deployment of automated fingerprint identification systems. In [6], an authentication method based on a new lighting compensation scheme when scanning fingerprints was developed. The proposed compensation scheme involves a new method of light decomposition and compensation of external lighting. Such approach enables the scholars to decently improve the image quality. The results of the experiments show that the image quality after applying the adaptive higher-order singular value decomposition on a tensor of wavelet subbands of a fingerprint (AHTWF) method is higher than the quality of the original images. The AHTWF method is recommended to apply to fingerprint images that have ridge and concavity structures, and also contain a lot of blurred areas. As the outputs of the fingerprint scanning can often be of the low quality, the AHTWF method is proposed to highly improve the higher accuracy of fingerprint classification. Multimodal biometrics is mainly used for certification and identity verification. Many biometric data are used for human authentication. Biometric recognition involves three vital phases, which include preprocessing, trait detection, and classification. Thus, preprocessing included using filters and image cropping, texture overlay, and shape creation [7]. Today, the security of personal information is very important. A huge amount of personal data and files are stored on servers with closed access. Access to private information is provided only to authorized users. Biometrics plays a big role in security. Nowadays, spe- cialists have begun to study the images of finger veins very actively. The great advantage of vein patterns on the fingers is that they cannot be copied or stolen. The use of basic fingers for authenticity is widely used in Japan as a database at airports, identity authentication, prisons, etc. The main attention is paid to the comparative study of existing methods of pattern recognition of veins, the technique of creating features [8]. A method based on the Gabor filter is introduced to extract fingerprint functions actively from images without pre-processing. The proposed method is more efficient than conventional methods for a small fingerprint recognition system. The results of experiments claim that the recognition coefficient of the k-nearest classifier of neighbors using the proposed functions based on the Gabor filter is 97.2% [9]. Improving image prints is an important step in pre-processing in fingerprint recognition programs. In [10], they proposed a new filter design method to improve the fingerprint image, mainly the traditional Gabor filter (TGF). Its parameter selection scheme does not depend on the image. The advantages are based on maintaining the structure of the fingerprint image and achieving a sequence of image enhancement. In [10], the Log-Gabor method is developed. It is used to solve the issue of improving fingerprint recognition and widen the usage of common Gabor filter. The presented method is important for the reliability of identification and other issues related to the fingerprint recognition. The Gabor filter can be divided into a band-pass filter and a low-pass filter. The orientation aspect is introduced to implement the quicker Gabor filter processing. The division into a band-pass filter and a low-pass filter enables to significantly reduce the computational complexity, by 1:2.8 times. Thus, the filter execution time decreases by (1:1.46), memory space lessens, and biometric image quality significantly improves compared to a traditional Gabor filter [11]. The works of Shelupanov [12] and Bidyuk [13] are devoted to the problem of biometric identification in which special attention is paid to the control and security of the system. Symmetry 2021, 13, 717 3 of 17

The work [14], where new approach to the protection of information systems is studied, deserves a special attention. In the works of [15], they considered attacks on biometric protection systems. A study [16] showed that biometric protection depends on many factors that will shape security pol- icy. Among foreign researchers, it is worth noting the works of Vacca [17], Nixon [18], and Nanavati [19]. The main contribution of this paper can be summarized as follows: • 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 Biometric systems have good performance with good input quality. However, their performance deteriorates sharply when poor quality input enters the system [20]. Examples of bad data can be noisy scans with low resolution fingerprints. Therefore, it will be promising to develop methods to improve the quality of such data so that the biometric content of protection could work faster and with fewer errors [21]. However, on the raw image due to noise, the print lines may be distorted, creating recognition errors [22]. To implement this, the image will be enhanced by applying filtering to reduce its noise. The existing Gabor filter based on Ateb-functions is effective for filtering [23]. Random noise is manifested in the form of chaotic granularities or extraneous points in the image. This noise is most noticeable in dark areas of the image, as the signal-to-noise ratio will be much lower than in bright areas. The difficulty of finding exact solutions gives rise to different variants of approximate methods. Filtration of digital images is based on the implementation of the physical process of absorption and reflection of light. The essence of the method is as follows. The original image is represented as a surface, each point of which corresponds to a pixel in the image. At these points, the reflection coefficients are calculated based on their proximity to the points that are black in the binary image of the print. For all points of the surface, this coefficient and the value of the intensity of the reflected light are determined.

2.2. Wavelet Transformation of the Gabor Function Let us consider the Fourier transformation of the simpliﬁed complex Gabor ﬁlter:

(( − ) +( − ) )2 2 (−( − ) +( − ) )2 2 − 1 [ ζ ζ0 cos θ υ υ0 sin θ σ + ζ ζ0 sin θ υ υ0 cos θ β ] 1 2 2σ2 2β2 ψ(ζ, υ, ζ0, υ0, θ, σ, β) = e , (1) pπσβ

where ζ, υ are the spatial frequencies along the x and y axes, σ is the standard deviation of the Gaussian nucleus, θ is the filter orientation angle, β is the filter phase. The center of the filter is located at the point (ξ = ξ0, υ = υ0) of the raster frequency plane. σ and β are scale parameters of elliptical Gaussians along the x and y axes [24]. Since these calculations are of great computational complexity, there are performed several simplifications. The parameters σ, β can be arbitrarily restricted β/σ = 2. Also:

ζ0 = ω0 cos θ,

ν0 = ω0 sin θ, Symmetry 2021, 13, 717 4 of 17 Symmetry 2021, 13, x FOR PEER REVIEW 4 of 17

q where frequency 𝜔 = 2𝜁 +𝜈2 where frequency ω0 = ζ0 + ν0 . FigureFigure1 shows 1 shows the the surface surface of theof the Gabor Gabor function functionψ( ζ𝜓,(υ𝜁,𝜐,𝜁, ζ0, υ0,𝜐, θ,,𝜃,𝜎,βσ, β) for) for different different valuesvalues of ofσ. When𝜎. When parameter parameterσ increases,σ increases, the the function function becomes becomes more more convex. convex. Therefore, Therefore, whenwhen we we apply apply this this function function for for ﬁltration, filtration, a biggera biggerσ meansσ means a signiﬁcanta significant change change in in the the brightnessbrightness of colorsof colors in thein the image. image. The The surfaces surfac ofes the of Gaborthe Gabor function function with with larger larger parameters parame- σ,ters in particular σ, in particularσ = 6, 7,σ = 8, 6, 9, 7, 10 8, are 9, 10 given are ingiven Appendix in AppendixA Figure A FigureA1. A1.

(a) (b)

(e)

Figure 1. Construction ofFigure wavelet-Gabor 1. Construction filter 𝜓( of𝜁,𝜐,𝜁 wavelet-Gabor,𝜐,𝜃,𝜎,β) in ﬁlter OX—ψ(𝜁ζ, ,inυ, OY—ζ0, υ0,υθ: ,(σa,) β𝜎) =in 1; OX— (b) 𝜎ζ , = in2; OY—(c) 𝜎 υ= :(3;a ()dσ) 𝜎= 1; = 4; (e) 𝜎 = 5. (b) σ = 2; (c) σ = 3; (d) σ = 4; (e) σ = 5.

