Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
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matrices, which can be organized in a matrix form as:<br />
⎡<br />
⎢<br />
SM = ⎢<br />
⎣<br />
S 1,1<br />
w<br />
S 2,1<br />
b<br />
S 1,2<br />
b . . . . . . S 1,C<br />
b<br />
S 2,2<br />
w . . . . . . S 2,C<br />
b<br />
. . . . . . . . . . . . . . .<br />
S i,1<br />
b . . . Si,i w . . . S i,C<br />
b<br />
. . . . . . . . . . . . . . .<br />
S C,1<br />
b<br />
The diagonal elements <strong>of</strong> SM, S i,i<br />
w<br />
S C,2<br />
b . . . . . . SC,C w<br />
⎤<br />
⎥<br />
⎦<br />
i = 1, 2, ...., C, represent the within-class scat-<br />
ter matrices for each <strong>of</strong> the C classes. Similarly, the non-diagonal elements <strong>of</strong> SM,<br />
S i,j<br />
b<br />
where i, j = 1, 2, ..., C and i �= j, are the between-class scatter matrices for two<br />
different classes i and j. Sb and Sw for C classes, denoted by SbC and SwC, can be<br />
obtained by summing up all non-diagonal and diagonal entries <strong>of</strong> SM respectively. So<br />
SwC =<br />
SbC =<br />
C�<br />
i=1<br />
C� C�<br />
S<br />
i=1 i�=j<br />
j=1<br />
i,j<br />
b<br />
S i,i<br />
w , (4.49)<br />
(4.50)<br />
Once SbC and SwC are obtained, optimum linear features can be selected by maxi-<br />
mizing tr(S −1<br />
wCSbC).<br />
4.4.3 Algorithm for Decision Fusion<br />
In this section, we describe the overall algorithm for our proposed method <strong>of</strong> decision<br />
fusion <strong>of</strong> two classifiers constructed on range space and null space <strong>of</strong> within-class<br />
scatter. We attempt to combine the complementary information present in null space<br />
and range space by decision fusion. The steps <strong>of</strong> our algorithm is given below:<br />
1. Construction <strong>of</strong> two classifiers (DNull and DRange) based on null space<br />
and range space.<br />
i) Compute Sw from training set X = [x 1 1, x 1 2, ..., x 1 N, x 2 1, ..., x C N] using Eqn. 4.1.<br />
ii) Perform eigen-analysis <strong>of</strong> Sw and select first r eigenvectors to form the<br />
basis vector set Q for range space V .<br />
90