Master Thesis - Department of Computer Science

The architecture of our proposed method is given in Fig. 4.3 where X i denote the
training set for class i. XNull and XRange are the two sets of means projected on null
space and range space separately. W Null
Opt
and W Range
Opt
are the sets of discriminative
directions available in null space and range space and W Dual
Opt represents the optimal
combined discriminative directions obtained from both spaces. The decision fusion
strategy is described in the following section.
4.4 Decision Fusion
The response vectors obtained from null and range classifiers are combined using
three decision fusion techniques. Among them sum rule and product rule [59], [60] are
adopted from already existing literature and third one is proposed by us. The response
vectors on a sample x obtained from null and range classifiers can be expressed as a
matrix of size 2 × C and is denoted by DP (x) [68],
⎡
⎢
DP (x) = ⎣ d1Null (x) . . . diNull (x) . . . dCNull (x)
d1 Range(x) . . . di Range(x) . . . dC ⎤ ⎡
⎥ ⎢
⎦ = ⎣
Range(x)
DNull(x)
⎤
⎥
⎦
DRange(x)
(4.29)
The task of a combination rule, F, is to construct ˜ D(x) = [ ˜ d 1 (x), . . . , ˜ d i (x), . . . ˜ d C (x)],
the fused soft class label provided by two classifiers as:
˜D(x) = F(DNull(x), DRange(x)) (4.30)
The fused soft class label can be hardened to a crisp class label c, by maximum/
minimum membership rule, based on the fact that measurements of a response vector
represent similarity/dissimilarity:
L(x) = arg max
c
where L(x) represents the class label assigned to x.
4.4.1 Existing Techniques for Decision Fusion
˜d c (x). (4.31)
Given DP (x), sum and product rules operate class-wise on each column of DP (x) to
produce ˜ D(x) [60, 68].
• Sum Rule: Sum Rule computes the soft class label vectors using:
˜d j (x) = d j
Null (x) + djRange(x),
j = 1, ...., C (4.32)
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