Subspace Methods for Visual Learning and Recognition

Transcription

1 Subspace Methods for Visual Learning and Recognition Horst Bischof Inst. f. Computer Graphics and Vision, Graz Univesity of Technology Austria Aleš Leonardis Faculty of Computer and Information Science, University of Ljubljana Slovenia

2 Outline Part 1 Motivation Appearance based learning and recognition Subspace methods for visual object recognition Principal Components Analysis (PCA) Linear Discriminant Analysis (LDA) Canonical Correlation Analysis (CCA) Independent Component Analysis (ICA) Non-negative Matrix Factorization (NMF) Kernel methods for non-linear subspaces 2

3 Outline Part 2 Robot localization Robust representations and recognition Robust PCA recognition Scale invariant recognition using PCA Illumination insensitive recognition Representations for panoramic images Incremental building of eigenspaces Multiple eigenspaces for efficient representation Robust building of eigenspaces Research issues 3

4 The name of the game complex objects/scenes varying pose (3D rotation, scale) cluttered background/foreground occlusions (noise) varying illumination 4

5 Object Representation High-Level Shape Models (e.g., Generalized Cylinders) Idealized Images Texture Less Mid-Level Shape Models (e.g. CAD models, Superquadrics) More Complex Well-defined geometry Low-level Appearance Based Models (e.g. Eigenspaces) Most complex Complicated shapes 5

6 Problems Segmentation: Pose/Shape: 6

7 Illumination 7

8 Example 8

9 Learning and recognition 3D reconstruction matching learning matching scene training images input image 9

10 Appearance-based approaches A renewed attention in the appearance-based approaches Encompass combined effects of: shape, reflectance properties, pose in the scene, illumination conditions. Acquired through an automatic learning phase. 10

11 A variety of successful applications: Appearance-based approaches Human face recognition e.g. [Beymer & Poggio, Turk & Pentland] Visual inspection e.g. [Yoshimura & Kanade] Visual positioning and tracking of robot manipulators, e.g. [Nayar & Murase] Tracking e.g., [Black & Jepson] Illumination planning e.g., [Murase & Nayar] Image spotting e.g., [Murase & Nayar] Mobile robot localization e.g., [Jogan & Leonardis] Background modeling e.g., [Oliver, Rosario & Pentland] 11

12 Appearance-based approaches Objects are represented by a large number of views: 12

13 Subspace Methods Images are represented as points in the N-dimensional vector space Set of images populate only a small fraction of the space Characterize subspace spanned by images Image set Basis images Representation 13

14 Subspace Methods Properties of the representation: Optimal Reconstruction PCA Optimal Separation LDA Optimal Correlation CCA Independent Factors ICA Non-negative Factors NMF Non-linear Extension Kernel Methods 14

15 Image Matching ρ = x T y x y > Θ Normalized images x y 2 < Ψ Compress images 15

16 16 Eigenspace representation Image set (normalised, zero-mean) We are looking for orthonormal basis functions: Individual image is a linear combination of basis functions ) ( ) ( )) ( ) ( ( ) ( ) ( y x u y x u y u x y x j j k j j j j k j j j k j j j q q q q q q = = = = =

17 Optimisation problem Best basis functions ν? Taking the k eigenvectors with the largest eigenvalues of PCA or Karhunen-Loéve Transform (KLT) 17

18 Efficient eigenspace computation n << m Compute the eigenvectors u' i, i = 0,...,n-1, of the inner product matrix The eigenvectors of XX T can be obtained by using XX T Xv i '=λ' i Xv i ': 18

19 Principal Component Analysis 19

20 Principal Component Analysis = q 1 + q 2 + q

21 Image representation with PCA u 1 u 2 u 3 21

22 Image presentation with PCA 22

23 Properties PCA Any point x i can be projected to an appropriate point q i by : q i = U T (x i - µ) and conversely (since U -1 = U T ) Uq i + µ = x i Y x i U T (x i -µ) Y X Uq i + µ q i X

24 Properties PCA It can be shown that the mean square error between x i and its reconstruction using only m principle eigenvectors is given by the expression : j= 1 j= 1 PCA minimizes reconstruction error N λ j m λ j = N λ j= m+ 1 j PCA maximizes variance of projection Finds a more natural coordinate system for the sample data.

