Recognition, resolution, and complexity of objects subject to affine transformations

The problem of recognizing objects subject to affine transformation in imag es is examined from a physical perspective using the theory of statistical estimation. Focusing first on objects that occlude zero-mean scenes with ad ditive noise, we derive the Cramer-Rao lower bound on the mean-square error in an estimate of the six-dimensional parameter vector that describes an o bject subject to affine transformation and so generalize the bound on one-d imensional position error previously obtained in radar and sonar pattern re cognition. We then derive two useful descriptors from the object's Fisher i nformation that are independent of noise level. The first is a generalized coherence scale that has great practical value because it corresponds to th e width of the object's autocorrelation peak under affine transformation an d so provides a physical measure of the extent to which an object can be re solved under affine parameterization. The second is a scalar measure of an object's complexity that is invariant under affine transformation and can b e used to quantitatively describe the ambiguity level of a general 6-dimens ional affine recognition problem. This measure of complexity has a strong i nverse relationship to the level of recognition ambiguity. We then develop a method for recognizing objects subject to affine transformation imaged in thousands of complex real-world scenes. Our method exploits the resolution gain made available by the brightness contrast between the object perimete r and the scene it partially occludes. The level of recognition ambiguity i s shown to decrease exponentially with increasing object and scene complexi ty. Ambiguity is then avoided by conditioning the permissible range of temp late complexity above a priori thresholds. Our method is statistically opti mal for recognizing objects that occlude scenes with zero-mean background.