A hierarchical Hough transform for fingerprint. matching. Liu Chaoqiang,

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1 A hierarchical Hough transform for fingerprint 1 matching Liu Chaoqiang, Temasek Laboratories and Centre for Wavelets, Approximation and Information Processing, National University of Singapore, tslliucq@nus.edu.sg Xia Tao, Centre for Wavelets, Approximation and Information Processing, National University of Singapore, iesxiat@nus.edu.sg Li Hui, Temasek Laboratories and Centre for Wavelets, Approximation and Information Processing, National University of Singapore, tsllh@nus.edu.sg Abstract This paper addresses the improvement on the generalized Hough transform for the biometric identification applications. The errors in generalized Hough transform for fingerprint matching are investigated and a new hierarchical Hough transform algorithm is proposed, which is faster and more accurate compared to conventional generalized Hough transform. Index Terms generalized Hough Transform, fingerprint minutiae, fingerprint matching, parameter space. I. Introduction THe biometrics is a technology that (uniquely) identifies a person based on his physiological or behavioral characteristics. Nowadays, biometrics identification is commonly in use for example at banks, airports, and government agencies. For identification signatures at present around the world, the popular methods are face images, fingerprint, voice print and iris pattern. Among them, the fingerprint is the most favorite for its uniqueness for

2 2 the individual. Accurate automatic personal fingerprint identification is becoming more and more important to the operation of our increasingly electronically interconnected information society. An automatic fingerprint authentication system has four main design components: acquisition, representation (template), feature extraction, and matching. Among them the feature extraction including the fingerprint image processing problem is well studied [1] [5]. However, no perfect feature extraction method to guarantee all fingerprint feature is extracted exactly, especially for the latent fingerprint with poor quality and incompleteness. Therefore, the better matching algorithm for the fingerprint authentication is a big problem in this society. Fingerprint matching has been approached from several different strategies, such as image-based [6] and ridge pattern matching of fingerprint representations. There also exist graph-based [7] schemes for fingerprint matching [8] and structual matching [9]. Also, given the minutiae representation of fingerprints, matching a fingerprint against a database reduces to the problem of point matching [10],[11]. In recent years, the Hough Transform [12], HT, has become an extremely popular method of 2D shape detection. The most important properties of HT are its immunity to noise and occlusion and its capacity to handle multiple occurrence of shapes being detected. Several novel ideas have been proposed to improve its computational efficiency and practicability [13],[14]. The generalized Hough transform-based approach to point pattern matching [15] converts point pattern matching to a problem of detecting peaks in the Hough space of transformation parameters. It discretizes the parameter space and accumulates evidence in the discretized space by deriving transformation parameters that relate two point patterns using a substructure or feature matching technique. However, the traditional generalized Hough transform used in fingerprint matching may lead to some errors due to the coarse quantization in the rotation parameter space. Acknowledging the inaccuracy of matching result due to the coarse quantization in the rotation parameter space, a novel fingerprint matching method based on hierarchical Hough transform is developed. Based on the analysis of this kind of errors, the generalized Hough transform in the translation parameter space is modified to ensure our candidate optimal solutions in the coarser level includes the real optimal solution in finer level. This paper is organized as follows. The fingerprint matching problem is briefly introduced in Section II. Our method

3 3 is described in Section III. The experimental results and discussions are presented in Section IV. Conclusions are drawn in Section V. II. Fingerprint Alignment and Matching As mentioned above, an automatic fingerprint authentication system is composed of four main design components: acquisition, representation (template), feature extraction, and matching. Although there are other representations such as pore [16], most representations are still minutiae (fingerprint features of ridge ending and ridge bifurcation [17]) based, and each minutia is described by its location (x,y coordinates) and the orientation. A feature extractor detects the minutiae for automated feature extraction and matching. In order to match two point sets with unknown orientation, scale, and translation, the two sets must be registered with respect to each other. The orientation, scale, and translation parameters are estimated using a generalized Hough Transform. Normally, all allowed transformations are discretized, and for each transformation, the matching score is computed. The transformation with the maximum score is taken as the correct one. The transformations F s,δx,δ y,δ θ : R 2 R 2 given below is the most common one in use for alignment. where R δθ = cos δ θ sin δ θ parameters, respectively. ( ) ( ) ( ) x x δx F s,δx,δ y,δ θ = sr δθ + y y δ y sin δ θ is the rotation transform. s, (δ x, δ y ), and δ θ cos δ θ (1) are the scale, shift, and rotation The generalized Hough transform-based approach to point pattern matching converts point pattern matching to a problem of detecting peaks in the Hough parameter space. Whereas, since only the set of the discretized parameter values are tested, the accuracy of optimum match depends on the quantization of the parameter space. In the fingerprint identification/authentication application, the scale difference is mainly caused by the dpi for different kind of sensors and it is supposed to be known in advance. Therefore in this paper we only deal with other two parameters only, namely rotation and translation. In practice, the parameter space is partitioned into the uniform cubic according to the specified precision of each parameters. Apparently, finer precision will cause more cubics and it will lead to the huge burden of computation cost. On the other hand, coarser precision sometimes will lead to the wrong result, especially for the fingerprint authentication applications. Therefore, the proper trade-off between computation cost and accuracy is indeed an