25 PCA for visual recognition and pose estimation Objects are represented as coordinates in an n-dimensional eigenspace. An example: 3-D space with points representing individual objects or a manifold representing parametric eigenspace (e.g., orientation, pose, illumination). 25

26 PCA for visual recognition and pose estimation Calculate coefficients Search for the nearest point (individual or on the curve) Point determines object and/or pose 26

27 Calculation of coefficients To recover a i the image is projected onto the eigenspace Complete image x i is required to calculate a i. Corresponds to Least-Squares Solution 27

28 Linear Discriminant Analysis (LDA) PCA minimizes projection error Best discriminating Projection PCA-Projection PCA is unsupervised no information on classes is used Discriminating information might be lost 28

29 LDA Linear Discriminance Analysis (LDA) Maximize distance between classes Minimize distance within a class Fisher Linear Discriminance ρ( w) = w w T T S S B W w w 29

30 LDA: Problem formulation n Sample images: c classes: { x,, } 1 L x n { χ,, } 1 L χ c Average of each class: Total average: µ i µ = = 1 n i 1 x k χ N n k = 1 i x k x k 30

31 31 LDA: Practice Scatter of class i: ( )( ) T i k x i k i x x S i k µ µ χ = = = c i S W S i 1 ( )( ) = = c i T i i i S B 1 µ µ µ µ χ B W T S S S + = Within class scatter: Between class scatter: Total scatter:

32 LDA: Practice After projection: y = W k T x k Between class scatter (of y s): ~ S B = W T S B W Within class scatter (of y s): ~ S W = W T S W W 32

33 LDA S 1 S B S W = S 1 + S 2 S 2 Good separation 33

34 Maximization of ρ( w) = w w T T S S B W w w LDA is given by solution of generalized eigenvalue problem S B w = λs w w For the c-class case we obtain (at most) c-1 projections as the largest eigenvalues of S B w = λs i ww i 34

35 LDA How to calculate LDA for high-dimensional images? Problem: S W is always singular Number of pixels in each image is larger than the number of images in the training set 1. Fischerfaces Reduce dimension by PCA and then perform LDA 2. Simultaneous diagonalization of S W and S B 35

36 LDA Fischerfaces (Belhumeur et.al. 1997) Reduce dimensionality to n-c with PCA U pca = arg max U U T QU = [ u 1 u 2... u n c ] Further reduce to c-1 with FLD W fld = T T W WpcaSBWpcaW arg max = [ w1 w2... wc 1] W T T W W SWW W pca pca The optimal projection becomes W opt = W Τ U Τ fld 36

37 LDA Example Fisherface of recognition Glasses/NoGlasses (Belhumeur et.al. 1997) 37

38 LDA Example comparison for face recognition (Belhumeur et.al. 1997) Superior performance than PCA for face recognition Noise sensitive Requires larger training set, more sensitive to different training data [Martinez&Kak2001] 38

39 Canonical Correlation Analysis (CCA) Also supervised method but motivated by regression tasks, e.g. pose estimation. Canonical Correlation Analysis relates two sets of observations by determining pairs of directions that yield maximum correlation between these sets. Find a pair of directions (canonical factors) w x R p, w y R q, so that the correlation of the projections c = w xt x and d = w yt y becomes maximal. 39

40 What is CCA? Canonical Correlation 0 r 1 Between Set Covariance Matrix ρ = E[ w w T x C E[ c T x w xx E[ cd] E[ w xx T x w 2 T C x ] E[ d xy T x w w x w xy T y y T ] E[ w C 2 ] yy w = y w T y ] y yy T w y ] = 40

41 What is CCA? Finding solutions w = w w x y A = 0 C yx C 0 xy, B = C 0 xx C 0 yy Rayleigh Quotient Generalized Eigenproblem r = w w T T Aw Bw Aw = µbw 41

42 CCA for images Same problem as for LDA Computationally efficient algorithm based on SVD A A = = C 1 2 xx C UDV xy T C 1 2 yy w w xi yi = = C C 1 2 xx 1 2 yy u v i i 42

43 Properties of CCA At most min(p,q,n) CCA factors Invariance w.r.t. affine transformations Orthogonality of the Canonical factors w w w T xi T yi T xi C C C xx yy xy w w w xj yj yj = = =

44 CCA Example Parametric eigenspace obtained by PCA for 2DoF in pose

45 CCA Example CCA representation (projections of training images onto w x1, w x2 )

46 Independent Component Analysis (ICA) ICA is a powerful technique from signal processing (Blind Source Separation) Can be seen as an extension of PCA PCA takes into account only statistics up to 2 nd order ICA finds components that are statistically independent (or as independent as possible) Local descriptors, sparse coding 46

