Page 220 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

LINEAR FEATURE EXTRACTION 209

where d i are the elements of the transformed difference of expectation,
T
T
i.e. d ¼ U 1/2 V m.
Equation (6.44) shows that in the transformed space the optimal
features are the ones with the largest contributions, i.e. with the largest
1 d 2 1 1 p 1

i ffiffiffiffi
þ ln
i þ p ffiffiffiffi ð6:45Þ
4 1 þ
i 2 2
i
The extraction method is useful especially when most class information
is contained in the covariance matrices. If this is not the case, then the
results must be considered cautiously. Features that are appropriate for
differences in covariance matrices are not necessarily also appropriate
for differences in expectation vectors.

Listing 6.2
PRTools code for calculating a Bhattacharryya distance feature extractor.

z ¼ gendatl([200 200],0.2); % Generate a dataset
J ¼ bhatm(z,0); % Calculate criterion values
figure; clf; plot(J, ‘r.-’); % and plot them
w ¼ bhatm(z,1); % Extract one feature
figure; clf; scatterd(z); % Plot original data
figure; clf; scatterd(z*w); % Plot mapped data

6.3.2 Feature extraction based on inter/intra class distance

The inter/intra class distance, as discussed in Section 6.1.1, is another
performance measure that may yield suitable feature extractors. The
starting point is the performance measure given in the space defined by
T
y ¼ 1/2 V z. Here, is a diagonal matrix containing the eigenvalues of
S w , and V a unitary matrix containing the corresponding eigenvectors. In
the transformed domain the performance measure is expressed as (6.10):

1 1
T
2
2
J INTER=INTRA ¼ traceð V S b V Þ ð6:46Þ
A further simplification occurs when a second unitary transform is
applied. The purpose of this transform is to decorrelate the between-
scatter matrix. Suppose that is a diagonal matrix whose diagonal
elements
i ¼ i, i are the eigenvalues of the transformed between-scatter