## TR#168: Texture Modeling: Temperature Effects on markov/Givvs Random Fields

### Rosalind W. Picard

Article available in:
SC.D. Thesis, MIT

Dept. of Elec. Eng. and Comp. Sci.

Advisor: A. P. Pentland

Temperature is a key parameter in the original physical formulation of
the Markov/Gibbs random field (MRF) model; however, it has been
neglected in most applications of these models to image processing and
computer vision. This thesis studies the effect of temperature on
texture and develops new characterizations of textures that are
synthesized with the MRF. The main contributions are as follows:
* We develop a new ``aura'' framework which describes the
neighborhood dependency of the MRF as morphological dilation. The
``aura matrix'' is used to write the Gibbs energy as a linear
combination of its co-occurrence matrices; this is the first time that
an MRF has been shown to be specified by a set of its co-occurrences.
* A new interpretation of MRF pattern formation based on the
physical mechanisms of mixing and separation is presented. For small
neighborhoods we relate this mechanism to boundary length
optimization. These results generalize the boundary length behavior
known to characterize the Ising model of statistical mechanics.
* New relationships between graylevel, temperature, and
MRF equilibrium are identified. These indicate that many of the
textures in the literature may have not been in equilibrium. In
particular, increasing the number of graylevels behaves similarly to
lowering the temperature; both require longer times to reach
equilibrium.
* We derive a new interpretation of the MRF bonding parameters
as annealing rate constants which control the graylevel mixing and
separation in different directions.
* Aura matrix features (related to co-occurrence features)
are shown to exhibit temperature dependent behavior similar to a phase
transition.
* The idea of ``transition temperature,'' analogous to critical
temperature, is developed for an MRF texture. The peaked region
of the specific heat is related to the bandwidth transitions of the
aura matrix.
* We develop the first characterization of texture produced
by an MRF in its ground state. The result shows how energy
minimization for an MRF may restrict the model from producing large
and important classes of natural textures.