Face%20Recognition%20using%20Tensor%20Analysis

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
Technique Resistance to Variations in Resistance to Variations in Resistance to Variations in
Pose Illumination Expression
EigenFaces Turk et al., 1991 Average Poor Average
Support Vector Machines Guo et al., 2001 Good Good Good
Multi-resolution analysis Ekenel et al., 2005 Good Good Very Good
TensorFaces Vasilescu et al., 2004 Very Good Very Good Very Good
4
Tensor Algebra Vasilescu et al., 2002

• Higher order generalization of vectors and
  matrices.
• An Nth order tensor is represented as
• A ? 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 ? 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 manifolds estimated geometry.