Geometric symmetry in the quadratic Fisher discriminant operating on image pixels

TL;DR

This paper explores how geometric symmetry can be incorporated into quadratic Fisher Discriminants operating on image pixels, using group theory to improve their generalisation and discrimination in symmetric image ensembles.

Contribution

It introduces a group theory-based procedure to identify and remove redundant QFD coefficients due to symmetry, generalising to higher order polynomial filters.

Findings

Symmetrisation improves QFD generalisation.

Redundant coefficients are effectively discarded.

Applicable to various lattice symmetries and polynomial filters.

Abstract

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.