Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
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two vectors can be expressed in the following way:<br />
d(yi, yj) = �yi − yj� 2 . (2.2)<br />
The desired class label for the probe image can be obtained by minimum mem-<br />
bership rule.<br />
L(xt) = arg min<br />
c rc. (2.3)<br />
• LDA (Linear Discriminant Analysis): The objective <strong>of</strong> LDA is to find the<br />
subspace that best discriminates different face classes by maximizing between-<br />
class scatter, while minimizing the within-class scatter. The eigenvectors cho-<br />
sen by LDA provide the best separation among the class distributions, while<br />
PCA selects eigenvectors which provide best representation <strong>of</strong> the overall sam-<br />
ple distribution. To illustrate the difference, Fig. 2.3 shows the first projection<br />
vector chosen by PCA and LDA for a two class problem. The eigenvectors for<br />
LDA can be obtained by computing the eigenvectors <strong>of</strong> S −1<br />
w Sb. Here, Sb and<br />
Sw are the between-class and within-class scatter matrices <strong>of</strong> training samples<br />
and are defined as:<br />
Sw =<br />
C�<br />
�<br />
i=1 xk∈Ci<br />
(xk − mi)(xk − mi) T , (2.4)<br />
C�<br />
Sb = ni(mi − m)(mi − m)<br />
i=1<br />
T . (2.5)<br />
where mi is the mean face for i th class and ni is the number <strong>of</strong> training samples<br />
in i th class. LDA subspace is spanned by a set <strong>of</strong> vectors W, which maximizes<br />
the criterion, J, defined as:<br />
J = tr(Sb)<br />
. (2.6)<br />
tr(Sw)<br />
W can be constructed by the eigenvectors <strong>of</strong> S −1<br />
w Sb. In most <strong>of</strong> the image pro-<br />
cessing applications, the number <strong>of</strong> training samples is usually less than the<br />
dimension <strong>of</strong> the sample space. This leads to the so-called small-sample-size<br />
(SSS) problem due to the singularity <strong>of</strong> the within-class scatter matrix. To<br />
overcome SSS problem, the following approaches are attempted: a two stage<br />
PCA+LDA approach [120], Fisherface method [12] and discriminant compo-<br />
nent analysis [143]. In all cases the higher dimension face data is projected<br />
14