Master Thesis - Department of Computer Science

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in our case. On the other hand, a between-class scatter matrix is the scatter of the expected vectors around the mixture mean as: C� Sb = Pi(Mi − M0)(Mi − M0) i=1 T . (4.43) where M0 represents the expected vector of the mixture distribution and is given by: C� M0 = E{x} = PiMi. (4.44) i=1 A linear transformation from an n-dimensional x to an m-dimensional y (m < n) is expressed by: y = A T x. (4.45) where A is an n × m rectangular matrix and the column vectors are linearly inde- pendent. The problem of feature extraction for classification is to find the A which optimizes J. It can be easily proved that J is optimized if A constitutes first C-1 eigenvectors of S −1 w Sb [39]. 4.4.2.2 Nonparametric Linear Discriminant Analysis The number of features, C-1, selected in LDA [39] is suboptimal in Bayes sense. Therefore, if the estimate of Bayes error in feature space (y space) is much larger than the one in the original space (x space), the feature extraction process should be augmented. If tr(S −1 w Sb) is used as a criterion, LDA selects the first (C-1)-dimensional subspace containing classification information using the scatter of mean vectors, while the second (n-C+1)-dimensional space containing information due to the covariance- differences is neglected, where n is the dimensionality of x space. Fig. 4.5 shows some of the cases where traditional LDA does not work. Therefore, we need to select additional features from the (n-C+1)-dimensional subspace. The basis of nonparametric LDA [39] is the nonparametric formulation of scatter matrices, using k-nearest neighbor (kNN) techniques, which measure between-class and within-class scatter on a local basis, both being generally of full rank. In addition, the nonparametric na- ture of the scatter matrix inherently leads to extracted features that preserve class structure important for accurate classification. 88
C 2 C 1 C 1 C 2 C X X 2 M X M X 2 1 C 1 (a) M1 = M2 = M0 (b) C 1 (c) C 2 M1 = M2 = M0 Figure 4.5: Examples where LDA based on the criteria tr(S −1 w Sb) fails to discriminate between classes. Let C1 and C2 be two classes, and x (i) refers to samples from Ci. The Ci-local mean of a sample x (l) from class Cl is denoted by mi(x (l) ) and is computed as: mi(x (l) ) = 1 k k� j=1 x (i) jNN , (4.46) where x (i) jNN is the jth NN from Ci for the sample x (l) , a sample of class Cl. Nonparametric between-class scatter matrix for two classes, denoted by Sb can be defined as: Sb = P1E{(x (1) − m2(x (1) ))(x (1) − m2(x (1) )) T |C1} + P2E{(x (2) − m1(x (2) ))(x (2) − m1(x (2) )) T |C2} (4.47) Similarly, nonparametric within-class scatter matrix for two-class is denoted by Sw and can be defined as: Sw = P1E{(x (1) − m1(x (1) ))(x (1) − m1(x (1) )) T |C1} + P2E{(x (2) − m2(x (2) ))(x (2) − m2(x (2) )) T |C2} (4.48) We calculate Sb and Sw for C classes using the above defined scatters for two classes (equations 4.47 and 4.48). For a set of C classes we thus get a total of C 2 scatter 89
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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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features (gradient coherence, inten
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Figure 2.7: Examples of minutiae; A
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according to the local orientation
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• Parallel Mode: This operational
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can address the problem of noisy se
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Page 63 and 64: 3. The final DS soft vector is calc
Page 65 and 66: on three face databases and compare
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Page 69 and 70: Image Rows h(.) g(.) 2 2 Columns Fi
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Page 75 and 76: Genuine Impostor Scores Scores 1 0
Page 77 and 78: Error d1 = (1 − TZeroF RR) d2 = |
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Page 83 and 84: Table 3.3: Peak Recognition Accurac
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Page 87 and 88: Table 3.7: Performance of Ekenel’
Page 89 and 90: Table 3.9: Performance of Ekenel’
Page 91 and 92: to our proposed method. Section 4.2
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Page 99 and 100: Thus, calculation of the QR factori
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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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