Our goal is to find the best features to classify hand signs. The best criterion to evaluate feature sets is the Bayes error. However, in practice, it is hard to obtain a posteriori probability functions, and since the sample size is small, their estimates have severe biases and variances. One frequently used criterion in practice is based on a family of functions of scatter matrics, which are conceptually simple and give systematic feature extraction algorithms.
Suppose samples of Y are m-dimensional
random vectors from c classes. The ith class has a probability
, a mean vector 
and a scatter matrix 
. The
within-class scatter matrix is defined by
The between-class scatter matrix is
where the grand mean m is defined as
.
The mixture scatter matrix is the covariance matrix of all
the samples regardless of their class assignments:
Suppose we use k-dimensional linear features 
where
W is an 
rectangular matrix whose column vectors are
linearly independent. The above mapping represents a linear projection
from m-dimensional space to k-dimensional space. The samples
project to a corresponding set of samples
whose within-class scatter, and between-class scatter
matrices are 
and 
, respectively.
Thus, the problem of feature extraction for classification is to find
which maximizes 
. 
is larger when
the between class scatter is larger or the within-class scatter is smaller.
For details of computing , the reader is referred to [26].
We call the feature extracted by the above method the
most discriminating features (MDFs).