## TR#160: On the Structure of Aura and Co-occurrence Matrices for the Gibbs Random Fields

### Rosalind W. Picard and Ibrahim M. Elfadel

Article available in:
Journal of Mathematical

Imaging & Vision

1992, No. 2, pp. 5-25.

The aura matrix of an image indicates how much of each graylevel is
present in the neighborhood 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 field models. We derive this structure by characterizing
configurations 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 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 anti-tridiagonal. 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.

PDF .
Full list of tech reports