47 Independent Component Analysis (ICA) m scalar variables X=(x 1... x m ) T They are assumed to be obtained as linear mixtures of n sources S=(s 1... s n ) T X = AS Task: Given X find A, S (under the assumption that S are independent) 47

48 ICA Example Original Sources Mixtures Recovered Sources 48

49 ICA Example ICA basis obtained from 16x16 patches of natural images (Bell&Sejnowski 96) 49

50 ICA Algorithms 1. Minimize (Maximize) function Complex matrix (tensor) functions 2. Adaptive Algorithms based on stochastic gradient Measure of independence Non-Gaussian, e.g. Kurtosis, Negentropy Fast ICA Algorithm (Hyvärinen) 1. W = W / Repeat until 3 2. W = 2 W WW 1 2 T convergence WW T 50

51 ICA Properties ICA works only for Non-Gaussian Sources Usually centering and Whiteing of data is performed We can not measure the variance of the components X = AP 1 PS ICA does not provide ordering ICA components are not orthogonal 51

52 ICA for Noise Suppression Sparse Code Shrinkage (similar to Wavelet Shrinkage Hyvärinen 99) 52

53 Face Recognition using ICA PCA vs. ICA on Ferret DB (Baek et.al. 02) PCA ICA 53

54 Non-Negative Matrix Factorization (NMF) How can we obtain part-based representation? Local representation where parts are added E.g. learn from a set of faces the parts a face consists of, i.e. eyes, nose, mouth, etc. Non-Negative Matrix Factorization (Lee & Seung 1999) lead to part based representation 54

55 Matrix Factorization - Constraints V WH PCA : W are orthonormal basis vectors W = [ 1 2, L w, w, w n ], w i w j = δ ij VQ : H are unity vectors H T 1 L = [ h, h2, L, hn], h j = [0,0,1,0,,0] NMF: V,W,H are non-negative Vij, Wij, H ij 0 i, j 55

56 NMF - Cost functions Euclidean distance between A and B A B 2 = ( A ij B ij ij ) 2 Divergence of A from B (Relative entropy) D( A B) = A ij ( Aij log Aij + B ij ij B ij ) 56

57 NMF - update rules V WH 2 is non-increasing under H aµ H aµ ( W ( W T T V ) aµ WH ) aµ W iµ W iµ ( VH ( WHH ) T iµ T ) iµ D( V WH ) is non-increasing under H aµ H aµ i W V ia iµ k W ( WH ) ka iµ W ia W ia H V µ aµ iµ iµ ν W ( WH ) aν We can start with random matrices for W and H and update each matrix iteratively until W and H are at a stationary point - the cost functions are invariant at this point. 57

58 Concrete example Handwritten Digits Training data set for learning process 58

59 Learning Find basis images from the training set Training images Basis images 59

60 Reconstruction New image Basis images Encoding X [Non-negative] [Non-negative] Approximation 60

61 Face features Basis images Encoding (Coefficients) Reconstructed image 61

62 Kernel Methods All presented methods are linear Can we generalize to non-linear methods in a computational efficient manner? 62

63 Kernel Methods Kernel Methods are powerful methods (introduced with Support Vector Machines) to generalize linear methods BASIC IDEA: 1. Non-linear mapping of data in high dimensional space 2. Perform linear method in high-dimensional space Non-linear method in original space 63

64 Kernel Trick Problem: High-dimensional spaces: E.g. N=16x16 polynomial of degree Can we avoid computing the non-linear mapping directly? E.g. polynomial and inner products Φ( x) Φ( y) = ( x 1, 2x1x2, x2 )( y1, 2 y1 y2, y2 ) = ( xy) = k( x, y) If algorithm can be specified in terms of dot products and non-linearity satisfies Mercers condition we can apply the kernel trick 64

65 Example kernels Gaussian: Polynomial : Sigmoid: x y k ( x, y ) = exp( ) 2 2σ d k ( x, y ) = ( x y + c), c 0 k ( x, y) = σ ( κ (x y) + Θ ), κ, Θ 2 R Nonlinear separation can be achieved. 65

66 Kernel Principal Component Analysis KPCA carries out a linear PCA in the feature space F The extracted features take the nonlinear form f k (x) = l i= 1 α k i k(x i x), The k αi are the components of the k-th eigenvector of the matrix ( k (x x )) i j ij 66

