Master Thesis - Department of Computer Science

where E[.] is the expectation operator and Ā is the mean image of the en-
tire training set. The column vector x is chosen such that the generalized
total scatter criterion is maximized. This means that total scatter of the pro-
jected vectors is maximized. It is shown in [141] that the scatter criterion is
satisfied when x is chosen as the orthonormal eigenvectors of the image co-
variance matrix Gt. For each eigenvector xs, there exists a projected feature
vector ys. Therefore, the principal components of 2D-PCA is a vector unlike
that of PCA which is a scalar. If S principal components are used (corre-
sponding to the largest S eigenvalues of Gt), the principal component vectors
obtained can be represented as, B = [y1, y2, ..., yS]. Thus, B is the feature
matrix of the image sample A. For recognition using the nearest neighbor
strategy, the distance between any two feature matrices Bi = [y (i)
1 , y (i)
2 , ..., y (i)
S ]
and Bj = [y (j)
1 , y (j)
2 , ..., y (j)
S ] is given as,
d(Bi, Bj) =
S�
s=1
�y (i)
s
− y(j)
s �2 (2.15)
where �y (i)
s − y (j)
s �2 is the Euclidean distance between the two principal com-
ponent vectors y (i)
s
and y(j)
s . A test image feature Bt is assigned to class Ck
if, d(Bt, Bl) = min
j d(Bt, Bj), and Bl ∈ Ck. Unlike the conventional PCA, the
2D-PCA does not involve computation of a large correlation matrix and there-
fore is relatively less computation intensive. But on the other hand, it requires
more memory for storing feature matrices.
Yang, Zhang et al. [141] tested 2D-PCA method on ORL, AR and Yale face
databases. For the ORL database, the authors used two strategies for experi-
ments: (a) 5 image samples per class for training and (b) leave one out strategy
for observing the average performance. In case (a) the recognition rate is 96%
and in case of (b) the same is reported to be 98.3% for ORL. Leave one out
strategy was adopted for Yale database and maximum accuracy is reported to
be 84.24%.
• 2D-LDA (Two dimensional Linear Discriminant Analysis): In the re-
cently proposed 2D-LDA [77], the image is not reordered as a column vector.
The process of projection of an image using 2D-LDA is given as follows. An
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