Master Thesis - Department of Computer Science

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matrices, which can be organized in a matrix form as: ⎡ ⎢ SM = ⎢ ⎣ S 1,1 w S 2,1 b S 1,2 b . . . . . . S 1,C b S 2,2 w . . . . . . S 2,C b . . . . . . . . . . . . . . . S i,1 b . . . Si,i w . . . S i,C b . . . . . . . . . . . . . . . S C,1 b The diagonal elements of SM, S i,i w S C,2 b . . . . . . SC,C w ⎤ ⎥ ⎦ i = 1, 2, ...., C, represent the within-class scatter matrices for each of the C classes. Similarly, the non-diagonal elements of SM, S i,j b where i, j = 1, 2, ..., C and i �= j, are the between-class scatter matrices for two different classes i and j. Sb and Sw for C classes, denoted by SbC and SwC, can be obtained by summing up all non-diagonal and diagonal entries of SM respectively. So SwC = SbC = C� i=1 C� C� S i=1 i�=j j=1 i,j b S i,i w , (4.49) (4.50) Once SbC and SwC are obtained, optimum linear features can be selected by maxi- mizing tr(S −1 wCSbC). 4.4.3 Algorithm for Decision Fusion In this section, we describe the overall algorithm for our proposed method of decision fusion of two classifiers constructed on range space and null space of within-class scatter. We attempt to combine the complementary information present in null space and range space by decision fusion. The steps of our algorithm is given below: 1. Construction of two classifiers (DNull and DRange) based on null space and range space. i) Compute Sw from training set X = [x 1 1, x 1 2, ..., x 1 N, x 2 1, ..., x C N] using Eqn. 4.1. ii) Perform eigen-analysis of Sw and select first r eigenvectors to form the basis vector set Q for range space V . 90
iii) Project all class means onto null space and range space using the basis vectors for range space only (see Eqn. 4.12-4.13). The sets of class means projected on null space and range space are denoted by XNull and XRange, respectively. iv) Compute the scatter matrices, SNull and SRange of XNull and XRange, respectively. v) Perform eigen-analysis of SNull and SRange to obtain discriminatory direc- tions W Null opt and W Null opt in null space and range space separately. Let, DNull and DRange be two classifiers defined on the null space and range space, respectively. 2. Decision Fusion of DNull and DRange. i) Construct DP (x) for a test sample, x. ii) Obtain ˜ D (using any one among (a) or (b) or (c)) (a) Apply sum rule to obtain ˜ D. (b) Apply product rule to obtain ˜ D. (c) Apply proposed method of decision fusion (see Fig. 4.4 given in section 4.4.2) to obtain ˜ D. iii) Use maximum membership rule to obtain crisp class labels for a test sample x. Experimental results for our proposed method is given in the following section. 4.5 Experimental Results and Discussion We observe the performance of our proposed method on three public face databases: Yale, ORL and PIE. Yale has 15 subjects with 11 samples per class. ORL and PIE consist of 40 and 60 subjects with 10 and 42 samples per class respectively. More details about these databases are provided in Section 3.4.1. 91
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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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Prior to Matching Sensor Level Feat
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Page 69 and 70: Image Rows h(.) g(.) 2 2 Columns Fi
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Page 73 and 74: (a) (a1) (a2) (a3) (b) (b1) (b2) (b
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 97 and 98: ΩNull and ΩRange represents the
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Page 115 and 116: Table 4.3: Effect of increasing num
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Page 127 and 128: 5.3.1 The Algorithm The steps of ou
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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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