Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
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a lower dimensional space using PCA and then LDA is applied to this PCA subspace.<br />
But the removed subspace may contain some useful discriminative information which<br />
is lost in this process.<br />
Chen et al. [25] suggested that the null space <strong>of</strong> the within-class scatter Sw<br />
contains most discriminative information, and hence they proposed LDA in null space<br />
<strong>of</strong> Sw called N-LDA. However, when the number <strong>of</strong> training samples is large, the null<br />
space becomes small resulting in loss <strong>of</strong> discriminative information outside this null<br />
space. The performance <strong>of</strong> the null space depends on the dimension <strong>of</strong> the null<br />
space. Thus any preprocessing that reduces original sample space should be avoided.<br />
Another drawback <strong>of</strong> this approach is that it involves solving the eigenvalue problem<br />
for a very high dimensional matrix. Yu et al. [138] proposed an algorithm which<br />
basically uses simultaneous diagonalization method [39]. First, the null space <strong>of</strong><br />
between-class scatter Sb is removed, assuming that the null space <strong>of</strong> Sb contains no<br />
discriminative information, and then the method seeks a projection to minimize Sw<br />
in the transformed subspace <strong>of</strong> Sb. As the rank <strong>of</strong> Sb is smaller than that <strong>of</strong> Sw,<br />
removing the null space <strong>of</strong> Sb may lose entire null space <strong>of</strong> Sw. So Sw is likely to be <strong>of</strong><br />
full rank after this removal [14, 28, 45]. One more drawback is that the whitening <strong>of</strong><br />
Sb is redundant in this method. Another novel method, proposed by Huang et al. [45]<br />
uses PCA+Null space approach. Here the core idea is that null space <strong>of</strong> Sw is useful for<br />
discrimination unlike that <strong>of</strong> Sb. Since the null space <strong>of</strong> total scatter matrix St is the<br />
intersection <strong>of</strong> the null spaces <strong>of</strong> Sb and Sw, this method applies PCA to remove the<br />
null space <strong>of</strong> St and then Null space method is used on reduced subspace <strong>of</strong> St. Hakan<br />
et al. [20] proposed a novel scheme to solve SSS problem called Discriminant common<br />
vectors (DCV) method. Among two algorithms to extract the discriminant common<br />
vectors for representing each person in the training set <strong>of</strong> face database, one algorithm<br />
uses within-class scatter matrix <strong>of</strong> the samples in the training set while the other uses<br />
the subspace methods and the Gramm-Schmidt orthogonalization procedure. These<br />
DCVs are used for classification <strong>of</strong> new faces.<br />
Most <strong>of</strong> the null space based methods like DCV [20], PCA+Null space approach<br />
[45] explored null space <strong>of</strong> Sw while others like PCA+LDA [120] and Fisherface [12]<br />
eliminated a part <strong>of</strong> null space to obtain discriminative features. None <strong>of</strong> them ex-<br />
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