Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
This means for a classifier D:<br />
r i D (x) = P (i|x), i = 1, 2, ....., C. (5.2)<br />
Response vector is also known as the s<strong>of</strong>t class label provided by a classifier. The<br />
decision <strong>of</strong> a classifier can be hardened to a crisp class label c, where c ∈ {1, 2, ......, C},<br />
by maximum (minimum) membership rule, based on the fact that elements <strong>of</strong> response<br />
vector represent similarity (dissimilarity):<br />
D(x) = arg max r<br />
c<br />
c D(x). (5.3)<br />
In a traditional multiple classifier system, a feature vector x is classified into one <strong>of</strong><br />
the C classes using L classifiers {D1, D2, ......, DL}, using the feature vectors xl, l =<br />
1, 2, ..., L respectively. Measurement level (also called response vector level) com-<br />
bination strategies give final decision by fusing the response vectors from multiple<br />
classifiers. Formally, for a feature vector x, response vectors from multiple classifiers<br />
can be organized as a matrix called decision pr<strong>of</strong>ile (DP):<br />
⎡<br />
⎤ ⎡ ⎤<br />
⎢ d1,1(x)<br />
⎢ . . .<br />
⎢<br />
DP (x) = ⎢ di,1(x)<br />
⎢ . . .<br />
⎣<br />
dL,1(x)<br />
. . .<br />
. . .<br />
. . .<br />
. . .<br />
. . .<br />
d1,j(x)<br />
. . .<br />
di,j(x)<br />
. . .<br />
dL,j(x)<br />
. . .<br />
. . .<br />
. . .<br />
. . .<br />
. . .<br />
d1,C(x) ⎥ ⎢ D1(x) ⎥<br />
⎥ ⎢ ⎥<br />
⎥ ⎢ ⎥<br />
. . . ⎥ ⎢ . . . ⎥<br />
⎥ ⎢ ⎥<br />
⎥ ⎢ ⎥<br />
di,C(x) ⎥ = ⎢ Di(x) ⎥<br />
⎥ ⎢ ⎥<br />
⎥ ⎢ ⎥<br />
. . . ⎥ ⎢<br />
⎥ ⎢ . . . ⎥<br />
⎦ ⎣ ⎦<br />
dL,C(x) DL(x)<br />
We denote i th row <strong>of</strong> the above matrix as Di(x) = [di,1(x), ....., di,C(x)], where<br />
di,j(x) is the degree <strong>of</strong> support given by classifier Di to the hypothesis that x belongs<br />
to class j. Di(x) is the response vector <strong>of</strong> classifier Di for the sample x. The task <strong>of</strong><br />
any combination rule is to construct ˜ D(x), the fused output <strong>of</strong> L classifiers as:<br />
˜D(x) = F(D1(x), . . . , Di(x), . . . , DL(x)). (5.4)<br />
Some fusion techniques known as class-conscious [68], do column-wise class-by-class<br />
operation on DP(x) matrix to obtain ˜ D(x). Example <strong>of</strong> this type <strong>of</strong> fusion techniques<br />
are: sum, product, min, max, etc [60]. Another fusion approach known as class-<br />
indifferent [68], use entire DP(x) to calculate ˜ D(x). The later needs training at fusion<br />
level in the sense that it has to create one or more than one templates per class from<br />
103