A mathematical model for whorl fingerprint
AA M
ATHEMATICAL M ODEL FOR W HORL F INGERPRINT
A P
REPRINT
Ibrahim Jawarneh
Department of MathematicsAl-Hussein Bin Talal UniversityMa’an, P.O. Box (20), 71111, Jordan [email protected]
Nesreen Alsharman
Computer ScienceThe World Islamic Sciences and Education UniversityAmman, Jordan [email protected]
October 2, 2020 A BSTRACT
In this paper, different classes of the whorl fingerprint are discussed. A general dynamical systemwith a parameter θ is created using differential equations to simulate these classes by varying thevalue of θ . The global dynamics is studied, and the existence and stability of equilibria are analyzed.The Maple is used to visualize fingerprint’s orientation image as a smooth deformation of the phaseportrait of a planar dynamical system. K eywords Whorl fingerprint · Concentric whorl · Spiral whorl · Composite whorl with "S" core · Simulations of thewhorl fingerprint.
Fingerprints are a set of raised lines that form unique patterns on the pads of the fingers and thumbs. Everyone leavesparts or entire fingerprints on many things through our daily activities by touching cups, doors, books, etc., so studyingfingerprints is important in security especially no two people have been found to have the same fingerprints. One of theearly studies about fingerprints appeared in 1892 by Sir Francis Galton in his book, finger prints [3]. There are threegeneral types of fingerprints; loop, whorl, and arch. The whorl type occurs in about 25–35% of all fingerprints, see[1, 3, 4].The whorl patterns display in the way that the ridges in the center tend to show a circular orientation with a core towhorl and two deltas in the right and left sides. The focus of this paper is on three basic categories of whorl fingerprintwhich are concentric whorl, spiral whorl, and composite whorl with "S" core:• Concentric whorl, this pattern represents the most basic form of a whorl in which the core is circular orelliptical in the center of the fingerprint, see picture (a) in figure 1.• Spiral whorl, the ridges flow are in winding way in the center making a spiral core, see picture (b) in figure 1.• Composite whorl with "S" core, this pattern twists its ridges in the way that forms a core in "S" shape in thecenter, see picture (c) in figure 1.Few studies talked about modeling of the whorl fingerprint specially using the phase portraits of a system of differentialequations. The idea of phase portraits in texture modelling can be seen in [7] where a characterizing oriented patternswas proposed using the qualitative differential equation theory to analyze real texture images, but no fingerprint imageseems to have been considered. In the thesis by Ford [2], the complex flows were divided into simpler componentswhich are modeled by linear phase portraits and then combined to obtain a model for the entire flow field, this idea wasapplied to fingerprints in [5].Fingerprint can be captured as graphical ridge and valley patterns, so the global representation of the above categoriesof fingerprint’s flow-like patterns as a smooth deformation of the phase portrait of a system of differential equationsis considered in this paper. Using differential equations in the formularization of the general dynamical system that a r X i v : . [ m a t h . D S ] O c t PREPRINT - O
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Figure 1: (a) Concentric whorl, (b) spiral whorl and (c) composite whorldescribes concentric, spiral, and composite whorl fingerprint requires understanding the behaviour of the ridges andinterpreting the deltas and cores that appear in these classes, and how the singular points that represent the core to whorland deltas in the patterns of phase portraits look like in the considered model.The system of differential equations basically has two types of singular points; first type is nondegenerate singular point(The Jacobian matrix of the system has no zero eigenvalues), the second type is degenerate singular point (The Jacobianmatrix has at least one zero eigenvalue). For the core to whorl can be represented as center or spiral singular point inthe system of differential equations, but the delta can be interpreted as a singular point with three heperbolic sectors,and there is no