Master Thesis - Department of Computer Science

This means for a classifier D:
r i D (x) = P (i|x), i = 1, 2, ....., C. (5.2)
Response vector is also known as the soft class label provided by a classifier. The
decision of a classifier can be hardened to a crisp class label c, where c ∈ {1, 2, ......, C},
by maximum (minimum) membership rule, based on the fact that elements of response
vector represent similarity (dissimilarity):
D(x) = arg max r
c
c D(x). (5.3)
In a traditional multiple classifier system, a feature vector x is classified into one of
the C classes using L classifiers {D1, D2, ......, DL}, using the feature vectors xl, l =
1, 2, ..., L respectively. Measurement level (also called response vector level) com-
bination strategies give final decision by fusing the response vectors from multiple
classifiers. Formally, for a feature vector x, response vectors from multiple classifiers
can be organized as a matrix called decision profile (DP):
⎡
⎤ ⎡ ⎤
⎢ d1,1(x)
⎢ . . .
⎢
DP (x) = ⎢ di,1(x)
⎢ . . .
⎣
dL,1(x)
. . .
. . .
. . .
. . .
. . .
d1,j(x)
. . .
di,j(x)
. . .
dL,j(x)
. . .
. . .
. . .
. . .
. . .
d1,C(x) ⎥ ⎢ D1(x) ⎥
⎥ ⎢ ⎥
⎥ ⎢ ⎥
. . . ⎥ ⎢ . . . ⎥
⎥ ⎢ ⎥
⎥ ⎢ ⎥
di,C(x) ⎥ = ⎢ Di(x) ⎥
⎥ ⎢ ⎥
⎥ ⎢ ⎥
. . . ⎥ ⎢
⎥ ⎢ . . . ⎥
⎦ ⎣ ⎦
dL,C(x) DL(x)
We denote i th row of the above matrix as Di(x) = [di,1(x), ....., di,C(x)], where
di,j(x) is the degree of support given by classifier Di to the hypothesis that x belongs
to class j. Di(x) is the response vector of classifier Di for the sample x. The task of
any combination rule is to construct ˜ D(x), the fused output of L classifiers as:
˜D(x) = F(D1(x), . . . , Di(x), . . . , DL(x)). (5.4)
Some fusion techniques known as class-conscious [68], do column-wise class-by-class
operation on DP(x) matrix to obtain ˜ D(x). Example of this type of fusion techniques
are: sum, product, min, max, etc [60]. Another fusion approach known as class-
indifferent [68], use entire DP(x) to calculate ˜ D(x). The later needs training at fusion
level in the sense that it has to create one or more than one templates per class from
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