4 4 important problem for the real application. Mainly, there are two kinds of errors/difficulties of the existing fingerprint matching and alignment, Missing the matched pair due to the error bound of translation changing caused by the coarse precision of the rotation parameter. Duplication of calculation on the minutiae to against more than one, i.e. there are more than one minutiae are matched against one minutia in the template and vice versa. In this paper, we will try to give a good solution for these two problems based on our analysis. For simplicity, in the following contents, the denotation v 1 v 2 for two vectors means for every element, this inequality holds and sometimes we use v to represent its x, y components. Also, when we state θ 1 θ 2 for two angles, we always compare θ 1, θ 2 under the modular 2π. III. Hierarchical Hough Transform The accuracy of alignment transform F δx,δ y,δ θ plays an important role in fingerprint matching process. To achieve higher accuracy, the precision of all parameters should be high even though that will lead to burden of computation cost. In this application, the parameter space consists of the transform of rotation and translation by which the two coordinate systems will be aligned. Let U be the feature set of template fingerprint saved in database, and V be the feature set of latent fingerprint. The element in a feature set is a triplet consisting of coordinate and orientation of one minutia. Suppose the template feature set has M minutiae, i.e. U = {u i, i = 1,, M} and the latent feature set has N minutiae, i.e. V = {v i, i = 1,, N}. Let M xy = (max( v x ), max( v y )) T be the maximum value of the fingerprint image of set V. xy = ( x, y ) T, θ are the expected precision of alignment parameters of the x, y translation and rotation angle θ respectively. α is the tolerate bound of orientation difference of two matched minutiae and xy is the tolerate bound of the location difference of two matched minutiae. [x min, x max ], [y min, y max ] and [θ min, θ max ] are the intervals of possible translation and rotation angles for the latent fingerprint against the template fingerprint in database respectively. One typical error in conventional generalized Hough transform is that the Hough transform will miss some matched pair of minutiae of two fingerprints. This is caused by the inaccuracy of the rotation precision of alignment transform. More precisely, the following theorem will give the necessary condition for such cases.

5 5 Theorem 1: The missing pair of the matched minutiae Suppose the error bound of the fingerprint is xy, θ, the rotation angle for the alignment of v j against u i is θ, if the rotation transform imposed on V is R θ, Suppose u i = (x u, y u ) T, v j = (x v, y v ) T, v 0 j = (x0 v, y 0 v) T, v 1 j = (x1 v, y 1 v) T, and v j R θ v 0 R j, v θ j vj 1 then we have, vj 1 u i xy + 2 sin( θ θ ) v j 2 According to Theorem 1, we know that the error bound of translation for different minutia in V should be different, especially when θ θ is large. This difference is significant and will cause some matched minutiae regarded as unmatched one. For the sake of simplicity, we can use M xy,ρ = 2 sin(ρ/2)m xy to replace in 2 sin( θ θ 2 ) v j in Theorem 1 to estimate the influence of this difference, where ρ is the precision of the rotation parameter in alignment transform, i.e. θ θ ρ. That will lead to algorithm 1. Given an array of precision sequence (ρ 1, ρ 2,, ρ K ), where ρ 1 > ρ 2 > > ρ K = θ and the threshold T of the minimum threshold for the number of matched minutiae regarded two fingerprint sets are same fingerprint. Algorithm 1: Hough Transform Algorithm for orientation angle θ 1) Initialize the rotation precision and rotation range by set j = 1, Θ = {[θ min, θ max ]}. 2) ρ = ρ j, Θ r =, n=0, where Θ r is the set of all possible intervals of the rotation angle and n is the maximum number of the matched minutiae. 3) For each interval [β, β ] Θ a) Partition [β, β ] = L 1 i=0 [β i, β i+1 ] where β 0 = β, β i = β i 1 + ρ, β L = β, i=0,. b) for interval [β i, β i+1 ], i) Q = {(k, l) (u k,θ v l,θ ) β i ρ 2 ρ + α, u k U, v l V } ii) If j = K goto step (3(b)vi) iii) then using algorithm 2 to get the maximum number of matched pair under rotation R βi, i.e. m = H 1 (Q, xy + M xy, ρ 2, β i) iv) If m T then Θ r = Θ r {[βi, β i+1 ]} v) goto step (3(b)vii). vi) Using algorithm 3 to get the maximum number of matched pair under rotation R βi, i.e. m = H 2 (Q, xy + M xy, ρ 2, β i, n) vii) If m > n, then m n,