2.3.2.3. Ateb-Functions Ateb-Functions as as a Newa New Tool Tool to Developto Develop Filtering Filtering InIn the the 1960s, 1960s, Rosenberg Rosenberg first first defined defined Ateb-functions Ateb-functions as as the the inversion inversion of of incomplete incomplete Beta-functionsBeta-functions [25 [25]] In In particular, particular, the the entry entry Ateb Ateb is ais reversea reverse permutation permutation of of the the letters letters of of thethe word word Beta. Beta. Rosenberg, Rosenberg, using usin Ateb-functions,g Ateb-functions, wrote wrote an analytical an analytical solution solution of differential of differen- equationstial equations with power with power nonlinearity nonlinearity to describe to describe oscillatory oscillatory motion. motion. SomewhatSomewhat later, later, Senyk Senyk [26 [26]] generalized generalized and and investigated investigated thethe functionalfunctional properties of ofAteb-functions. Ateb-functions. The The introduced introduced periodic periodic and and hyperbolic hyperbolic Ateb-functions Ateb-functions are are used used to tocon- constructstruct solutions solutions of of systems systems of of no nonlinearnlinear differential differential equations. equations. TheThe solution solution of of these these equations equations was was built built using using periodic periodic symmetrical symmetrical Ateb-functions. Ateb-functions. SenykSenyk used used solutions solutions of of Ateb-functions Ateb-functions inin the study study of of stationary stationary oscillations oscillations in inessentially essen- tiallynonlinear nonlinear systems systems that that interact interact with with the theenergy energy source. source. In particular, In particular, periodic periodic Ateb-func- Ateb- functionstions were were used used to toconstruct construct resonant resonant modes. modes. InIn the the 1970s, 1970s, a studenta student of of Senyk—A.M. Senyk—A.M. Vozny Vozny [27 [27]—for]—for the the first first time time decomposed decomposed hyperbolic and periodic Ateb-functions into Taylor series in the vicinity of the initial value hyperbolic and periodic Ateb-functions into Taylor series in the vicinity of the initial value of the argument ω = 0 at n = 1. Vozny used Ateb-functions to study the motion of an of the argument 𝜔 = 0 at n = 1. Vozny used Ateb-functions to study the motion of an object with constant mass under the action of frontal resistance forces proportional to object with constant mass under the action of frontal resistance forces proportional to the nonlinear velocity, the dynamic part of the problem was solved, which is reduced to minimizing the integral.

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the nonlinear velocity, the dynamic part of the problem was solved, which is reduced to minimizing the integral.

2.4. Models of the Periodic Symmetrical Ateb-Functions Let n, m be parameters deﬁned by the relations:

...... 2Q1 + 1 2Q2 + 1 n = .. , m = .. , Q1, Q1, Q2, Q2 = 0, 1, 2, . . . (2) 2Q1 + 1 2Q2 + 1 Symmetry 2021, 13, x FOR PEER REVIEW 5 of 17 Let us consider the expression: 2.4. Models of the Periodic Symmetrical Ateb-Functions Z −1≤u≤1 Let n, m be parameters defined by1 the relations:− n − m ω = t n+1 (1 − t) m+1 dt (3) 2 1 𝑛 = , 𝑚 = , 𝑄 ,𝑄 ,𝑄,𝑄 = 0,1,2,… (2) Substituting variables of the form t = 1 − um+1 from formula (3) we obtain the ratio: Let us consider the expression: n Z−1≤u≤1 − 1 m + 1 m+1 n+1 𝜔 = 𝑡ω = −(1 − 𝑡) 𝑑𝑡 1 − u du (3) (4) 2 2 1

+ SubstitutingThe dependence variables uofon theω formfor thetu=−1 integralm 1 from (4) formula is a function (3) we obtainn and them andratio: is called the Cos(ω) of the Ateb-function𝑚+1 and is denoted by: 𝜔 = − (1−𝑢) 𝑑𝑢 (4) 2 u = ca(m, n, ω) (5) The dependence u on ω for the integral (4) is a function 𝑛 and m and is called the Cos(ω) Ifof nthe= Ateb-functionm = 1, then we and obtain is denotedu = Cos( by: ω). It was also proved𝑢 = in𝑐𝑎(𝑚, [27] that 𝑛,𝜔) the Ateb-functions introduced in this way(5) are periodic with period Π (m, n) where [28]: If n = m = 1, then we obtain u = Cos (ω). It was also proved in [27] that the Ateb-functions introduced in this way are periodic 1 1 with period Π (m, n) where [28]: Γ n+1 Γ m+1 Π(m, n)= . (6) 1 1 Γ + + + Π(𝑚, 𝑛) = . n 1 m 1 (6) In In(6) (6)Γ (1Γ /(1/( (n +n 1)),+ 1)),Γ (1 Γ/ (1/((m +m 1))+ is 1)) a gamma is a gamma function. function. Therefore,Therefore, if necessary, if necessary, the calculation the calculation of functions of functions is sufficient is sufﬁcientfor the argument for the ω argument ω thatthat varies varies on the on thesegment: segment: 1 0 ≤1ω ≤ Π(m, n) 0≤𝜔≤ Π(𝑚, 𝑛) 2 2 UsingUsing the theproperties properties of periodicity, of periodicity, you can you continue can continue a series of acalculations. series of calculations. Thus, it Thus, is iteasy is easyto obtain to obtain the value the of value the entire of the period entire of period the Ateb-function. of the Ateb-function.

3. 3.Wavelet Wavelet Transformation Transformation of the of Ateb–Gabor the Ateb–Gabor Function Function 3.1.3.1. New New Type Type of Filtering of Filtering WeWe have have proposed proposed a new a newtype of type filtering, of filtering, which is which built by is developing built by developing a new method a new method of oftransforming transforming the wavelet-Ateb–Gabor, the wavelet-Ateb–Gabor, which whichwill allow will the allow use of the a wider use of range a wider of range of filtrationfiltration sets sets and and better better filter filterbiometric biometric images. images. A new method A new methodfor classifying for classifying fingerprint fingerprint imagesimages by bytypes types of papillary of papillary patterns patterns based based on the on usage the usageof the Ateb–Gabor of the Ateb–Gabor filter, which filter, which is is implementedimplemented by by wavelet wavelet transforms, transforms, is prop is proposed.osed. The functional The functional scheme scheme of identifica- of identification tionis shownis shown in in Figure Figure2 2..

FigureFigure 2. Basic 2. Basic steps steps of wavelet-Ateb–Gabor of wavelet-Ateb–Gabor filtering. ﬁltering.

The functional identification scheme allows you to reduce the time spent on computing Ateb–Gabor functions and on image filtering and reduce computational costs. We can influence the decision of results by time and frequency. This is used in Fourier transforms.

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The functional identification scheme allows you to reduce the time spent on computing Ateb–Gabor functions and on image filtering and reduce computational costs. We can influence the decision of results by time and frequency. This is used in Fourier transforms.

