Local Image Structure and Procrustes Metrics

The differential geometry of images (one of numerous applications but certainly an important one) involves singly isotropic space, rather than Euclidean space, the reason being that image intensity is not commensurate with the dimensions of the image plane. The Procrustes root mean square nonuniformity measure in such spaces immediately leads to a principled definition of curvedness, shape index, and orientation of local second-order structure. However, it is categorically different from the conventional curvature-based measures. One obtains a natural Euclidean metric of shape space that is readily extended to the cubic and quartic orders of approximation. In this setting, it is simple to derive the marginal probability densities for curvedness, shape index, and orientation for isotropic and anisotropic Gaussian random fields. For slight anisotropies, the marginals are much closer to those empirically found in natural images than the conventional formalisms suggest. The main articulation fits the edge and ridge structure imposed by the linear and quadric orders. The cubic and quartic structures contribute in a natural manner to these edge and ridge structures, whereas their higher-order saddle structures contribute little to the variance and can generally be ignored in applications.