67 KPCA and Dot Products Find eigenvectors V and eigenvalues λ of the covariance matrix C = 1 l Φ ( x i ) Φ ( x i ) T l i = 1. Again, replace with Φ(x) Φ(y). k(x,y). 67

68 Kernel PCA Toy Example Artificial data set from three point sources, 100 point each. 68

69 De-noising in 2-dimensions A half circle and a square in the plane De-noised versions are the solid lines 69

70 Kernel-CCA Reformulation of CCA for finite sample size n X p n matrix of training images Y q n matrix of pose parameters Aˆ T T = 1 0 XY 1 XX 0 = T, Bˆ n 1 XY 0 n 1 0 YY T 70

71 Kernel-CCA Theorem The component vectors w x *, w y * of the extremum points w* of r = w w T T Aw ˆ Bw ˆ lie in the span of the training data X, Y, i.e., * * f g : w = Xf, w =, x y Yg 71

72 Kernel-CCA III T K= X X T L= Y Y n n Inner Product (Gram) Matrix r = ( T T f g ) 0 LK KK KL f 0 g 0 f ( T T f g ) 0 LL g 72

73 Kernel-CCA Apply non-linear transformations φ(), θ() to the data φ(x) = < φ(x 1 ),, φ(x n ) > θ(y) = < θ(y 1 ),, θ(y n ) > The Kernel Trick: K ij = φ(x i ) T φ(x j ) = k φ (x i,x j ) L ij = θ(y i ) T θ(y j ) = k θ (y i,y j ) Inner Product in Feature Space Kernel Evaluation in Input Space 73

74 Experiments Hippo: Rotated through 360 o (1 DOF) in 2 o steps. CCA-Factor Linear Y-Encoding: y i = turntable position α i in degrees. Estimates for y i w x1 74

75 Experiments Trigonometric Y-Encoding: y i = <sin(α i ),cos(α i )> CCA-Factors Estimates for y i1,y i2 atan2 w x 1 w x 2 75

76 Experiments Using Kernel-CCA, optimal output features can be found automatically Application of a RBF-kernel to the scalar output parameters α i yielded two factors pairs with a canonical correlation of 1. 76

77 Outline Part 1 Motivation Appearance based learning and recognition Subspace methods for visual object recognition Principal Components Analysis (PCA) Linear Discriminant Analysis (LDA) Canonical Correlation Analysis (CCA) Independent Component Analysis (ICA) Non-negative Matrix Factorization (NMF) Kernel methods for non-linear subspaces 77

78 Outline Part 2 Robot localization Robust representations and recognition Robust recognition using PCA Scale invariant recognition using PCA Illumination insensitive recognition Representations for panoramic images Incremental building of eigenspaces Multiple eigenspaces for efficient representation Robust building of eigenspaces Research issues 78

79 Mobile Robot 79

80 Panoramic image 80

81 Environment map environments are represented by a large number of views localisation = recognition 81

82 Compression with PCA 82

83 Image representation with PCA 83

84 Localisation 84

85 Distance vs. similarity 85

86 Robot localisation Interpolated hyper-surface represents the memorized environment. The parameters to be retrieved are related to position and orientation. Parameters of an input image are obtained by scalar product. 86

87 Localisation 87

88 Enhancing recognition and representations Occlusions, varying background, outliers Robust recognition using PCA Scale variance Multiresolution coefficient estimation Scale invariant recognition using PCA Illumination variations Illumination insensitive recognition Rotated panoramic images Spinning eigenimages Incremental building of eigenspaces Multiple eigenspaces for efficient representations Robust building of eigenspaces 88

89 Occlusions 89

90 Standard recovery of coefficients 90

91 Non-robustness 91

92 Robust method 92

93 Robust algorithm 93

94 Selection 94

95 Robust recovery of coefficients 95

96 Robustness Experimental results 96

97 Recognition and pose estimation 97

98 Robust localisation under occlusions 98

99 Robust localisation at 60% occlusion Standard approach Robust approach 99

100 Mean error of localisation Mean error of localisation with respect to % of occlusion 100

101 Multiresolution coefficient estimation Multiresolution a well-known technique to reduce computational complexity a search for the solution at the coarsest level and then a refinement through finer scales Standard eigenspace method cannot be applied in an ordinary multiresolution way it relies on the orthogonality of eigenimages. 101

