Master Thesis - Department of Computer Science

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Gramm-Schmidt Orthonormalization is given bellow. 4.3.1.1 Covariance Sum Method Here the problem deals with the computation of an eigenmodel from two different sets of feature vectors XNull and XRange with the aim to combine the eigenmod- els constructed separately on the two sets. As XNull and XRange contain the class means projected on null space and range space, this combination will merge the discriminative directions from both spaces. Let the combined feature vector set be Z = [XNull, XRange] and total number of feature vectors in Z is L = 2 ∗ C. The combined mean can be expressed as, µZ = 1 L (CµNull + CµRange) = 1 2 (µNull + µRange) (4.24) Then the combined covariance matrix can be written as, SZ = 1 � C� x L i=1 i Null (xiNull )T + Using Eqn. 4.14-4.15 in the above equation, we get, C� x i=1 i Range (xiRange )T � − µZµ T Z (4.25) SZ = 1 � CSNull + CµNullµ L T Null + CSRange + CµRangeµ T � Range − µZµ T Z = 1 2 SNull + 1 2 SRange + 1 4 (µNull − µRange)(µNull − µRange) T . (4.26) Last term in the Eqn. 4.25 allows for a change of mean. Now the eigenvectors Φ = [β1, ..., βd] of SZ can be computed from, SZΦ = ΦΘ and Φ T Φ = I. (4.27) The range space of SZ will give W Dual (= [β1, ..., βk]), where k is the rank of SZ. 4.3.1.2 Gramm-Schmidt Orthonormalization Gramm-Schmidt Orthonormalization technique [42] is used to compute the QR factorization. Given A, a m-by-n (m > n) matrix, it’s QR factorization can be written as A = QR, where Q ∈ R m×m is orthogonal and R ∈ R m×n is upper triangular. If A has full column rank then the first n columns of Q form an orthogonal basis for range(A). 78
Thus, calculation of the QR factorization is one way to compute an orthogonal basis for a set of vectors. In our case A = [W Null opt , W Range opt ] ∈ R d×2(C−1) , m = d and n = 2(C − 1), where d >> 2(C − 1). The following algorithm computes the factorization A = QR where the columns of Q constitute the orthogonal basis for A and form W Dual . Algorithm for Gramm-Schmidt Orthonormalization for k = 1 : n end R(k, k) = �A(1 : m, k)�2 Q(1 : m, k) = A(1 : m, k)/R(k, k) for j = k + 1 : n end R(k, j) = Q(1 : m, k) T A(1 : m, j) A(1 : m, j) = A(1 : m, j) − Q(1 : m, k)R(k, j) Finally, Q is assigned to W Dual . 4.3.2 Search Criterion and Techniques for Optimal Feature Selection After obtaining W Dual , we select an optimal feature set from W Dual to form W Dual opt maximizing a class separability criterion. A set of features providing maximum value of separability criterion on a validation set (disjoint from training and testing set) is selected for evaluating on the testing subset of the database. The class separability criterion used in our method is given by, J = tr(Sb) . (4.28) tr(Sw) where Sb and Sw are between-class and within-class scatter given in Eqn. 4.1-4.2. We use backward and forward selection techniques which avoid exhaustive enumeration [39] as used in branch and bound selection technique. Even though these methods do not guarantee the selection of best possible feature subset, we use them for their simplicity. 79 by
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DEVELOPMENT OF EFFICIENT METHODS FO
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To My Parents, Sisters and Dearest
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Mirnalinee, Shreyasee, Lalit, Manis
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LDC Linear Discriminant Classifier
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3 An Efficient Method of Face Recog
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A.6 Minutiae Matching . . . . . . .
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4.1 Effect of increasing number of
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List of Figures 2.1 Summary of appr
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3.7 Area difference between genuine
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Abstract Biometrics is a rapidly ev
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CHAPTER 1 Introduction The issues a
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1.1.1 Applications Biometrics has b
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• Spoof Attacks: An impostor may
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• A new approach for multimodal b
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CHAPTER 2 Literature Review This re
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ange [23], infrared scanned [137] a
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u 2 x 2 u 1 x 1 (a) PCA basis (b) P
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PCA Projection Class 1 Class 2 LDA
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F DIFS DFFS F (a) (b) F 1 L Figure
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image A of m rows and n columns is
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faces of 40 subjects. • Others: A
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malizing the vector components, the
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Figure 2.5: A fingerprint image wit
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(i) For each block centered at (i,
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(a) (b) Figure A.7: Enhanced images
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(a) (b) Figure A.10: (a) and (b) sh
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(a) (b) Figure A.12: Binarized imag
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(a) (b) Figure A.14: Thinned images
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(a) (b) Figure A.16: Thinned finger
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Paired Minutiae Minutiae with unmat
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Bibliography [1] FVC 2002 website.
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[25] Li-Fen Chen, Hong-Yuan Mark Li
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[49] Lin Hong, Yifei Wan, and Anil
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[71] L. Lam and C. Y. Suen. Applica
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IEEE Tran. on Pattern Analysis and
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of-the-Art Systems. IEEE Tran. on P
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and Human Communication, pages 374-
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Master Thesis - Department of Computer Science

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