Page 218 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

LINEAR FEATURE EXTRACTION 207

a two-dimensional measurement space with samples from two classes.
The covariance matrices of both classes are depicted as ellipses.
T
Figure 6.8(b) shows the result of the operation 1/2 V . The operation
V T corresponds to a rotation of the coordinate system such that the
ellipse of class ! 1 lines up with the axes. The operation 1/2 corres-
ponds to a scaling of the axes such that the ellipse of ! 1 degenerates into
a circle. The figure also shows the resulting covariance matrix belonging
to class ! 2 .
T
The result of the operation 1/2 V on z is that the covariance matrix
associated with ! 1 becomes I and the covariance matrix associated with
T
! 2 becomes 1/2 V C 2 V 1/2 . The Bhattacharyya distance in the trans-
formed domain is:

2 3
1 1 1 2 T 1 2
T 6 jI þ V C 2 V j 7
2 q ð6:38Þ
2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5
J BHAT ð V zÞ¼ ln4
1
1
T
2 N j V C 2 V j

2
2
The second step consists of decorrelation with respect to ! 2 . Suppose
that U and are matrices containing the eigenvectors and eigenvalues of
T
the covariance matrix 1/2 V C 2 V 1/2 . Then, the operation
T
T
U 1/2 V decorrelates the covariance matrix with respect to class ! 2 .
The covariance matrices belonging to the classes ! 1 and ! 2 transform
T
into U IU ¼ I and , respectively. Figure 6.8(c) illustrates the decorrel-
ation. Note that the covariance matrix of ! 1 (being white) is not affected
T
by the orthonormal operation U .
The matrix is a diagonal matrix. The diagonal elements are denoted
T
T

i ¼ i, i . In the transformed domain U 1/2 V z, the Bhattacharyya
distance is:
" # N 1
1
1
T
T
J BHAT ðU V zÞ¼ ln jI þ j 1 X ln 1 p ffiffiffiffi 1 ð6:39Þ

2
ffiffiffiffiffiffi ¼

i þ p
2 2 N p j j 2 i¼0 2 ffiffiffiffi

i
The expression shows that in the transformed domain the contribution
to the Bhattacharyya distance of any element is independent. The con-
tribution of the i-th element is:
1 1 p 1

ln ffiffiffiffi ð6:40Þ

i þ p
2 2 ffiffiffiffi

i