102 Standard multiresolution coefficient estimation Eigenimages in each resolution layer are computed from a set of templates in that layer Computationally costly and requires additional storage space 102

103 Robust multiresolution coefficient estimation Robust method requires only a single set of eigenimages obtained on the finest resolution. Linear system of equations: does not require orthogonality. 103

104 Multiresolution coefficient estimation 104

105 Scaled images 105

106 Scale estimation 106

107 Numerical demonstration 107

108 Multiresolution approach Estimate scale & coefficients simultaneously in the pyramid Efficient search structure 108

109 Experimental results test image 109

110 Experimental results 110

111 Illumination insensitive recognition Recognition of objects under varying illumination global illumination changes highlights shadows Dramatic effects of illumination on objects appearance Training set under a single ambient illumination 111

112 Illumination insensitive recognition Our Approach Global eigenspace representation Local gradient based filters Efficient combination of global and local representations Robust coefficient recovery in eigenspaces 112

113 Eigenspaces and filtering 113

114 Filtered eigenspaces 114

115 Gradient-based filters Global illumination Gradient-based filters Steerable filters [Simoncelli] 115

116 Robust coefficient recovery Highlights and shadows Robust coefficient recovery Robust solution of linear equations = a + a + a = a + a + a = a + a + a M M a + a + = a L +L +L Hypothesize & Select +L 116

117 Experimental results Test images Our approach Standard method Demo 117

118 Experimental results Robust filtered method - all eigenvectors used obj % ang avg Standard method - all eigenvectors used obj % ang avg

119 Illumination invariant localisation Illumination variations and occlusions 119

120 Filtered eigenvectors 120

121 Experimental results Training set: straight path, uniform illumination 121

122 Experimental results Test sets T/1/2/3 without occlusion Test sets 4/5/6/7 with occlusion 122

123 Experimental results Comparison with standard method 123

124 Experimental results Comparison with standard method 124

125 Experimental results Comparison with standard method Coefficient error, test sets 2 and 6 125

126 Experimental results Comparison with standard method Coefficient error, test sets 3 and 7 126

127 Experimental results Average localisation error (in cm). 127

128 Rotated panoramic images 128

129 Unwrapping 129

130 rotated/shifted n times A rotated panoramic image Inner product matrix Q=X T X symmetric, Toeplitz, circulant 130

131 Eigenvectors of a circulant matrix Shift theorem: the eigenvectors of a general circulant matrix are the N basis vectors from the Fourier matrix F=[u 0 ', u 1 ',..., u n-1 '], where The eigenvalues can be calculated simply by retrieving the magnitude of the DFT of one row of Q 131

132 From u i ' to u i The eigenvectors of XX T can be obtained by using XX T Xu i '=λ' i Xu i ': eigenvectors u i same frequency as u i ', phase and amplitude may change 132

133 Eigenvectors 133

134 Generalisation to several locations 134

135 A set of rotated images P different locations, each shifted n times Every Q ij is circulant (but in general not symmetric!) Is it possible to exploit these properties? It is still possible to compute the eigenvectors without performing the SVD decomposition of A. 135

136 Rotated panoramic images 136

137 Eigenvalue problem Solution of the problem Aw' = µw' matrix blocks Q jl of A are circulant matrices every circulant matrix can be diagonalised in the same basis by Fourier matrix F all the submatrices Q jk have the same set of eigenvectors 137

138 Aw' = µw' written blockwise: Derivations Since u i ' is an eigenvector of every Q jl, λ i jk is an eigenvalue of Q jl corresponding to u i '. 138

139 Derivations This implies a new eigenvalue problem Λα i = µα ι, where and 139

140 Computational complexity Since Q jk =Q T kj, it can be proved that Λ is Hermitian and we have P linearly independent eigenvectors α i, which provide P linearly independent eigenvectors w' i. Since the same procedure can be performed for every v' i, we can obtain N P linearly independent eigenvectors of A. It is therefore possible to solve the eigen-problem using N decompositions of order P (as opposed to decomposition of P N). 140

141 Complex eigenspace of spinning images Real and imaginary part of one of the vectors: Real and imaginary part of one of the w vectors: 141

142 Eigenvectors 142

143 Energy distribution compressing efficiency of the eigenspace 62 images, each in 50 different orientations 143

144 Timings for building the eigenspace Images of dimensions 40x68, each image was rotated/shifted 68 times, i.e., for 40 locations we got 2720 images. This is also the number of image elements (the border case when the covariance matrix is of the same size as the inner product matrix, and the complexity of the SVD method reaches its upper bound). 144

