Phase transitions and multi-scale Markov random fields: application to texture synthesis

The increased use of Markov random field (MRF) models in image processing applications makes it important to understand their behavior under scale transformations. In this work, a multi-scale MRF is viewed as a parameter trajectory on a surface whose topological structure depends on the coarsening scheme employed in the scale transformation. These transformations are based on renormalization theory and are often nonlinear. Their fixed points play an important role in predicting the statistical properties of MRFs under a wide range of coarsening schemes. The concept of phase transition is introduced and its importance in model parameter selection is demonstrated in the context of texture synthesis. Specifically, we show that for grey-level images, the Markov assumption implies nontrivial phase transitions around which small parameter changes lead to very different textures