  This paper discusses a new type of discriminant analysis based on the
orthogonal projection of data onto a generalized difference subspace (GDS). In
our previous work, we have demonstrated that GDS projection works as the
quasi-orthogonalization of class subspaces, which is an effective feature
extraction for subspace based classifiers. Interestingly, GDS projection also
works as a discriminant feature extraction through a similar mechanism to the
Fisher discriminant analysis (FDA). A direct proof of the connection between
GDS projection and FDA is difficult due to the significant difference in their
formulations. To avoid the difficulty, we first introduce geometrical Fisher
discriminant analysis (gFDA) based on a simplified Fisher criterion. Our
simplified Fisher criterion is derived from a heuristic yet practically
plausible principle: the direction of the sample mean vector of a class is in
most cases almost equal to that of the first principal component vector of the
class, under the condition that the principal component vectors are calculated
by applying the principal component analysis (PCA) without data centering. gFDA
can work stably even under few samples, bypassing the small sample size (SSS)
problem of FDA. Next, we prove that gFDA is equivalent to GDS projection with a
small correction term. This equivalence ensures GDS projection to inherit the
discriminant ability from FDA via gFDA. Furthermore, to enhance the
performances of gFDA and GDS projection, we normalize the projected vectors on
the discriminant spaces. Extensive experiments using the extended Yale B+
database and the CMU face database show that gFDA and GDS projection have
equivalent or better performance than the original FDA and its extensions.
