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

Rosalind W. Picard

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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.