Equivalence of Julesz and Gibbs texture ensembles

Research on texture has been pursued along two different lines. The first line of research, pioneered by Julesz (1962), seeks the essential ingredients in terms of features and statistics in human texture perception. This leads us to a mathematical definition of texture as a Julesz ensemble. A Julesz ensemble is the maximum set of images that share the same value of some basic feature statistics as the image lattice Λ→Z², or equivalently it is a uniform distribution on this set. The second line of research studies statistical models, in particular, Markov random field (MRF) and FRAME models (Zhu et al., 1997), to characterize texture patterns locally. In this article, we bridge the two lines by the fundamental principle of equivalence of ensembles in statistical mechanics (Gibbs, 1902). We prove that 1) the conditional probability of an arbitrary image patch given its environment, under the Julesz ensemble or the uniform model, is inevitably a FRAME (MRF) model, and 2) the limit of the FRAME (MRF) model, which we called the Gibbs ensemble, is equivalent to a Julesz ensemble as Λ→Z². Thus the advantages of the two methodologies can be fully utilized