3.2. Mathematical Model of Wavelet Transform Ateb–Gabor Function The wavelet transform of the function f (t) will look like:

Z ∞ 1 ∗ t − b W(a, b) = p f (t)ψ dt, (7) |a| −∞ a

∗( t−b ) where ψ a —is a complex conjugation of the function ψ(t) the parameter b ∈ R corresponds to the time shift, and is called shift parameter, and the parameter a speciﬁes the scaling and is a stretch parameter.

2 ψ(t) = 1 − t2 e−t /2, (8)

Wavelet functions are characterized by independent frequency and time values. There- fore it is possible to simulate processes in time and frequency domains with their help. The best filters for image preprocessing are Haar wavelets, Daubechies, WAVE, MHAT, which are described in [29]. When solving classification problems, it is possible to change the shape of the wavelet transformation by introducing a discrete wavelet transformation [30].

N−1 1 ∗ i − b Wψ(a, b) = p ∑ f (t)ψ , |a| i=0 a

Using (7) and (8) we deﬁne ψa,b(t) as:

(t−b)2 (t−b)2 − 1 − ·e 2a2 1 t − b a2 ψ (t) = √ ψ = √ , (9) a,b a a a

The function f (m, n, t), which is an Ateb–Gabor, has the form:

2 − t f (m, n, t) = e 2σ2 Ca(m, n, 2Π, θ, t), (10)

2 − t where Ca(m, n, 2Π, θ, t) is an Ateb function, e 2σ2 is the deviation of the Gaussian nucleus, which determines the amplitude of the function, σ—is standart deviation of the outer Gaussian function. The integral transformation of the wavelet function Ateb–Gabor ATEBG(a, b, m, n, θ, σ, t) according to (7) and (10) can be calculated as:

Z ∞ 2 1 − t ∗ t − b ATEBG(a, b, m, n, θ, σ, t) = p e 2σ2 Ca(m, n, 2Π, θ, t)ψ dt (11) |a| −∞ a

After some transformations, we get:

2 t2 Z ∞ t2 4σ − 2 − n ATEBG(a, b, m, n, θ, σ, t) = e 2σ2 Ca(m, n, 2Π, θ, t) + e 2σ2 Sa (n, m, 2Π, θ, t)dt. (12) t m + 1 −∞

3.3. Wavelet-Ateb–Gabor Function ATEBG(a, b, m, n, θ, σ, t) with Different Parameters σ As a result of visualization of formulas (1) and (12), the surfaces of the wavelet transform Ateb–Gabor function ATEBG(a, b, m, n, θ, σ, t) with different parameters were constructed (Figures2–5 and AppendixA Figure A2), which helped in the future to conduct better ﬁltration. Symmetry 2021, 13, 717 7 of 17

Symmetry 2021, 13, x FOR PEER REVIEW 7 of 17

(a) (b)

(e)

Figure 3. Construction ofFigure wavelet-Ateb–Gabor 3. Construction filter of wavelet-Ateb–Gabor 𝐴𝑇𝐸𝐵𝐺(𝑎, 𝑏, 𝑚, ﬁlter 𝑛, 𝜃,𝜎,𝑡) inATEBG OX—𝑎(, ain, b OY—, m, n,bθ: ,(σa), t𝜎) in = 1; OX— (b) 𝜎a , = in 2; OY— (c) 𝜎 b: = 3; (d) 𝜎 = 4; (e) 𝜎 = 5. Symmetry 2021, 13, x FOR PEER REVIEW(a) σ = 1; (b) σ = 2; (c) σ = 3; (d) σ = 4; (e) σ = 5. 8 of 17

3.4. Simulation of Wavelet-Ateb–Gabor Function with Parameters n, 0 < n < 1 The numerical values of the period of the wavelet of the Ateb–Gabor function were calculated (Table 1). The values of m = 1 were taken to calculate the period, and n was changed from 0.1 to 1.

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.

𝒎 𝒏 The Period of the Wavelet Function Ateb–Gabor (a) (b) 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( c) 0.9 (d) 3.03237316197 1 ( 1 ) 3.14159265359 Figure 4. Construction ofFigure wavelet-Ateb–Gabor 4. Construction filter of wavelet-Ateb–Gabor𝜓 𝜁,𝜐,𝜁,𝜐,𝜃,𝜎,β with ﬁlter parametersψ(ζ, υ, ζ0, υm0, =θ ,1,σ ,𝜎β ) =with 1; in parametersOX—𝑎, in OY—m = 1, b: (a) n = 0.1; (b) 𝑛 = 0.2σ; (=c) 1;𝑛 in OX—For = 0.3; the(ad,) in 𝑛case OY— = when 0.4b:(. a) nm= = 0.1;1, σ (b =) n1 =and0.2; the (c) parametern = 0.3; (d) n =acquires0.4. numbers less than one, we obtain such plots of ATEBG (a, b, m, n, θ, σ, t): as shown in Figure 4, or in Appendix A FigureWhen A3. n is greater than one, we obtain such plots of ATEBG (a, b, m, n, θ, σ, t) as shown in Figures 5 and 6, or in the Appendix A Figures A4 and A5.

3.5. Simulation of Wavelet-Ateb–Gabor Function with Parameters m, 1 < m < 10 The numerical values of the period of the wavelet of the Ateb–Gabor function were calculated, when m is changing from 1 to 10, and n = 1 (Table 2).

Table 2. The period of the wavelet function Ateb–Gabor when m is changing from 1 to 10, and n = 1.

The Period of the Wavelet Function Ateb–Ga- 𝒎 𝒏 bor 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

(a) (b)

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(a) (b)

Figure 4. Construction of wavelet-Ateb–Gabor filter 𝜓(𝜁,𝜐,𝜁,𝜐,𝜃,𝜎,β) with parameters m = 1, 𝜎 = 1; in OX—𝑎, in OY— b: (a) n = 0.1; (b) 𝑛 = 0.2; (c) 𝑛 = 0.3; (d) 𝑛 = 0.4.

When n is greater than one, we obtain such plots of ATEBG (a, b, m, n, θ, σ, t) as shown in Figures 5 and 6, or in the Appendix A Figures A4 and A5.

Table 2. The period of the wavelet function Ateb–Gabor when m is changing from 1 to 10, and n = 1.

Symmetry 2021, 13, x FOR PEER REVIEW 9 of 17 (a) (b)

Figure 5. Construction ofFigure wavelet-Ateb–Gabor 5. Construction offilter wavelet-Ateb–Gabor 𝜓(𝜁,𝜐,𝜁,𝜐,𝜃,𝜎,β ﬁlter) withψ(ζ parameters, υ, ζ0, υ0, θ, nσ ,=β 1;) with𝜎 = parameters1; in OX—𝑎n, in= 1;OY—σ = 1; b: (a) m = 1; (b) 𝑚 = ; 2 (inc) OX—𝑚 =a , 3; in(d OY—) 𝑚 b:(= a. 4) m = 1; (b) m = 2; (c) m = 3; (d) m = 4.

