Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
the projection <strong>of</strong> µi (mean <strong>of</strong> i th class) on null space and range space respectively and<br />
can be obtained by,<br />
x i Null = ¯ Q ¯ Q T µi = µi − QQ T µi, (4.12)<br />
x i Range = QQ T µi. (4.13)<br />
The scatter matrices SNull and SRange can be obtained from XNull = [x 1 Null , .., xC Null ]<br />
and XRange = [x 1 Range, .., x C Range] respectively as,<br />
SNull =<br />
SRange =<br />
C�<br />
(x<br />
i=1<br />
i Null − µNull)(x i Null − µNull) T , (4.14)<br />
C�<br />
(x<br />
i=1<br />
i Range − µRange)(x i Range − µRange) T . (4.15)<br />
where µNull and µRange are the mean <strong>of</strong> XNull and XRange respectively,<br />
µNull = 1<br />
C<br />
µRange = 1<br />
C<br />
C�<br />
x<br />
i=1<br />
i Null, (4.16)<br />
C�<br />
x i Range . (4.17)<br />
i=1<br />
Now to obtain the discriminative directions in null space and range space, we need<br />
to maximize the following criteria,<br />
Since, W Null<br />
opt<br />
and W Range<br />
opt<br />
J(W Null<br />
opt ) = arg max<br />
W |W T SNullW |, (4.18)<br />
J(W Range<br />
opt ) = arg max<br />
W |W T SRangeW |. (4.19)<br />
are two independent sets <strong>of</strong> vectors we need to merge for<br />
obtaining maximum discriminability present in a face space. The merging techniques<br />
are described in the Section 4.3.1. The requirements for feature fusion strategy are<br />
two sets <strong>of</strong> discriminatory directions (i.e W Null<br />
opt<br />
and W Range<br />
opt ), whereas for decision<br />
fusion we have to formulate the problem <strong>of</strong> designing two distinct classifiers, based<br />
on these two sets obtained separately from null space and range space.<br />
For decision fusion, we project all the class means µi’s on W Null<br />
opt<br />
to obtain<br />
ΩNull = [ω1 Null , ...., ωC Null ]. Similarly, the projection <strong>of</strong> µi’s on W Range<br />
opt , gives ΩRange =<br />
[ω1 Range , ...., ωC Range ], where:<br />
ω i Null<br />
= W Null<br />
opt µi, i = 1, ..., C (4.20)<br />
ω i Range = W Range<br />
opt µi, i = 1, ..., C. (4.21)<br />
76