Face Recognition using Tensor Analysis

1. Face Recognition using Tensor Analysis Presented by Prahlad R Enuganti

2. Face Recognition • Why is it necessary? • Human Computer Interaction • Authentication • Surveillance • Problems include change in • Illumination • Expression • Pose • Aging

3. Existing Techniques

4. Tensor Algebra[Vasilescu et al., 2002] • Higher order generalization of vectors and matrices. • An Nth order tensor is represented as A Є R I1 x I2 x…. IN and each element by aijk….N The mode n vectors of a tensor are obtained by varying index n while keeping other indices fixed. They are obtained by flattening the tensor A and are represented by A(n) Example of flattening a 3rd order tensor

5. Tensor Decomposition • In case of 2-D, a matrix D can be decomposing using SVD D = U1 ∑ U2T , where ∑ is a diagonal singular matrix, U1 and U2 are column and row orthogonal space respectively In terms of mode - n vectors,the product can be rewritten as D = (∑ ) X1 (U1)X2 (U2) In case of a Tensor of dimension N, the N-mode SVD can be expressed as D = (Z ) X1 (U1)X2 (U2) … … XN (UN) Where Z is known as the core tensor and is analogous to diagonal singular value matrix in 2-D SVD

6. N – mode SVD Algorithm • For n = 1 , 2 … N, compute matrix Un by calculating the SVD of flattened matrix D(n) and setting Un to be the left matrix of the SVD. • Core Tensor can be solved as Z = (D) X1 (U1T) X2 (U2T) ...... XN(UNT)

7. TensorFaces • Our data here consists of 5 variables: people, pixels, pose, illumination and expression. • Therefore we perform the N –mode decomposition of the 5th order tensor and obtain • D = Z X1 Upeople X2 Uviews X3 Uillum X4 Uexpr X5 Upixels The main advantage of tensor analysis is that it maps all images of a person regardless of other variables to the same coefficient vector giving zero inter-class scatter.

8. ISOMAP (Isometric Feature Mapping)[Tenenbaum et al. ] • Finds meaningful low-dimensional manifold of higher dimensional data by preserving the geodesic distances. • Unlike PCA or MDS, ISOMAP is capable of discovering even the nonlinear degrees of freedom. • It is guaranteed to converge to the true structure.

9. ISOMAP : How does it work? • Calculates the weighted neighborhood graph for every point by either the Є neighborhood rule or the k nearest neighbor rule. • Estimates the geodesic distances between all pairs of points on the lower dimensional manifold by computing the shortest path distances in the graph • Applies classical MDS to construct an embedding in the lower dimensional space that best preserves the manifold’s estimated geometry.