such singular point exist. So the closer singular point for the delta is the cusp with cutting appropriateedge of it. To know more information about the cusp, we present the following theorem, including determining sometypes of degenerate equilibrium points of the planar system that are found in [6]. Assume that the origin is an isolatedcritical point of the planar system ˙ x = P ( x, y ) , ˙ y = Q ( x, y ) . (1)where P ( x, y ) and Q ( x, y ) are analytic in some neighborhood of the origin, and consider the case when the matrix A = Df (0) has two zero eigenvalues i.e., det ( A ) = 0 , tr ( A ) = 0 , but A (cid:54) = 0 , in this case the system (1) can be put inthe normal form ˙ x = y, ˙ y = a k x k [1 + h ( x )] + b n x n y [1 + g ( x )] + y R ( x, y ) . (2)where h ( x ) , g ( x ) and R ( x, y ) are analytic in a neighborhood of the origin, h (0) = g (0) = 0 , k ≥ , a k (cid:54) = 0 and n ≥ . Theorem 1.1 (Theorem 2 and theorem 3, page 151 [6]) . .1. Let k = 2 m + 1 with m ≥ in (2) and let λ = b n + 4( m + 1) a k . Then if a k > , the origin is a (topological)saddle. If a k < , the origin is • a focus or a center if b n = 0 and also if b n (cid:54) = 0 and n > m or if n = m and λ < , • a node if b n (cid:54) = 0 , n is an even number and n < m and also if b n (cid:54) = 0 , n is an even number, n = m and λ ≥ , • a critical point with an elliptic domain if b n (cid:54) = 0 , n is an odd number and n < m and also if b n (cid:54) = 0 , n isan odd number, n = m and λ ≥ .2. Let k = 2 m with m ≥ in (2) . Then the origin is • a cusp if b n = 0 and also if b n (cid:54) = 0 and n ≥ m , • a saddle-node if b n (cid:54) = 0 and n < m . It is clear now from the above theorem that if the Jacobian matrix Df ( x ) has two zero eigenvalues, then the criticalpoint x is either a focus, a center, a node, a (topological) saddle, a saddle-node, a cusp, or a critical point with anelliptic domain. 2 PREPRINT - O
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2, 2020In [8], Zinoun used the Taylor polynomials and the normal forms then applied the theorem (1.1) to formulate somesystems of the classes of the whorl fingerprint such as the concentric whorl ˙ x = y, ˙ y = − x ( x − . (3)and the spiral whorl fingerprint ˙ x = y, ˙ y = ( y − x/ x − . (4)In this research, the spiral whorl fingerprint is developed, a new model for composite whorl with "S" core is suggestedand all above categories of the whorl fingerprint are generalized in a dynamical system with a parameter θ as follows ¨ x − ( θ ˙ x − x )( x − = 0 , θ ∈ R . (5)Equation (5) can be written as a first order system ˙ x = y, ˙ y = − x ( x − + θy ( x − , θ ∈ R . (6)This paper is organized as follows: In the next section, we study the stability of the equilibria of the system (6). Insection 3 an interesting simulations are shown for the three categories of the whorl fingerprint. A brief results aresummarized in section 4. To study the stability of the system (6), we find the equilibria first, which are the solutions of the following equations: y, (7) − x ( x − + θy ( x − . (8)and are given by E = (0 , , E (1 , ,and E = ( − , which are called equilibrium points or singular points. TheJacobian matrix of (6) takes the form J = (cid:20) − ( x − + 4( θy − x )( x − x θ ( x − (cid:21) . (9)The Jacobian matrix evaluated at the equilibrium point E = (0 , is J ( E ) = (cid:20) − θ (cid:21) . (10)We summarize the stability of E in the following theorem. Theorem 2.1.
The equilibrium point E = (0 , is stable if θ < , unstable if θ > , and center if θ = 0 .Proof. From the Jacobian matrix (10), the eigenvalues of J ( E ) are λ , = θ ± √ θ − . (11)Notice that the eigenvalues of the equilibrium point E = (0 , depend only on the parameter θ . When θ < , thereal parts of the eigenvalues are negative, so E is stable. Similarly, when θ > , the real parts of the eigenvalues arepositive, and E is unstable. Finally, when θ = 0 , we get λ , = ± i , and so E is center.The equilibrium points E = ( − , and E = (1 , have the same Jacobian matrix J ( E ) = J ( E ) = (cid:20) (cid:21) . (12)We summarize the stability of E = ( − , and E = (1 , in the following theorem Theorem 2.2.
The equilibrium points E = ( − , and E = (1 , are cusps. PREPRINT - O
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Proof.