FigureFrom3 shows Figure the 5 surfaceit follows of thethat wavelet-Ateb–Gabor the surface of the wavelet-Ateb–Gabor function ATEBG(a, functionb, m, n, θ, ATEBGσ, t) with(a, the b, m parameters, n, θ, σ, t)m with= 1, parametersn = 1. When n = we 1, increaseσ = 1; when parameter we increaseσ, the functionthe parameter becomes m we moreobserve convex, different so the ﬁltering shape and effect value. will be At applied m = 1 and to a largerm = 2 areathere of is the a image.surface The with greater positive σ (asvalues, seen inbut Figure when2 ),m the= 3 largerand m the= 4, more the image signiﬁcant will be change illumin inated brightness in those in areas the imagewhere the cansurface be observed. is located. The Investigation plots for σ = 6, of 7, the 8, 9,wavelet-Ateb–Gabor 10 are given in Appendix functionA Figure ATEBG A2 (a., b, m, n, θ, σ, t) with parameters n = 1, σ = 1, and m = 5, 6 are given in Appendix A Figure A4. 3.4. Simulation of Wavelet-Ateb–Gabor Function with Parameters n, 0 < n < 1 3.6.The Simulation numerical of valuesWavelet-Ateb–Gabor of the period Function of the wavelet with Parameters of the Ateb–Gabor n = m = 3, 1 function < σ < 4 were calculatedExperiments (Table1). Theof the values wavelet-Ateb–Gabor of m = 1 were taken function to calculate ATEBG ( thea, b, period,m, n, θ, andσ, t) weren was per- changedformed from separately 0.1 to 1. with equal parameters m and n. To do this, the period presented in Table 3 was calculated. Table 1. The period of the wavelet function Ateb–Gabor, which is calculated by (6) when n is changing fromTable 0.1 to 3. 1, The and periodm = 1. of the wavelet function Ateb–Gabor with equal parameters m = n.

m n The PeriodThe of the period Wavelet Of Functionthe Wavelet Ateb–Gabor Function 𝒎 𝒏 1 0.1 2.12142061299Ateb–Gabor 13 0.23 2.24050260067 7.41629870921 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

For the case when m = 1, σ = 1 and the parameter n acquires numbers less than one, we obtain such(a plots) of ATEBG(a, b, m, n, θ, σ, t):( asb) shown in Figure4, or in AppendixA Figure A3. When n is greater than one, we obtain such plots of ATEBG(a, b, m, n, θ, σ, t) as shown in Figures5 and6, or in the AppendixA Figures A4 and A5.

Figure 6. Construction of wavelet-Ateb–Gabor filter 𝜓(𝜁,𝜐,𝜁,𝜐,𝜃,𝜎,β)with parameters n = 3, m = 3; in OX—𝑎, in OY—b: (a) 𝜎 = 1; (b) 𝜎 = 2; (c) 𝜎 = 3; (d) 𝜎 = 4.

From Figure 6 we can see that for large and equal values of the parameter n = m = 3 the surface of the wavelet-Ateb–Gabor function ATEBG (a, b, m, n, θ, σ, t) changes with spiked peaks with increasing deviation of the Gaussian nucleus, σ = 1; 2; 3; 4. As can be

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Figure 5. Construction of wavelet-Ateb–Gabor filter 𝜓(𝜁,𝜐,𝜁,𝜐,𝜃,𝜎,β) with parameters n = 1; 𝜎 = 1; in OX—𝑎, in OY— b: (a) m = 1; (b) 𝑚 = ; 2 (c) 𝑚 = 3; (d) 𝑚 = . 4

From Figure 5 it follows that the surface of the wavelet-Ateb–Gabor function ATEBG (a, b, m, n, θ, σ, t) with parameters n = 1, σ = 1; when we increase the parameter m we observe different shape and value. At m = 1 and m = 2 there is a surface with positive values, but when m = 3 and m = 4, the image will be illuminated in those areas where the surface is located. Investigation of the wavelet-Ateb–Gabor function ATEBG (a, b, m, n, θ, σ, t) with parameters n = 1, σ = 1, and m = 5, 6 are given in Appendix A Figure A4.

3.6. Simulation of Wavelet-Ateb–Gabor Function with Parameters n = m = 3, 1 < σ < 4 Experiments of the wavelet-Ateb–Gabor function ATEBG (a, b, m, n, θ, σ, t) were performed separately with equal parameters m and n. To do this, the period presented in Table 3 was calculated.

Table 3. The period of the wavelet function Ateb–Gabor with equal parameters m = n.

The period Of the Wavelet Function 𝒎 𝒏 Symmetry 2021, 13, 717 Ateb–Gabor 9 of 17 3 3 7.41629870921

(a) (b)

(c) (d) 𝜓(𝜁,𝜐,𝜁 ,𝜐 ,𝜃,𝜎,β) 𝑎 Figure 6. Construction ofFigure wavelet-Ateb–Gabor 6. Construction filter of wavelet-Ateb–Gabor with ﬁlter parametersψ(ζ, υ, ζ0, nυ 0=, θ3,, σm, β=) 3;with in OX— parameters, in OY—n =b: 3, (a) 𝜎 = 1; (b) 𝜎 = 2; (c) 𝜎m == 3; 3; ( ind) OX—𝜎 = 4.a, in OY—b:(a) σ = 1; (b) σ = 2; (c) σ = 3; (d) σ = 4.

3.5. SimulationFrom Figure of Wavelet-Ateb–Gabor 6 we can see that Function for large with and Parameters equal values m, 1 of < mthe < 10parameter n = m = 3 theThe surface numerical of the values wavelet-Ateb–Gabor of the period of function the wavelet ATEBG of the (a, Ateb–Gabor b, m, n, θ, σ function, t) changes were with calculated,spiked peaks when with m is increasing changing from devia 1tion to 10, of and the nGaussian= 1 (Table nucleus,2). σ = 1; 2; 3; 4. As can be

Table 2. The period of the wavelet function Ateb–Gabor when m is changing from 1 to 10, and n = 1.

m 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

From Figure5 it follows that the surface of the wavelet-Ateb–Gabor function ATEBG(a, b, m, n, θ, σ, t) with parameters n = 1, σ = 1; when we increase the parameter m we observe different shape and value. At m = 1 and m = 2 there is a surface with positive values, but when m = 3 and m = 4, the image will be illuminated in those areas where the surface is located. Investigation of the wavelet-Ateb–Gabor function ATEBG(a, b, m, n, θ, σ, t) with parameters n = 1, σ = 1, and m = 5, 6 are given in AppendixA Figure A4.

3.6. Simulation of Wavelet-Ateb–Gabor Function with Parameters n = m = 3, 1 < σ < 4 Experiments of the wavelet-Ateb–Gabor function ATEBG(a, b, m, n, θ, σ, t) were performed separately with equal parameters m and n. To do this, the period presented in Table3 was calculated.

