Structure of aura and co-occurrence matrices for the Gibbs texture model

Abstract

The aura matrix of an image indicates how much of each gray level is present in the neighborhood of each other gray level and generalizes the popular texture-analysis tool, the co-occurrence matrix. In this paper we show that interesting structure appears in both the aura and co-occurrence matrices for textures that are synthesized from Gibbs random-field models. We derive this structure by characterizing configurations of the distribution that are most likely to be synthesized when the Gibbs energy is minimized. This minimization is an important part of applications that use the Gibbs model within a Bayesian estimation framework for maximum a posteriori (MAP) estimation. In particular, we show that the aura matrix will become tridiagonal for an attractive autobinomial field when suitable constraints exist on the histogram, neighborhood, and image sizes. Under the same constraints, but where the field is repulsive instead of attractive, the matrix will become antitridiagonal. The interpretation of this structure is especially significant for modeling textures with minimum-energy configurations: zeros in the matrix prohibit certain colors from occurring next to each other, thus prohibiting large classes of textures from being formed.