145 Eigenspace of spinning images K-L expansion of a set of rotated panoramic images SVD on the complete covariance matrix is not necessary Instead, we solve a set of smaller eigen-problems The final eigenvectors are composed of locally varying harmonic functions (analytic functions!) 145

146 Other approaches Images stored in arbitrary orientation Images stored in a reference orientation (e.g. gyrocompass) Autocorrelation FFT power spectra Zero Phase Representation Eigenspace of spinning-images 146

147 Batch computation of PCA i i + 1 k l 147

148 Incremental computation of PCA i i + 1 k 148

149 Incremental computation of PCA Algorithm increase dimensionality discard a dimension preserve dimensionality 149

150 Determining the number of eigenvectors Increase the number of dimensions by one if the distance between the last image and its projection is high. Increase the number of dimensions by one if the cumulative distance between the images and their projections is high. 150

151 Incremental PCA in detail Extend U with a residual h n of the new image y and rotate by R Rotation matrix R is a result of the eigenproblem Λ' is the new eigenvalue matrix ; 151

152 Incremental PCA in detail D is assembled from the current eigenvalues Λ, the new image y and its corresponding coefficients a 152

153 Comparison with batch method 153

154 Localization with incremental method Matched nate Location [m] of location a training test image image 154

155 Reconstruction error through time 155

156 Multiple Eigenspaces - Motivation A single eigenspace high dimensionality low-dimensional structure of data is ignored poor generalisation Ad-hoc partitioning of the image set is not efficient 156

157 Multiple eigenspaces our goal Systematically construct multiple low-dimensional eigenspaces from a set of training images Each image is described as a linear combination Design a numerically feasible and robust procedure 157

158 Eigenspace growing and selection 158

159 Multiple eigenspaces - experiments 159

160 Eigenspace growing and selection 160

161 Eigenimages of individual eigenspaces 161

162 Mean images of individual eigenspaces 162

163 "Box" images in four eigenspaces 163

164 "Block" images in five eigenspaces 164

165 Multiple eigenspaces 165

166 Robust Subspace Learning Subspace learning from data containing outliers: Detect outliers Learn using only inliers. [D. Skočaj, A. Leonardis, H. Bischof: A robust PCA algorithm for building representations from panoramic images, ECCV 2002] 166

167 EM algorithm for learning Solving systems of linear equations Only in nonmissing pixels E-step: Estimate coefficients A using given principal vectors U. M-step: Estimate principal vectors U using given coefficients A. Smoothing in missing pixels 167

168 Energy function Minimization of reconstruction error in non-missing pixels (the main property of PCA). Smoothing reconstructed values in missing pixels (additional constraint to prevent over-fitting). 168

169 Robust learning algorithm Input: Learning images containing outliers and occlusions. 1. Compute PV using SVD on the whole image set. 2. Detect outliers (pixels with large reconstruction error). 3. Compute PV from inliers using EM algorithm. 4. Repeat until change in outlier set is small. Output: Principal subspace, learning images without outliers, detected outliers and occlusions. 169

170 Experimental results synthetic data ground truth added outliers standard PCA 2PC standard PCA 8PC robust PCA 8PC 170

171 Experimental results real data input standard PCA robust PCA outliers 171

172 Research issues Comparative studies (e.g., LDA versus PCA, PCA versus ICA) Robust learning of other representations (e.g. LDA, CCA) Integration of robust learning with modular eigenspaces Local versus Global subspace represenations Combination of subspace representations in a hierarchical framework 172

173 Further readings Recognizing objects by their appearance using eigenimages (SOFSEM 2000, LNCS 1963) Robust recognition using eigenimages (CVIU 2000, Special Issue on Robust Methods in CV) Hierarchical top down enhancement of robust PCA (SSPR 2002) Illumination insensitive eigenspaces (ICCV 2001) Mobile robot localization under varying illumination (ICPR 2002) Eigenspace of spinning images (OMNI 2000, ICPR 2000, ICAR 2001) Incremental building of eigenspaces (ICRA 2002, ICPR 2002) Multiple eigenspaces (Pattern Recognition, In press) Robust building of eigenspaces (ECCV 2002) Generalized canonical correlation analysis (ICANN 2001) 173