Rigid-motion-invariant classification of 3-D textures

This paper studies the problem of 3-D rigid-motion-invariant texture discrimination for discrete 3-D textures that are spatially homogeneous by modeling them as stationary Gaussian random fields. The latter property and our formulation of a 3-D rigid motion of a texture reduce the problem to the study of 3-D rotations of discrete textures. We formally develop the concept of 3-D texture rotations in the 3-D digital domain. We use this novel concept to define a "distance" between 3-D textures that remains invariant under all 3-D rigid motions of the texture. This concept of "distance" can be used for a monoscale or a multiscale 3-D rigid-motion-invariant testing of the statistical similarity of the 3-D textures. To compute the "distance" between any two rotations R(1) and R(2) of two given 3-D textures, we use the Kullback-Leibler divergence between 3-D Gaussian Markov random fields fitted to the rotated texture data. Then, the 3-D rigid-motion-invariant texture distance is the integral average, with respect to the Haar measure of the group SO(3), of all of these divergences when rotations R(1) and R(2) vary throughout SO(3). We also present an algorithm enabling the computation of the proposed 3-D rigid-motion-invariant texture distance as well as rules for 3-D rigid-motion-invariant texture discrimination/classification and experimental results demonstrating the capabilities of the proposed 3-D rigid-motion texture discrimination rules when applied in a multiscale setting, even on very general 3-D texture models.