Curvature Analysis of Pattern Transformation Manifolds

Transformation manifolds are quite attractive for image analysis applications that require transformation invariance properties. The geometric structure of a transformation manifold has a profound influence on the design of processing algorithm, and the curvature is a major parameter in the characterization of the manifold geometry. We propose here a procedure for the computation of an upper bound for the maximum principal curvature of a pattern transformation manifold. We provide an analytical formulation of the curvature bound and show that the numerical computation of this bound is mostly dependent on the rotation parameters. Experimental results indicate that the curvature bound of the manifold has considerable dependence on the spatial complexity and smoothness of the generating pattern. Moreover, experiments with discretization of manifolds suggest that the curvature of the manifold is likely to affect the accuracy of compact representation and sampling algorithms.