مرکز منطقه ای اطلاع رساني علوم و فناوري - A class of feature extraction criteria and its relation to the Bayes risk estimate

Abstract :

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.