Master Thesis - Department of Computer Science

the projection of µi (mean of i th class) on null space and range space respectively and
can be obtained by,
x i Null = ¯ Q ¯ Q T µi = µi − QQ T µi, (4.12)
x i Range = QQ T µi. (4.13)
The scatter matrices SNull and SRange can be obtained from XNull = [x 1 Null , .., xC Null ]
and XRange = [x 1 Range, .., x C Range] respectively as,
SNull =
SRange =
C�
(x
i=1
i Null − µNull)(x i Null − µNull) T , (4.14)
C�
(x
i=1
i Range − µRange)(x i Range − µRange) T . (4.15)
where µNull and µRange are the mean of XNull and XRange respectively,
µNull = 1
C
µRange = 1
C
C�
x
i=1
i Null, (4.16)
C�
x i Range . (4.17)
i=1
Now to obtain the discriminative directions in null space and range space, we need
to maximize the following criteria,
Since, W Null
opt
and W Range
opt
J(W Null
opt ) = arg max
W |W T SNullW |, (4.18)
J(W Range
opt ) = arg max
W |W T SRangeW |. (4.19)
are two independent sets of vectors we need to merge for
obtaining maximum discriminability present in a face space. The merging techniques
are described in the Section 4.3.1. The requirements for feature fusion strategy are
two sets of discriminatory directions (i.e W Null
opt
and W Range
opt ), whereas for decision
fusion we have to formulate the problem of designing two distinct classifiers, based
on these two sets obtained separately from null space and range space.
For decision fusion, we project all the class means µi’s on W Null
opt
to obtain
ΩNull = [ω1 Null , ...., ωC Null ]. Similarly, the projection of µi’s on W Range
opt , gives ΩRange =
[ω1 Range , ...., ωC Range ], where:
ω i Null
= W Null
opt µi, i = 1, ..., C (4.20)
ω i Range = W Range
opt µi, i = 1, ..., C. (4.21)
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