  This article examines the design of Quadratic Fisher Discriminants (QFDs)
that operate directly on image pixels, when image ensembles are taken to
comprise all rotated and reflected versions of distinct sample images. A
procedure based on group theory is devised to identify and discard QFD
coefficients made redundant by symmetry, for arbitrary sampling lattices. This
procedure introduces the concept of a degeneracy matrix. Tensor representations
are established for the square lattice point group (8-fold symmetry) and
hexagonal lattice point group (12-fold symmetry). The analysis is largely
applicable to the symmetrisation of any quadratic filter, and generalises to
higher order polynomial (Volterra) filters. Experiments on square lattice
sampled synthetic aperture radar (SAR) imagery verify that symmetrisation of
QFDs can improve their generalisation and discrimination ability.
