A class of feature extraction criteria and its relation to the Bayes risk estimate

Feature extraction criteria of the form f(D_{l}, \cdots ,D_{M},\Sigma_{1}, \cdots , \Sigma_{M}) are considered where D_{i} and \Sigma_{i} are the conditional means and covariances. The function f is assumed to be invariant under nonsingular linear transformations and coordinate shifts. For the case f(D_{1},D_{2},\Sigma_{o}) , f is shown to depend only upon the distance (D_{2}-D_{1})^{T} \Sigma_{o}^{-1}(D_{2}-D_{1}) between classes. The M class case, f(D_{1}, \cdots ,D_{M}, \Sigma_{o}) , is shown to depend upon the (M-1)M/2 between-class distances (D_{j}-D_{k})^{T} \Sigma^{-1}_{O}(D_{j}-D_{k}) . This criterion is also shown to be equivalent to the mean-square-error of the general Bayes risk estimate. The most general f is reduced to a function of M sets of between-class distances with metrics induced by the conditional covariances. When M=2 this dependence is reduced to two parameters which may be regarded as different between-class distance measures. Finally, the linear mapping is reduced to a one-parameter problem as in the work of Peterson and Mattson.