Recognition, Resolution, and Complexity of Objects Subject to Affine Transformations

Abstract

The problem of recognizing objects subject to affine transformation in images is examined from a physical perspective using the theory of statistical estimation. Focusing first on objects that occlude zero-mean scenes with additive 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 object subject to affine transformation and so generalize the bound on one-dimensional position error previously obtained in radar and sonar pattern recognition. We then derive two useful descriptors from the object's Fisher information that are independent of noise level. The first is a generalized coherence scale that has great practical value because it corresponds to the width of the object's autocorrelation peak under affine transformation and so provides a physical measure of the extent to which an object can be resolved under affine parameterization. The second is a scalar measure of an object's complexity that is invariant under affine transformation and can be used to quantitatively describe the ambiguity level of a general 6-dimensional affine recognition problem. This measure of complexity has a strong inverse 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 perimeter and the scene it partially occludes. The level of recognition ambiguity is shown to decrease exponentially with increasing object and scene complexity. Ambiguity is then avoided by conditioning the permissible range of template complexity above a priori thresholds. Our method is statistically optimal for recognizing objects that occlude scenes with zero-mean background.