This correspondence 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 symmetrization of any quadratic filter, and generalizes to higher order polynomial (Volterra) filters. Experiments on square lattice sampled synthetic aperture radar (SAR) imagery verify that symmetrization of QFDs can improve their generalization and discrimination ability.

Stochastic background models incorporating spatial correlations can be used to enhance the detection of targets in natural terrain imagery, but are generally difficult to apply when the statistics are non-Gaussian. Chapple and Bertilone (see Opt. Commun., vol.150, p.71-76, 1998) proposed a simple stochastic model for images of natural backgrounds based on the pointwise nonlinear transformation of Gaussian random fields, and demonstrated its effectiveness and computational efficiency in modeling the textures found in natural terrain imagery acquired from airborne IR sensors. In this paper, we show how this model can be used to design algorithms that detect small targets (up to several pixels in size) in natural imagery by identifying anomalous regions of the image where the statistics differ significantly from the rest of the background. All of the model-based algorithms described here involve nonlinear spatial processing prior to the final decision threshold. Monte Carlo testing reveals that the model-based algorithms generally perform better than both the adaptive threshold filter and the generalized matched filter for detecting low-contrast targets, despite the fact that they do not require the target statistics needed for constructing the matched filter.

I derive two-stage, statistically suboptimal target detectors for images. The first, or transformation, stage is a "generalized matched filter" (GMF) that linearly transforms the input image. I propose three rational signal-to-noise-ratio criteria whose maximization yields the three GMFs. The second, or detection, stage is a univariate "Neyman-Pearson detector" (NPD), which executes a pointwise likelihood ratio test on the GMF transformed images. Experiments on infrared and synthetic-aperture radar imagery compare GMF/NPDs with several established detectors.