1. Subspace Representation for Face Recognition Presenters: Jian Li and Shaohua Zhou

2. Overview 4 different subspace representations PCA, PPCA, LDA, and ICA 2 options Kernel v.s. Non-Kernel 2 databases with 3 different variations Pose, Facial expression, and Illumination

3. Subspace representations Training data X (d,n) X = [x 1, x 2, …, x n ] Subspace decomposition matrix W (d,m) W = [w 1, w 2, …, w m ] Representation Y (m,n) Y = W’ * X

4. PCA, PPCA, LDA and ICA PCA, in an unsupervised manner, minimizes the representation error ||X – Y||. LDA, in a supervised manner, minimizes the within-class distance while maximizing the between-class distance. ICA, in an unsupervised manner, maximizes the independence between Y ’s. Probabilistic PCA, coming late …

5. Kernel or Non-Kernel Often somewhere reduces to some forms related to dot product Kernel trick Replacing dot product by kernel function Mapping the original data space into a high-dimensional feature space K(x,y) = Gaussian kernel: exp(- 0.5 |x – y|^2/sigma^2)

6. Gallery, Probe, Pre-processing Training dataset Testing dataset Gallery: Reference images in testing Probe: Probe images in testing Pre-processing Down-sampling Zero-mean-unit-variance x = { x - mean(x) } / var(x) Crop face region only

7. AT&T Database Pose variation 40 classes, 10 images/class, 28 by 23 Set1 Set2 (Mirror of Set1)

8. FERET Database Facial expression and illumination variation 200 classes, 3 images/class, 24 by 21 Set1 Set2 Set3

9. Probabilistic PCA (PPCA) -- I PCA only extracts PCs thereby losing probabilistic flavor PPCA add this by interpreting the reconstruction error as confidence level y = u + W * x + e Different choices of e Factor analysis, PPCA (Tipping and Bishop ’99) PCA

10. Probabilistic PCA (PPCA) -- II Assume e has covariance matrix, pho*I R = U * D * U’ W = U m * (D m – pho*I) ^(1/2) Pho = mean of the remaining eigenvalues Implemented algorithm B. Moghaddam ’01 W = U m * (D m) ^(1/2) - 2log P(y) = sum (Pci^2/Di) + e^2 / pho + const Construct inter-person space

11. Probabilistic KPCA (PKPCA) Replace PCA by KPCA in the PPCA algorithm Estimating e by computing sum of all remaining PC’s.

12. ICA Independent face PCA pre-whitening: X1 = U’ * X Y = W * X1 Independent facial expression Y = W * X’

13. Kernel ICA F. Bach and M. I. Jordan ‘01 ‘Kernel trick’ is played when measuring independence Canonical correlation -- independence

14. Experimental Setup Training Ranking the gallery based on the distance or probability CMS curve

15. Distance Metric SAD, SQD, Correlation (mean removed)

16. Tweaking Gaussian kernel width

17. Eigenfaces & Fisherfaces Eigenfaces Fisherfaces

18. Independent Basis Faces & Facial Features Ind. Faces Ind. Facial Features

19. Performance on pose variation

20. Performance on facial expression variation

21. Performance on illumination variation

22. Comparison of 4 methods

23. Comparison of Kernel/Non- kernel methods

24. Computational load Training time: PCA < LDA < PPCA < ICA KPCA < KLDA < PKPCA << KICA Testing time: PCA = LDA = ICA < PPCA KPCA = KLDA = KICA < PKPCA