Page 219 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

208 FEATURE EXTRACTION AND SELECTION

Therefore, if in the transformed domain D elements have to be selected,
the selection with maximum Bhattacharyya distance is found as the set
with the largest contributions. If the elements are sorted according to:

p ffiffiffiffiffi 1 p ffiffiffiffiffi 1 p ffiffiffiffiffiffiffiffiffi 1

0 þ p ffiffiffiffiffi
1 þ p ffiffiffiffiffi
N 1 þ p ffiffiffiffiffiffiffiffiffi ð6:41Þ

0
1
N 1
then the first D elements are the ones with optimal Bhattacharyya
distance. Let U D be an N D submatrix of U containing the D corres-
T
ponding eigenvectors of 1/2 V C 2 V 1/2 . The optimal linear feature
extractor is:

1
T
W ¼ U V T ð6:42Þ

2
D
and the corresponding Bhattacharyya distance is:

1 D 1 1 p 1
X
J BHAT ðWzÞ¼ ln ffiffiffiffi ffiffiffiffi ð6:43Þ

i þ p
2
i¼0 2
i
Figure 6.8(d) shows the decision function following from linear feature
extraction backprojected in the two-dimensional measurement space.
Here, the linear feature extraction reduces the measurement space to a
one-dimensional feature space. Application of Bayes classification in this
space is equivalent to a decision function in the measurement space defined
by two linear, parallel decision boundaries. In fact, the feature extraction is
a projection onto a line orthogonal to these decision boundaries.

The general Gaussian case

If both the expectation vectors and the covariance matrices depend on
the classes, an analytic solution of the optimal linear extraction problem
is not easy. A suboptimal method, i.e. a method that hopefully yields
a reasonable solution without the guarantee that the optimal solution
will be found, is the one that seeks features in the subspace defined by
the differences in covariance matrices. For that, we use the same simul-
taneous decorrelation technique as in the previous section. The Bhatta-
charyya distance in the transformed domain is:

1
T
X
X
T
J BHAT ðU V zÞ¼ 1 N 1 d 2 i þ 1 N 1 ln 1 p ffiffiffiffi 1 ffiffiffiffi ð6:44Þ

2

i þ p
4
i¼0 1 þ
i 2 i¼0 2
i