6 6 viii) If i < L, then i + 1 i, goto step (3b) 4) if Θ r = goto step (6) 5) If j < K, Θ r Θ, j + 1 j, goto step (2) 6) If n < T, These two fingerprint can not be matched, end. 7) These two fingerprint are matched with n pairs of minutiae, corresponding score is computed accordingly. Remarks 1) In the algorithm 1, if K = 1, then this algorithm will be the normal one for the generalized Hough transform. 2) The precision sequence (ρ 1, ρ 2,, ρ K ) can be chosen arbitrarily to meet the requirement of trade-off between the computation cost and the accuracy. The algorithm proposed here is a hierarchical one in the sense of in algorithm 1. We compute the parameter space of orientation. Based on that partition, the algorithm 2 and 3 will generate different parameter space for different angle precision. The searching algorithm for computing the exact number of matching pairs between U and V can be obtained using the following algorithm, Algorithm 2: Coarse Matching Algorithms in parameter space of θ named H 1 (Q, D, β) 1) Quantizing the parameter space [x min, x max ] [y min, y max ] into an accumulator P (i, j), i = x min,, x max, j = y min,, ymax and P (i, j) = 0, where. are the quantization size. 2) Q = {(x, y) (x, y) = u i R β v j, (i, j) Q} 3) For each point (x, y) Q, i 0 = x (D x )/2, i 1 = x+(d x )/2 ; j 0 = y (D y )/2, j 1 = y+(d y )/2 P (i, j) = P (i, j) + 1, i = i 0,, i 1, j = j 0,, j 1 4) (i max, j max ) = arg max(p (i, j)). i,j 5) return P (i max, j max ). Remarks 1) In step 2, if one point in set V is transformed already, it will no longer be processed. 2) In algorithm 2, the quantization size, will differ to x, y, if = x, = y. It will be the most common matching algorithm, if = x /2, = y /2 as we used here, it is the more accurate matching

7 7 algorithm than usual ones. 3) In fingerprint identification application, the transform of latent fingerprint feature set V can be done once only, then let U varies over all the fingerprint feature sets in template database to speed up the identification process. Algorithm 2 is a very coarse algorithm for the estimation of the number of matched pair between two fingerprint. We use this coarse one for the rough rotation precision to speed up the matching process but a fine one should be used in the last rotation precision case. That is algorithm 3. The precise number of matched pair between two fingerprint is impossible to get but the following theorem will tell us when the precise number of matched pair can be obtained. Theorem 2: The duplicate calculation of the matched pair Suppose two minutiae v j and v k match to u i simultaneously with the error bound xy then v j v k 2 xy The Theorem 2 shows that if u j u k 2 xy, and v j v k 2 xy, j, k, then there is no duplicate count of the matched pair, in other words, no minutiae is matched to more than one minutiae in the other set. Algorithm 3: Fine Matching Algorithms in parameter space of θ named H 2 (Q, D, β, n) 1) Quantizing the parameter space [x min, x max ] [y min, y max ] into an accumulator P (i, j), i = xmin,, xmax, j = y min,, y max and P (i, j) = 0, where. are the quantization size. 2) V = {v v = R β v, v V } 3) Q = {(x, y) (x, y) = u i v j, (i, j) Q} 4) For each point (x, y) Q, i 0 = x (Dx )/2, i 1 = x+(dx )/2 ; j 0 = y (Dy )/2, j 1 = y+(d y )/2 P (i, j) = P (i, j) + 1, i = i 0,, i 1, j = j 0,, j 1 5) (i max, j max ) = arg max(p (i, j)) and δ x = (i max ), δ y = (j max ), i,j 6) if P (i max, j max ) n, return 0; 7) Generating the matching matrix M(U, V ) = {m ij } ij, i = 1,, M, j = 1,, N, where m ij = 0, if u i v j (δ x, δ y ) T ( x, y ) T, otherwise m ij = 1. 8) return m = min( (1 i j set U, V. m ij ), (1 j i m ij )) as the estimation of the matched pair of minutiae between two Remarks 1) In last step the product can be replaced by the binary product operation with the fact that m ij {0, 1}.