The eigenvalues in this case are λ , = 0 which means degenerate equilibrium points, i.e., det ( J ) = 0 , tr ( J ) = 0 ,but J (cid:54) = 0 . In this case it is shown in the introduction that the system can be put in the normal form to which functionscan be reduced in a neighbourhood of the degenerate critical points E = ( − , and E = (1 , as the following ˙ x = y, ˙ y = ± α ( x ± + o (( x ± ) . (13)Apply theorem (1.1), k = 2 , b n = 0 , h ( x ) = 0 and α > which gives that both E = ( − , and E = (1 , arecusps. In this section, we display the phase portrait of the system (6) using particular values of the parameter θ and matchthem with the images in the above categories of the whorl fingerprint. We get similar shape to the concentric whorlat the value θ = 0 , around and closed to θ = 0 , we get shapes look like to the spiral whorl and when we increase thevalue of the parameter θ to be around one, we get phase portrait closed to the composite whorl with "S" core. For moredetails let us go over these cases using Maple software. The basic features of the concentric whorl are the existence of the circular or elliptical ridges in the middle whichare represented by the center in the phase portrait, the two deltas which are represented by the two cusps in the phaseportrait in the right and left sides, and we can draw two connections between the deltas through the ridges which arerepresented by separatrices between the cusps from above and below half planes in the phase portrait. Let us put θ = 0 in the system (6), we get the following system Example 3.1. ˙ x = y, ˙ y = − x ( x − . (14)Figure 2 shows the phase portrait of the example 3.1 using Maple and the image that represents the concentric whorl. Iteasy to compare and matching center with circular ridges in the middle , the cusps with the deltas, and the separatriceswith the connections in both pictures. (a) (b) Figure 2: Simulation of the concentric whorl fingerprint: (a) phase portrait of example 3.1 and (b) image of theconcentric whorl fingerprint. 4
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In this type, we go over two kinds of the spiral whorl:• UR-LL spiral whorl in which there exist an upper right connection (UR connection) between the spiral coreand the right delta and a lower left connection (LL connection) between the spiral core and the left delta, seepicture (b) in figure 3.• LR-UL spiral whorl in which there exist a lower right connection (LR connection) between the spiral core andthe right delta and an upper left connection (UL connection) between the spiral core and the left delta, seepicture (b) in figure 4.To get the first kind of spiral, substitute θ = 0 . in the system (6), we get the following system Example 3.2. ˙ x = y, ˙ y = − x ( x − + 0 . y ( x − . (15)The phase portrait of the example 3.2 is shown in figure 3. We can see the similarity between the phase portrait and theimage that represents UR-LL spiral whorl by comparing the focus with the spiral core, the cusps with deltas, the upperright orbit (UR orbit) between the focus and the right cusp with the upper right connection (UR connection), and thelower left orbit (LL orbit) between the focus and the left cusp with the lower left connection (LL connection). (a) (b) Figure 3: Simulation of the UR-LL spiral whorl fingerprint: (a) phase portrait of example 3.2 and (b) image of theUR-LL spiral whorl fingerprint.To get the kind LR-UL spiral whorl, we should think how to switch the connections in the figure 3 up side down, thiscan be happened if we use negative value of θ , we use θ = − . in the system (6) which is explained in example 3.3. Example 3.3. ˙ x = y, ˙ y = − x ( x − − . y ( x − . (16)The phase portrait of example 3.3 is illustrated in figure 4 which achieves our target.5 PREPRINT - O
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Figure 4: Simulation of the LR-UL spiral whorl fingerprint: (a) phase portrait of example 3.3 and (b) image of theRU-LL spiral whorl fingerprint.
The composite whorl with "S" core is characterized with a center looks like an "S", and two cusps in the two sides. If θ in the system (6) grows up around one, the flow is twisted enough to create "S" in the middle with keeping the cusps inboth sides, for example consider system (6) with θ = 0 . we get the following system Example 3.4. ˙ x = y, ˙ y = − x ( x − + 0 . y ( x − . (17)Notice the flow in the phase portrait of the example 3.4 and the friction ridges of the composite whorl with "S" arealmost similar. (a) (b) Figure 5: Simulation of the composite whorl with S core fingerprint: (a) phase portrait of example (3.4) and (b) imageof the composite whorl with "S" core. 6
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The dynamical system (6) with the parameter θ is a good source to generate different categories of the whorl fingerprint.We have noticed that the shape of the flow of this dynamical system at a particular value of the parameter θ and theshape of the ridges in the corresponding image of a category of the whorl fingerprint are almost identical to each other.The flexibility of the system (6) enable us to simulate a class of whorl depending on how much this class is twisted ineither a clockwise direction or counterclockwise direction. References [1] de Jongh A, Lubach AR, Lie Kwie SL, Alberink Measuring the Rarity of Fingerprint Patterns in the DutchPopulation Using an Extended Classification Set.
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39, pages 102–114, 2006.[6] Perko, Lawrence. Differential equations and dynamical systems.
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Springer Series in Perception Engineering, Springer, New York, NY. arXiv:1802.05671arXiv:1802.05671