# Exploring the Julesz Ensemble by Efficient Markov Chain Monte Carlo

### S.C. Zhu, X. Liu and Y.N. Wu

This article presents a mathematical definition of texture pattern--the {\em Julesz ensemble}, which is the maximum set of images that share identical statistics $\phi$ defined on infinity lattice $\plane^2$. Then texture modeling is posed as an inverse problem: given a set of images sampled from an unknown Julesz ensemble $\Omega_\ast$, we search for the sufficient and necessary statistics $\phi_\ast$ which define $\Omega_\ast$. Associated with a Julesz ensemble is a probability distribution $q(\I; \phi)$ which is uniform over the images in the ensemble and has zero probability outside. It has been proven\cite{Wu98} that $q(\I; \phi)$ is the {\em limit distribution} of the Gibbs distribution (FRAME model or Markov random field) derived by the minimax entropy principle\cite{Zhu_neuro}. Therefore, not only the engineering practice of synthesizing textures by matching statistics has been put on a firm mathematical foundation, but also we are released from the burden of learning the expensive FRAME model in feature pursuit, model selection and texture synthesis. The paper also proposed an algorithm for sampling Julesz ensembles by efficient Markov chain Monte Carlo, which generates texture images by moveing along filter coefficients and significantly extends the traditional single site Gibbs sampler. The paper shall also compare four popular statistical measures in the literature, namely, moments, rectified functions, marginal histograms and joint histograms of linear filter responses in terms of their descriptive abilities. We demonstrate that a small number of bins in the marginal histograms are sufficient in capturing a variety of texture patterns, so the full joint histograms appear to be an over-fit. We illustrate our theory and algorithm by successfully synthesizing a large number of natural textures.