8 8 2) The actual number matched pair m 0 satisfies the relation rank(m) m 0 m. 3) When the application prefers the strict criteria for the matched fingerprint, m = rank(m) could be used in last step. 4) The theorem 2 shows that if u j u k 2 xy, and v j v k 2 xy, j, k, then rank(m) = m 0 = m. That also implies m 0 = P (i, j), therefore if we find the two fingerprint feature sets U, V satisfies the condition in theorem 2, we will use algorithm 2 to replace algorithm 3. IV. Experimental Results The performance of the proposed algorithm can be convinced by the comparison with the generalized Hough transform. We conduct the simulation result using the test set of Fingerprint Verification Competition [18]. We choose DB1 of image size , which means the M xy = 150 and maximum rotation is in the range [ 15, 15], x = y = 7, α = 10. Because of the limit of space, we only give the simulation result for different choice of rotation precision array (ρ 1, ρ 2,, ρ K ). Obviously, for the larger ρ, the speed is faster. In the following table, instead of using ρ, we list M xy ρ for different cases as M xy, ρ M xyρ, that means when M xy ρ = 1, corresponding possible distortion of the error bound using in algorithm 1 will be less than 0.5 and can be negligible. The first three columns are the cases without using algorithm 1, for the fair comparison, all the other parts for different cases are same. It shows that the accuracy of two precision elements case is equal to the accuracy of the finest precision case for one precision element and the speed of matching is around 8 times faster and as fast as the coarsest precision case for one precision element. This shows the power of our new algorithm and also the EER (equal error rate, defined in [18]) row shows that if we do not consider the problem of distortion caused by the coarse rotation precision, it will really damage our matching performance in the sense of matching accuracy. M xy ρ (20, 1) EER 2.12% 1.98% 1.59% 1.59% time (ms) A figure of the matching accuracy in the term of FNMR (False Non-Match Rate) and FMR (False Match Rate) are showed in Figure 1, also defined in [18].

9 9 Figure. 1 The matching accuracy for different M xy ρ. V. Conclusions In this paper we analysised two kinds of errors in fingerprint matching, namely missing matched pair and duplicate matching problem and proposed a new algorithm to overcome these difficulties encountered by conventional generalized Hough transform algorithm. The proposed algorithm can achieve the accurate matching result in an efficient way. Experiment results show that the error will degrade the matching performance and our hierarchical Hough transform will overcome this problem in the efficient way. References [1] B.G. Sherlock, D.M. Monro and K. Millard, Fingerprint enhancement by directional Fourier filtering, IEE Proc.-Vis. Image Signal Process., vol. 141, no. 2, pp , [2] J. Nickerson and L. O Gorman, Matched Filter design for fingerprint image enhancement, pattern Recognition, vol. 22, no. 1, pp , [3] N. Ratha, S. Chen, and A.K. Jain, Adaptive flow orientation based feature extraction in fingerprint images, Pattern Recognition, vol. 28, no. 11, , [4] B.M. Mehtre, Fingerprint image analysis for automatic identification, Machine Vision and Applications, vol. 6, , [5] Dario Maio and Davide Maltoni, Direct gray-scale minutiae detection in fingerprints, IEEE Trans. Pattern Anal. Machine Intell., vol. 19, no. 1, pp.27 40, 1997, [6] R. Bahuguna, Fingerprint verification using hologram matched filterings, Proc. Biometric Consortium Eighth Meeting, San Jose, CA, June [7] M. Eshera and K.S. Fu, A graph distance measure for image analysis, IEEE Trans. Syst., Man, Cybern., vol. SMC-13, no. 3, [8] D.K. Isenor, and S.G. Zaky, Fingerprint identification using graph matching, Pattern Recognition, vol. 19, no. 2, pp , [9] K. Hrechak, and J.A. McHugh, Automated fingerprint recognition using structural matching, Pattern Recognition, vol. 23, no. 8, pp , 1990.

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