Table 3. The period of the wavelet function Ateb–Gabor with equal parameters m = n.

m n The Period of the Wavelet Function Ateb–Gabor 3 3 7.41629870921 Symmetry 2021, 13, 717 10 of 17

Symmetry 2021, 13, x FOR PEER REVIEW From Figure6 we can see that for large and equal values of the parameter n = m =10 3 ofthe 17 surface of the wavelet-Ateb–Gabor function ATEBG(a, b, m, n, θ, σ, t) changes with spiked seenpeaks from with Figure increasing 6, the deviation greater σ of, the the more Gaussian peaks nucleus, are observed.σ = 1; 2;Investigation 3; 4. As can beof the seen wave- from let-Ateb–GaborFigure6, the greater functionσ, the ATEBG more ( peaksa, b, m, are n, θ observed., σ, t) with Investigation parameters m of = then = wavelet-Ateb–3, and σ = 5; 6; 7;Gabor 8; 9; 10 function are givenATEBG in Appendix(a, b, m, n ,Aθ ,Figureσ, t) with A5. parameters m = n = 3, and σ = 5; 6; 7; 8; 9; 10 are givenFrom inthe Appendix followingA figuresFigure A5it follows. that changing the filter parameters allows us to selectFrom different the followingfiltering options figures that it follows differ thatin shape changing and numerical the filter parameters values. As allowscan be usseen to fromselect the different above filteringdependencies, options the that introduction differ in shape of the and Ateb–Gabor numerical filt values.er makes As can it possible be seen from the above dependencies, the introduction of the Ateb–Gabor filter makes it possible to to obtain more control on the effect of filtering comparing to the traditional Gabor filter. obtain more control on the effect of filtering comparing to the traditional Gabor filter. Thus, Thus, the Gabor filter depends on the parameters 𝜁,𝜐,𝜁 ,𝜐 ,𝜃,𝜎,β. The value of the Gabor the Gabor filter depends on the parameters ζ, υ, ζ , υ , θ, σ,β. The value of the Gabor filter filter depends on the coordinates of the filter0 0center 𝑥 = 𝑥 , 𝑦 = 𝑦 and points depends on the coordinates of the filter center x = x , y = y and points ξ= ξ , υ = υ 𝜉 = 𝜉 ,𝜐 = 𝜐 of the raster frequency plane, also, it depends0 0 on σ and β—scale0 parame-0 of the raster frequency plane, also, it depends on σ and β—scale parameters of elliptical ters of elliptical Gaussians along the x and y axes, θ- filter orientation angle, 𝜎—filter Gaussians along the x and y axes, θ-filter orientation angle, σ—filter phase. Hence the phase. Hence the Gabor function has six modes of freedom: 𝜁 ,𝜐 ,𝜃,𝜌,𝜎,β. Gabor function has six modes of freedom: ζ , υ , θ, ρ, σ, β. The introduction of the Ateb–Gabor filter0 depends0 on the ATEBG parameters (a, b, m, The introduction of the Ateb–Gabor filter depends on the ATEBG parameters (a, b, n, θ, σ, t), where in addition to the basic parameters, rational parameters m and n are m, n, θ, σ, t), where in addition to the basic parameters, rational parameters m and n are added, which significantly affect the shape of the function and result of filtering. added, which significantly affect the shape of the function and result of filtering.

4.4. Modeling, Modeling, Results Results 4.1.4.1. Dataset Dataset for for Filtering Filtering TheThe transformation and and determination determination of of the the efficiency efficiency of the proposed filteringfiltering methodmethod based based on on wavelet-Ateb–G wavelet-Ateb–Gaborabor fingerprint fingerprint filtering filtering was was performed performed on on the the basis basis of theof the freely freely available available database database NIST NIST Special Special Database Database 302. 302. The Theexamples examples of images of images are dis- are playeddisplayed in Figure in Figure 7. 7.

(a) (b) (c) (d) (e) (f)

Figure 7. TheThe examples examples of images ( a)) Sample1; Sample1; ( (b) Sample 2; ( c) Sample 3; ( d) Sample 4; (e) Sample 5; (f) Sample 6.

4.2.4.2. Wavelet-Ateb–Gabor Wavelet-Ateb–Gabor Fingerprint Fingerprint Image Filtering Wavelet-GaborWavelet-Gabor transformationtransformation is is used used in in the the filtering filtering of biometricof biometric images, images, because because this thisfeature feature allows allows you you to improveto improve the the contours contours of of the the ridges ridges of of biometrics. biometrics. TheThe developed methodmethod of thethe wavelet-Ateb–Gaborwavelet-Ateb–Gabor filter filter allows allows to to select select parameters parameters as it as contains it contains more more vari- variablesables and and to carry to carry out aout qualitative a qualitative filtering filtering in one in pass.one pass. In further In further research, research, it is necessary it is nec- essaryto develop to develop a technique a technique of selection of selection of parameters of parameters of the Ateb–Gaborof the Ateb–Gabor filter. The filter. developed The de- velopedmethod worksmethod a littleworks longer a little than longer the usual than wavelet-Gabor the usual wavelet-Gabor filter, because filter, the mathematical because the mathematicalcalculations are calculations more complex are more and thecomplex complexity and the of complexity the calculation of the algorithm calculation is higher. algo- rithmHowever, is higher. we can However, choose suchwe can rational choose parameters such rational that parameters will allow high-qualitythat will allow filtering high- qualityand identification filtering and in identification one pass. The in relationship one pass. The between relationship the frequency between and the the frequency width of andthe wavelet-Ateb–Gaborthe width of the wavelet-Ateb–Gabor filter was determined, filter was which determined, allowed to which automatically allowed to execute auto- maticallyfiltering to execute find the filtering edges ofto objectsfind the with edges different of objects frequencies, with different sizes frequencies, and directions. sizes and directions.In conclusion, a method for replacing the average filtering frequencies of Ateb–Gabor withIn close-to-zero conclusion, a values method has for been replacing developed. the average The filtering results offrequencies numerous of experimentsAteb–Gabor withdemonstrate close-to-zero satisfying values edges has of been the biometricdeveloped. images, The obtainedresults of after numerous applying experiments the chosen demonstrateparameters of satisfying Ateb–Gabor edges filter. of the The biometric results of images, the experiments obtained after are presented applying inthe Table chosen4. parameters of Ateb–Gabor filter. The results of the experiments are presented in Table 4. Pre-image processing and wavelet-Ateb–Gabor filtering were performed. A comparison was made based on the signal-to-noise ratio of the wavelet-Ateb–Gabor-filtered image and the input image. The following results were obtained.

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4.3. Comparison of the Efficiency of the Wavelet-Ateb–Gabor Filter with the Existing Ones The original Gabor filter is defined by ten frequency characteristics. The wavelet- Ateb–Gabor transformation developed and described in this article can significantly increase the number of frequency characteristics. It is possible to further increase the number to 900. Therefore, 900 filtration samples could be conducted. When selecting parame-

