Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
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The architecture <strong>of</strong> our proposed method is given in Fig. 4.3 where X i denote the<br />
training set for class i. XNull and XRange are the two sets <strong>of</strong> means projected on null<br />
space and range space separately. W Null<br />
Opt<br />
and W Range<br />
Opt<br />
are the sets <strong>of</strong> discriminative<br />
directions available in null space and range space and W Dual<br />
Opt represents the optimal<br />
combined discriminative directions obtained from both spaces. The decision fusion<br />
strategy is described in the following section.<br />
4.4 Decision Fusion<br />
The response vectors obtained from null and range classifiers are combined using<br />
three decision fusion techniques. Among them sum rule and product rule [59], [60] are<br />
adopted from already existing literature and third one is proposed by us. The response<br />
vectors on a sample x obtained from null and range classifiers can be expressed as a<br />
matrix <strong>of</strong> size 2 × C and is denoted by DP (x) [68],<br />
⎡<br />
⎢<br />
DP (x) = ⎣ d1Null (x) . . . diNull (x) . . . dCNull (x)<br />
d1 Range(x) . . . di Range(x) . . . dC ⎤ ⎡<br />
⎥ ⎢<br />
⎦ = ⎣<br />
Range(x)<br />
DNull(x)<br />
⎤<br />
⎥<br />
⎦<br />
DRange(x)<br />
(4.29)<br />
The task <strong>of</strong> a combination rule, F, is to construct ˜ D(x) = [ ˜ d 1 (x), . . . , ˜ d i (x), . . . ˜ d C (x)],<br />
the fused s<strong>of</strong>t class label provided by two classifiers as:<br />
˜D(x) = F(DNull(x), DRange(x)) (4.30)<br />
The fused s<strong>of</strong>t class label can be hardened to a crisp class label c, by maximum/<br />
minimum membership rule, based on the fact that measurements <strong>of</strong> a response vector<br />
represent similarity/dissimilarity:<br />
L(x) = arg max<br />
c<br />
where L(x) represents the class label assigned to x.<br />
4.4.1 Existing Techniques for Decision Fusion<br />
˜d c (x). (4.31)<br />
Given DP (x), sum and product rules operate class-wise on each column <strong>of</strong> DP (x) to<br />
produce ˜ D(x) [60, 68].<br />
• Sum Rule: Sum Rule computes the s<strong>of</strong>t class label vectors using:<br />
˜d j (x) = d j<br />
Null (x) + djRange(x),<br />
j = 1, ...., C (4.32)<br />
83