On the Structure of Aura and Co occurrence Matrices for the Gibbs Texture Model

The aura matrix of an image indicates how much of each graylevel is present in the neigh borhood of each other graylevel 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 which are synthesized from Gibbs random eld models We derive this struc ture by characterizing con gurations of the distribution which are most likely to be synthesized when the Gibbs energy is minimized This minimization is an important part of applications which 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 auto binomial eld when suitable constraints exist on the histogram neighborhood and image sizes Under the same con straints but where the eld is repulsive instead of attractive the matrix will become anti tridiagonal The interpretation of this structure is especially signi cant for modeling textures with minimum en ergy con gurations zeros in the matrix prohibit certain colors from occurring next to each other thus prohibiting large classes of textures from being formed Index Terms co occurrence matrix energy minimization pattern recognition texture modeling Gibbs Markov random eld This work was supported by the National Science Foundation and the Defense Advanced Research Projects Agency DARPA under Grant No MIP the National Science Foundation under Grant No IRI the Rome Air Development Center RADC of the Air Force System Command and the Defense Advanced Research Projects Agency DARPA under contract No F C Introduction Since the equivalence between Markov and Gibbs random elds GRF s was established by the Hammersley Cli ord theorem there has been a great interest in using random eld models for images and image texture patterns A nice variety of textures have been shown to exist as samples of a Gibbs random eld GRF s are also frequently incorporated into a Bayesian framework where often by simulated annealing a maximization is performed of an a posteriori probability In these cases and others where textures are being synthesized as samples of Gibbs models with low energy it is helpful to know the kinds of patterns that are likely to be formed In a recent paper we have shown that the Gibbs energy can be computed for a large class of GRF models by using a generalized form of graylevel co occurrences that we call aura measures When organized in a matrix indexed by the graylevels the aura measures form an aura matrix that is a generalization of the co occurrence matrix a popular texture analysis tool We say that a ground state aura matrix is the aura matrix corresponding to the minimum energy pattern The purpose of the present paper is to show that the structure of the ground state aura matrix can be determined algebraically for a large class of GRF texture models We provide a detailed analysis of the auto binomial GRF case and indicate how the methodology can be applied to the case of another popular texture model the Potts model The results described in this paper are signi cant because of the following reasons Knowing the structure of the ground state aura matrix helps characterize the minimum energy pattern and when it is attained These results indicate apparently for the rst time a relationship between a texture synthesis model and its co occurrence matrix Identifying properties of a texture which correspond to structure in co occurrence matrices has been an important pursuit Most importantly we show for the rst time that the ground state aura matrix structure implies strong restrictions about which classes of patterns can and cannot be generated as minimum energy con gurations of the auto binomial GRF The paper is organized as follows Section contains the notation assumptions and basic de nitions of the paper It is also motivational in that it describes the basic experimental discovery that has led to the theory developed in this paper Background for the aura framework is provided in Section Sections are the mathematical groundwork of the paper Based on Birkho s theorem about the convex hull of permutation matrices we show why what has been observed experimentally is theoreti cally bound to happen given the assumptions and the constraints of the problem We have assigned the proofs to a set of appendices that can be consulted as desired our results can be understood without reading these appendices In Section we comment on some of the applications and implications of the results obtained in this paper Finally the last section is devoted to a summary of our ndings Background Notation and assumptions Let an image be represented by a nite rectangular M N lattice S with a neighborhood structure N fNs s Sg where Ns S is the set of neighbors of the site s S Every site has a graylevel value xs f n g Let x be the vector xs s jSj of site graylevel values and be the set of all con gurations taken by x A neighborhood structure is said to be symmetric if s r S s Nr if and only if r Ns The set of all sites with graylevel g is Sg fs Sjxs gg g The vector of graylevel values is g n T The basic methodology for GRF texture synthesis is the following For the nite periodic lattice S with the symmetric neighborhood structure fNs s Sg we de ne the local two site interaction potentials between neighboring pixels Vsr xs xr r Ns and the Gibbs energy E x X