Symmetry 2021, 13, 717 ters of Ateb functions that are not, their maximum number is 2900. Therefore,11 of 17 a new filtering method is proposed. It can be considered as universal filtering method, as such method can be applied to any biometric image. Filtration can be performed once and effi- Table 4. Comparisonciently of filteredby selecting images by the Gabor correct and wavelet-Ateb–Gabor wavelet-Ateb–Gabor filter PSNR parameters. and MSE. Thus, the usage of the proposed method with selected parameters for filtering allows the achievement of a single Ateb Comparison, Filtration Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Filtering, m m TimeeffectivePSNR universal MSE PSNR filtering MSE of PSNR biometric MSE images. 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 sSix 38.77 samples, 10 39.59displayed 10.96 in 31.71 the Figure 8.78 7, 34.43 were 9.53 selected 31.49 for 8.72the experiment. 31.86 8.82 All of them 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 54were s 29.66subjected 8.21 to 30.82Gabor 8.53filtration 26.60 and 7.36 prop 26.06osed wavelet-Ateb–Gabor 7.21 26.37 7.87 25.37 filter. 7.02 The outcomes 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 57of s the 24.95 comparison 6.91 26.91 are displayed 7.45 23.84 in the 6.60 Table 20.34 4. 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 sComparison 22.20 6.14 24.79of images 6.86 with 2.65 different 9.58 19.75parameters 5.47 12.70m, n = 3.521, σ = 19.09pi and 5.28Gabor filter are 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 11presented s 19.35 as 5.35 correlation 23.10 6.39 measures 3.27 of 11.83 coinciden 2.65ce 9.58 of these 19.54 images 5.41 according 2.65 9.58 to PSNR [31] and MSE [32] filtered by Gabor filter and wavelet-Ateb–Gabor. Pre-image processing and wavelet-Ateb–Gabor filtering were performed. A compari- Table 4. Comparisonson was of madefiltered based images on the by signal-to-noiseGabor and wavelet-Ateb–Gabor ratio of the wavelet-Ateb–Gabor-filtered filter PSNR and MSE. image and the input image. The following results were obtained. Com- Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Ateb Fil- Filtration pari- 4.3. Comparison of the Efficiency of the Wavelet-Ateb–Gabor Filter with the Existing Ones tering, m Time PSNR MSE PSNR MSE PSNR MSE PSNR MSE PSNR MSE PSNR MSE son, m The original Gabor filter is defined by ten frequency characteristics. The wavelet-Ateb– Gabor transformation developed and described in this article can significantly increase 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 the number of frequency characteristics. It is possible to further increase the number to 0.9 1 1 min 55 s 38.77900. Therefore, 10 39.59 900 filtration 10.96 samples 31.71 could 8.78 be 34.43conducted. 9.53 When selecting 31.49 parameters 8.72 31.86 of 8.82 0.8 1 1 min 54 s 33.06Ateb functions 9.15 34.03 that are 9.42 not, their 28.75 maximum 7.96 number 30.18 is 2900. 8.36 Therefore, 28.43 a new 7.87 filtering 28.43 7.87 0.7 1 1 min 54 s 29.66method 8.21 is proposed. 30.82 It 8.53 can be 26.60 considered 7.36 as universal 26.06 filtering 7.21 method, 26.37 as such 7.87 method 25.37 7.02 can be applied to any biometric image. Filtration can be performed once and efficiently 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 by selecting the correct wavelet-Ateb–Gabor parameters. Thus, the usage of the proposed 0.5 1 1 min 57 s 24.95method 6.91 with 26.91selected 7.45parameters 23.84 for filtering 6.60 allows 20.34 the achievement 5.63 12.51 of a single 3.46 effective 21.23 5.88 0.4 1 2 min 1 s 23.35universal 6.46 filtering 25.47 of biometric 7.05 15.0 images. 7.75 19.96 5.52 12.52 3.47 19.70 5.45 0.3 1 1 min 53 s 22.20 Six 6.14 samples, 24.79 displayed 6.86 in 2.65 the Figure 9.587, were 19.75 selected for 5.47 the experiment. 12.70 All 3.52 of them 19.09 5.28 0.2 1 1 min 55 s 20.83were subjected 5.77 24.07 to Gabor 6.66 filtration 2.73 and proposed 9.86 19.41 wavelet-Ateb–Gabor 5.37 22.28 filter. The 6.17 outcomes 18.51 5.13 of the comparison are displayed in the Table4. 0.1 1 2 min 11 s 19.35 Comparison 5.35 23.10 of images 6.39 with 3.27 different 11.83 parameters 2.65 m, n 9.58= 1, σ = pi 19.54and Gabor 5.41 filter are 2.65 9.58 presented as correlation measures of coincidence of these images according to PSNR [31] and MSEAccording [32] filtered to the by Gaborresults filter of the and experiments, wavelet-Ateb–Gabor. the smaller is the parameter m, the bigger is theAccording discrepancy to the is results between of the the experiments, image filtered the smaller by the is theGabor parameter filter. m,Such the biggerThis dependence canis the be discrepancy noticed in is Figure between 8 theas well. image It filtered shows by the the dependence Gabor filter. Such of the This change dependence in the distortion can be noticed in Figure8 as well. It shows the dependence of the change in the distortion of thethe wavelet-Ateb–Gaborwavelet-Ateb–Gabor filter filter with with the Gabor the Gabor filter according filter according to PSNR dependingto PSNR depending on on thethe changechange of of the the parameters parametersm and m nand. The n comparison. The comparison according according to the MSE to depending the MSE depending on thethe changechange of of the the parameter parameterσ is shownσ is shown in Figure in 9Figure. 9.

Figure 8.8.Dependence Dependence of theof changethe change of distortion of distortion level of wavelet-Ateb–Gaborlevel of wavelet-Ateb–Gabor ﬁlter with Gabor filter ﬁlter with Gabor filteraccording according to PSNR to depending PSNR depending on the change on the of the change parameters of them parametersand n. m and n.

