Face Recognition Using Contourlet Transform and Multidirectional Illumination from a Computer Screen - Advanced Concepts for Intelligent Vision Systems - page 336

eigenvalues. Let λ n (where n =1 , 2 ,...N G ) represent the eigenvalues in de-

creasing order. We select the subspace dimension (i.e. number of eigenvectors) so
as to retain 90% energy and project the Contourlet coecients to this subspace.
If

×

U sk then the subspace Contourlet
coe cients at scale s and orientation k are given by
B sk =(

U s L

represents the first L eigenvectors of

U s L ) T ( A sk − µ sk p ) , (4)

U s L
represents the subspace for Contourlet coecients at scale s and orientation k .
Similar subspaces are calculated for different scales and orientations using the
training data and each time, the subspace dimension is chosen so as to retain
90% energy. In our experiments, we considered three scales and a total of 15
orientations along with the low pass sub-band image. Fig. 3 shows samples of a
sub-band image and Contourlet coecients at two scales and seven orientations.
The subspace Contourlet coecients were normalized so that the variance
along each of the L dimensions becomes equal. This is done by dividing the
subspace coecients by the square root of the respective eigenvalues. The nor-
malized subspace Contourlet coecients at three scales and 15 orientations of
each image are stacked to form a matrix of feature vectors

µ sk .Notethat where p is a row vector of all 1's and equal in dimension to

where each column
is a feature vector of the concatenated subspace Contourlet coe cients of an
image. These features are once again projected to a linear subspace however,
this time without subtracting the mean. Since the feature dimension is usually
large compared to the size of the training data,

B

BB T is very large. Moreover,

at most N 1 orthogonal dimensions (eigenvectors and eigenvalues) can be

calculated for a training data of size N

× G −

× G .The( N × G )th eigenvalue is always

B T B zero. Therefore, we calculate the covariance matrix C = instead and find

the N × G − 1 dimensional subspace as follows

U SV T = C , (5)

BU / diag( U = S ) . (6)

AU ) is divided by the square root
of the corresponding eigenvalue so that the eigenvectors in

In Eqn. 6, each dimension (i. e. column of

U (i. e. columns) are

AU is ignored to avoid division by zero.

of unit magnitude. The last column of

U

defines an N

× G −

Thus 1 dimensional linear subspace. The feature vectors

are projected to this subspace and used for classification

U T B F = (7)

3 Classification

We tested three different classification approaches. In the first approach, the cor-
relation between the features of the query and the training images was calculated
by

n tq − t q

γ = n (

) 2 n (

, (8)

( t ( q

t ) 2 − q ) 2 − ) 2