Randomized Invariant Features for Shape Classification

Our new notion of a continuous coding for a probability density starts from a general mathematical intuition: if densities are matched, rather than evaluated (as in classical learning theory), then they need not be represented explicitly, but merely need to be coded. The two criteria of uniqueness and continuity that a coding must satisfy are not very restrictive. Indeed, this is exactly why a coding is easier than an explicit representation. Given a density to be coded, is it possible to characterize a minimal coding? More realistically, a useful coding, in order to be small, is likely to satisfy uniqueness and continuity only approximately. Can the quality of a coding be measured, in this sense? To address this question, we can invoke basic principles of the theory of classification: suppose that ideal decision surfaces can be defined for the original (noninvariant) feature vector f. Mapping f to its trajectory under a given set of transformations will perturb these surfaces, because a trajectory that happens to cross a decision surface (that is, a trajectory that contains features that map to different classes) must be relocated in only one class. This perturbation comes with a price, that is, a misclassification rate. An additional price of the same nature is paid when the trajectories (or rather the randomized-feature densities defined over them) are coded. We plan to study misclassification rates for simple families of features and transformations. Theoretical analysis will give insights in elementary cases, and empirical studies will provide data for more realistic scenarios. Next, reversible imaging transformations can be naturally defined for three-dimensional images, as our example in section 2.4 shows in detail. For two dimensional images, the set of reversible transformations is more limited. How can irreversible transformations be approximated by reversible ones for two-dimensional images? For instance, a translation along the optical axis of the camera is an irreversible transformation, because scene details can appear and disappear as the viewpoint changes. However, within relatively wide limits, a simple scaling approximates the effects of translation along the optical axis (or, more rigorously, scaling and translation are exactly equivalent under orthographic projection). Similar considerations hold for rotations outside the image plane.