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Figure 9. Dependence of the change of distortion level of wavelet-Ateb–Gabor filter with Gabor filterFigureFigure according 9.9. DependenceDependence to MSE of depending of the the change change on of the distortion of change distortion levelof the ofleve parameters wavelet-Ateb–Gaborl of wavelet-Ateb–Gabor m and n. filter with filter Gabor with filter Gabor filteraccording according to MSE to depending MSE depending on the change on the of change the parameters of the parametersm and n. m and n. Comparison of the images filtered by wavelet-Ateb–Gabor filter using various pa- rametersComparison of σ and Gabor of the filter images is displa filteredyed by as wavelet-Ateb–Gabor correlation measures filterof coincidence using various of these pa- Comparison of the images filtered by wavelet-Ateb–Gabor filter using various pa- imagesrameters according of σ and to Gabor PSNR filter and isMSE displayed in the Tabl as correlatione 5. The comparison measures according of coincidence to the of PSNR these betweenrametersimages according the of wavelet-Ateb–Gaborσ and to Gabor PSNR andfilter MSE is filter displa in thewithyed Table th eas5 Gabor. correlation The comparison filter depending measures according on of thecoincidence to thechange PSNR of of these theimagesbetween parameter according the wavelet-Ateb–Gabor σ is shown to PSNR in Figure and MSE filter 10. The in with the comparison the Tabl Gabore 5. The filteraccording comparison depending to the onMSE according the depending change to ofthe PSNR onbetweenthe the parameter change the wavelet-Ateb–Gaborofσ isthe shown parameter in Figure σ is shown10 filter. The in with comparison Figure the 11. Gabor according filter todepending the MSE depending on the change of theon theparameter change of σ the is shown parameter in Figureσ is shown 10. The in Figure comparison 11. according to the MSE depending Table 5. Comparisonon the changeof filtered of images the parameter by Gabor and σ iswavelet-Ateb–Gabor shown in Figure PSNR 11. and MSE. Table 5. Comparison of filtered images by Gabor and wavelet-Ateb–Gabor PSNR and MSE. Ateb Filter- Ateb ComparisonTable 5. Comparison m = Filtration of filtered images by Gabor and wavelet-Ateb–Gabor PSNR and MSE. ing m = 1, n Comparison Filtration Sample 1 Sample 2 Sample 3 Sample 4 Filtering 1, n = 1, σ Time Sample 1 Sample 2 Sample 3 Sample 4 Ateb =Filter- 1, σ m = 1, n = 1, σ Time m = 1, n = 1,Comparisonσ m = Filtration ing m = 1, n PSNRPSNRSample MSE MSE 1 PSNR PSNR SampleMSE MSE 2 PSNR PSNR SampleMSE MSE 3PSNR PSNR MSE Sample 4 π /4 1, n =π 1, σ 2 minTime 18 s 12.77 3.14 4.56 17.01 4.76 17.21 17.59 4.87 = 1, σπ /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/3 π 2 min 3 s PSNR 3.61 3.54MSE 4.18PSNR 15.09MSE 4.28PSNR 15.49 MSE 17.40 PSNR 4.82 MSE π π/2/2 π π 2 min2 min 15 15 s s4.03 4.03 14.55 14.55 4.00 4.00 16.06 16.06 4.50 4.50 16.26 16.26 17.22 17.22 4.76 π 2*/42 ×π πππ π 2 2 min2min min 15 18 15 s s 4.724 4.72412.77 17.07 17.073.14 4.61 4.61 4.56 16.67 16.6717.01 4.61 4.61 4.7616.67 16.67 17.2117.04 17.04 17.594.72 4.87 × π π π 3*/33 π ππ 1 2min1 min min 58 583 s s s 4.25 4.25 3.61 15.35 15.35 3.54 4.63 4.63 4.18 16.72 16.72 15.09 4.63 4.63 4.2816.72 16.72 15.4917.00 17.00 17.404.71 4.82 4 × π π 2 min 3 s 3.94 14.25 4.55 16.70 4.55 16.70 17.00 4.71 π 4*/2 π ππ 2 min 315 s s 3.944.03 14.2514.55 4.554.00 16.7016.06 4.55 4.5016.70 16.2617.00 17.224.71 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

FigureFigure 10. 10. DependenceDependence of of PSNR PSNR between between wavelet-Ateb–Gabor wavelet-Ateb–Gabor filter ﬁlter with with Gabor Gabor filter ﬁlter on on change change of of parameterparameter σσ..

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

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

IndirectIndirect sections sections in in Figures 88–11–11 indicateindicate thatthat aa broaderbroader filteringfiltering methodmethod hashas beenbeen found.found. And And that that method method is is different different from the Gabor filter. filter. Ateb–Gabor wavelet filteringfiltering showsshows good good results results and and significantly significantly expands expands the the Gabor Ga- borfiltering filtering method. method. The The characteristics characteristics of the of waveletthe wavelet Ateb Ateb Gabor Gabor filtration filtration are investigated, are investigated,which iswhich shown is shown in numerous in numerous graphs. graphs. Filtered Filtered by a new by wavelet a new filterwavelet Ateb filter Gabor Ateb filtration. Gabor filtration.Filtration Filtration results are results compared are compared with Gabor with filtration Gabor basedfiltration on PSNRbased andon PSNR MSE estimates,and MSE estimates,which show which good show recognition good recognition results. The results. time characteristics The time characteristics of the filtering of are the estimated. filtering are estimated. 5. Conclusions 5. ConclusionsAteb–Gabor filtering allows changing of the intensity of the whole image, as well as the intensity in certain ranges, and thus make certain areas of the image more con- Ateb–Gabor filtering allows changing of the intensity of the whole image, as well as trasting. Ateb-functions vary from two rational parameters, which, in turn, allow more the intensity in certain ranges, and thus make certain areas of the image more contrasting. flexible control of filtering. The properties of the Ateb function, as well as the possibility of Ateb-functions vary from two rational parameters, which, in turn, allow more flexible changing the amplitude of the function, the oscillation frequency on the Ateb–Gabor filter control of filtering. The properties of the Ateb function, as well as the possibility of chang- are investigated. The development of filtration based on two-dimensional Ateb–Gabor is ing the amplitude of the function, the oscillation frequency on the Ateb–Gabor filter are shown. These dependencies are analyzed and appropriate experiments performed. The re- investigated. The development of filtration based on two-dimensional Ateb–Gabor is lationship between the frequency and the width of the Ateb–Gabor filter are determined, shown.which will These allow dependencies filters to be are found analyzed to find an thed edgesappropriate of objects experiments with different performed. frequencies The relationshipand sizes. Appropriate between the filtering frequency software and the are width developed. of the Fingerprints Ateb–Gabor are filter filtered are deter- using mined,the developed which will Ateb–Gabor allow filters filter. to be The found efficiency to find of the its edges use is of proved, objects which with different consists offre- a quencieslarger number and sizes. of options Appropriat for filteringe filtering the processedsoftware are images. develop Theed. results Fingerprints of numerous are filtered exper- usingiments the demonstrate developed theAteb–Gabor successful filter. selection The effici of edgesency inof theits use image is proved, based on which the obtained consists ofparameters a larger number of the Ateb–Gabor of options for filter. filtering the processed images. The results of numerous experimentsIn the future, demonstrate a method the forsuccessful selecting selectio rationaln of parametersedges in the for image wavelet-Ateb–Gabor based on the ob- tainedfiltering parameters will be developed of the Ateb–Gabor on the basis filter. of the wavelet Ateb–Gabor functions developed in thisIn study.the future, A method a method of constructingfor selecting ration a fieldal ofparameters directions for of wavelet-Ateb–Gabor papillary lines will also fil- teringbe developed, will be developed and according on the to basis this of field the of wavelet directions, Ateb–Gabor the developed functions filtering developed will bein thiscarried study. out. A method of constructing a field of directions of papillary lines will also be developed, and according to this field of directions, the developed filtering will be carried out.Author Contributions: Conceptualization, M.N.; methodology, Y.V.; software, Y.V.; validation, N.K., M.N. and Y.V.; formal analysis, M.N.; investigation, Y.V.; resources, N.K.; data curation, Y.V.; writing— Authororiginal Contributions: draft preparation, Conceptualization, N.K.; writing—review M.N.; and methodology, editing, N.K.; Y.V.; visualization, software, Y.V.;Y.V.; supervision, validation, N.K.,Y.V.; projectM.N. and administration, Y.V.; formal analysis, Y.V.; funding M.N.; acquisition, investigation, Y.V. Y.V.; All authors resources, have N.K.; read data and curation, agreed to Y.V.; the writing—originalpublished version draft of the preparation, manuscript. N.K.; writing—review and editing, N.K.; visualization, Y.V.; supervision, Y.V.; project administration, Y.V.; funding acquisition, Y.V. All authors have read and Funding: This research did not receive any specific grant from funding agencies in the public, agreed to the published version of the manuscript. commercial, or not-for-profit sectors. Funding: This research did not receive any specific grant from funding agencies in the public, com- Institutional Review Board Statement: Not applicable. mercial, or not-for-profit sectors. Informed Consent Statement: Not applicable. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable.

