Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
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where E[.] is the expectation operator and Ā is the mean image <strong>of</strong> the en-<br />
tire training set. The column vector x is chosen such that the generalized<br />
total scatter criterion is maximized. This means that total scatter <strong>of</strong> the pro-<br />
jected vectors is maximized. It is shown in [141] that the scatter criterion is<br />
satisfied when x is chosen as the orthonormal eigenvectors <strong>of</strong> the image co-<br />
variance matrix Gt. For each eigenvector xs, there exists a projected feature<br />
vector ys. Therefore, the principal components <strong>of</strong> 2D-PCA is a vector unlike<br />
that <strong>of</strong> PCA which is a scalar. If S principal components are used (corre-<br />
sponding to the largest S eigenvalues <strong>of</strong> Gt), the principal component vectors<br />
obtained can be represented as, B = [y1, y2, ..., yS]. Thus, B is the feature<br />
matrix <strong>of</strong> the image sample A. For recognition using the nearest neighbor<br />
strategy, the distance between any two feature matrices Bi = [y (i)<br />
1 , y (i)<br />
2 , ..., y (i)<br />
S ]<br />
and Bj = [y (j)<br />
1 , y (j)<br />
2 , ..., y (j)<br />
S ] is given as,<br />
d(Bi, Bj) =<br />
S�<br />
s=1<br />
�y (i)<br />
s<br />
− y(j)<br />
s �2 (2.15)<br />
where �y (i)<br />
s − y (j)<br />
s �2 is the Euclidean distance between the two principal com-<br />
ponent vectors y (i)<br />
s<br />
and y(j)<br />
s . A test image feature Bt is assigned to class Ck<br />
if, d(Bt, Bl) = min<br />
j d(Bt, Bj), and Bl ∈ Ck. Unlike the conventional PCA, the<br />
2D-PCA does not involve computation <strong>of</strong> a large correlation matrix and there-<br />
fore is relatively less computation intensive. But on the other hand, it requires<br />
more memory for storing feature matrices.<br />
Yang, Zhang et al. [141] tested 2D-PCA method on ORL, AR and Yale face<br />
databases. For the ORL database, the authors used two strategies for experi-<br />
ments: (a) 5 image samples per class for training and (b) leave one out strategy<br />
for observing the average performance. In case (a) the recognition rate is 96%<br />
and in case <strong>of</strong> (b) the same is reported to be 98.3% for ORL. Leave one out<br />
strategy was adopted for Yale database and maximum accuracy is reported to<br />
be 84.24%.<br />
• 2D-LDA (Two dimensional Linear Discriminant Analysis): In the re-<br />
cently proposed 2D-LDA [77], the image is not reordered as a column vector.<br />
The process <strong>of</strong> projection <strong>of</strong> an image using 2D-LDA is given as follows. An<br />
18