Master Thesis - Department of Computer Science

x
DNull
DRange
DP (x)
DNull(x)
DRange(x)
LDA−based
Eigenmodel
LDA−based
Eigenmodel
¯
DP (x)
HNull(x)
HRange(x)
Sum
Rule
˜D(x)
Maximum
Membership
Rule
Crisp Class Label
Figure 4.4: The fusion architecture of the proposed method of decision fusion.
defined in Eqn. 4.29) is replaced by ¯
DP (x), as:
¯
DP (x) =
⎡
⎢
⎣ HNull(x)
HRange(x)
⎤
⎥
⎦ (4.40)
where, H can be D or ¯ D based upon the success of the eigensubspaces constructed
from LDA or nonparametric LDA-based eigenmodel. Figure 4.4 gives the diagram of
the proposed decision fusion technique.
The general discussion on LDA and nonparametric LDA are given in the following:
4.4.2.1 Linear Discriminant Analysis
The objective in LDA [39] [35] (which is also known as Fisher’s Linear Discriminant)
is to find the optimal projection such that the between-class scatter is maximized
and within-class scatter is minimized. It is thus required to maximize the Fisher’s
criterion (J) as shown below:
J = tr(S −1
w Sb) (4.41)
where tr(.) is the trace of the matrix and Sb and Sw are the between-class and within-
class scatter matrices, respectively. A within-class scatter matrix shows the scatter
of samples around their respective class expected vectors, and is expressed by:
C�
Sw = PiE{(x − Mi)(x − Mi)
i=1
T C�
|i} = PiΣi. (4.42)
i=1
Mi and Σi are mean and covariance matrix of i th class, respectively and C is total
number of classes. Pi is the number of training sample for i th class and is equal to N
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