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Symmetry 2021, 13, x FOR PEER REVIEWData Availability Statement: Not applicable. 14 of 17 Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest. ConflictsAppendix of AInterest: The authors declare no conflict of interest. Appendix A Appendix A

(a) (b)

(e) (e) ( ) FigureFigure A1.A1. ConstructionConstruction Figure ofof Wavelet-GaborWavelet-Gabor A1. Construction filterfilter 𝜓 of𝜓(𝜁𝜁,𝜐,𝜁 Wavelet-Gabor, 𝜐, 𝜁,,𝜐𝜐,,𝜃,𝜎,β𝜃, 𝜎,β) ﬁlter inin OXOX—ψ—(ζ𝜁𝜁,,υ, in,inζ 0 OYOY—, υ0—, θυ,υσ:: ,((aβa)) 𝜎𝜎in == OX— 6;6; ((bb)ζ) ,𝜎𝜎 in == OY— 7;7; ((cc))υ 𝜎𝜎:( =a= )8;8;σ = 6; ((dd)) 𝜎𝜎 = = 9;9; ((ee)) 𝜎𝜎 == 10.10. (b) σ = 7; (c) σ = 8; (d) σ = 9; (e) σ = 10.

(a) (b) (a) (b)

Figure A2. Cont.

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(e) (e)

Figure A2. Construction of wavelet-Ateb–Gabor filter 휓(휁, 휐, 휁0, 휐(0e,)휃 , 𝜎, β) with parameters m = 1, n = 1 in OX—푎, in OY— Figure A2. Construction of wavelet-Ateb–Gabor filter 𝜓(𝜁,𝜐,𝜁 ,𝜐 ,𝜃,𝜎,β) with(e) parameters m = 1, n = 1 in OX—𝑎, in OY— b: (a) 𝜎 = 6; (b) 𝜎 = 7; (c) 𝜎 = 8; (Figured) 𝜎 = A2.9; (eConstruction) 𝜎 = 10. of wavelet-Ateb–Gabor ﬁlter ψ(ζ, υ, ζ0, υ0, θ, σ, β) with parameters m = 1, b: (a) 𝜎 = 6; (b) 𝜎 = 7;Figure (c) 𝜎 = A2. 8; ( Constructiond) 𝜎 = 9; (e) 𝜎 of = wavelet-Ateb–Gabor 10. filter 𝜓(𝜁,𝜐,𝜁,𝜐,𝜃,𝜎,β) with parameters m = 1, n = 1 in OX—𝑎, in OY— n = 1 in OX—a, in OY—b:(a)𝜓σ(𝜁=, 𝜐 6;, 𝜁 (b, 𝜐) σ, 𝜃=, 𝜎 7;,β ()c) σ = 8; (d) σ = 9; (e) σ = 10. 𝑎 Figureb: (a) 𝜎 A2. = 6; Construction (b) 𝜎 = 7; (c )of 𝜎 wavelet-Ateb = 8; (d) 𝜎 = 9;– (Gabore) 𝜎 = filter10. with parameters m = 1, n = 1 in OX— , in OY— b: (a) 𝜎 = 6; (b) 𝜎 = 7; (c) 𝜎 = 8; (d) 𝜎 = 9; (e) 𝜎 = 10.

(b) (a) ( b) (b) (a) (a) (b) Figure A3. Construction of wavelet-Ateb–Gabor(a) filter 휓(휁, 휐, 휁 , 휐 , 휃, 𝜎, β) with parameters m = 1, 𝜎 = 1; in OX—푎, in Figure A3. Construction𝜓(𝜁,𝜐,𝜁 ,𝜐 ,𝜃,𝜎,β of wavelet-Ateb–Gabor) 0 0 ﬁlter ψ(𝜎ζ, υ, ζ0, υ0, θ, σ, β𝑎) with parameters m = 1, Figure A3. ConstructionFigure of A3.wavelet-Ateb–Gabor Construction of wavelet-Ateb–Gabor filter filter 𝜓 (with𝜁,𝜐,𝜁 parameters,𝜐,𝜃,𝜎,β) withm = parameters1, = 1; in m OX— = 1, 𝜎, in= 1; in OX—𝑎, in OY—b: (a) n = 0.5; (b) n = 1. 𝜓(𝜁, 𝜐, 𝜁 , 𝜐 , 𝜃, 𝜎,β) 𝜎 𝑎 OY—b: (a) n = 0.5; (bFigure)OY— n =b A3.: 1(a. ) Constructionn = 0.5; (b) nσ of == wavelet-Ateb1 1;. in OX—a,– inGabor OY— filterb:(a) n = 0.5; (b) n = 1. with parameters m = 1, = 1; in OX— , in OY—b: (a) n = 0.5; (b) n = 1.

(a) (a) (b) (b) (a) (a) (b) (b) Figure A4.Figure Construction A4. Construction of Wavelet of Wavelet-Ateb–Gabor-Ateb–Gabor filter 휓filter(휁, 휐 𝜓, 휁(0𝜁,𝜐,𝜁, 휐0, 휃,,𝜐𝜎,,𝜃,𝜎,ββ) with) withparameters parameters n = 1, n =𝜎 1, = 𝜎1; =in 1; OX in OX——푎, in𝑎, OYin OY—— Figure A4. Construction of Wavelet-Ateb–Gabor ﬁlter ψ(ζ, υ, ζ0, υ0, θ, σ, β) with parameters n = 1, Figure A4. ConstructionFigure of A4. Wavelet-Ateb–Gabor Construction of Wavelet-Ateb filter 𝜓(𝜁,𝜐,𝜁–Gabor,𝜐 filter,𝜃,𝜎,β 𝜓)( 𝜁with, 𝜐, 𝜁 parameters, 𝜐 , 𝜃, 𝜎,β) with n = 1, parameters 𝜎 = 1; in OX— n = 1,𝑎 ,𝜎 in = 1;OY— in OX—𝑎, in OY— b: (a) m = b5;: (ab)) m 푚 = =5; ( 6b.) 𝑚 = 6. b: (a) m = 5; (b) 𝑚 = b6. : (a) m = 5; (b) 𝑚 = .6 σ = 1; in OX—a, in OY—b:(a) m = 5; (b) m = 6.

(a) (b) (a) (a ) (b) (b) (a) (b)

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(e) (f)

(g) Figure A5. Construction of wavelet-Ateb–Gabor filter 𝜓(𝜁,𝜐,𝜁 ,𝜐 ,𝜃,𝜎,β) with parameters n = 3, m = 3; in OX—𝑎, in OY— Figure A5. Construction of wavelet-Ateb–Gabor ﬁlter ψ(ζ, υ, ζ0, υ0, θ, σ, β) with parameters n = 3, 𝜎 ( 𝜎 𝜎 𝜎 𝜎 𝜎 𝜎 b: (a) = 4; b) = 5; (c)m = 6; 3; ( ind) OX— = 7;a (,e in) OY— = 8; (bf:() a ) =σ 9;= ( 4;g) (b )=σ 10.= 5; (c) σ = 6; (d) = 7; (e) = 8; (f) = 9; (g) σ = 10.

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