From Conceptualization to Modeling:
 Equivalence of Julesz Ensemble and Gibbs Models
Reference
[1] Y.N. Wu, S.C. Zhu and X.W. Liu, "Equivalence of Julesz Ensemble and FRAME Models", Int'l Journal of Computer Vision, 38(3), pp247265, 2000. [2] S.C. Zhu, Y.N. Wu, and D.B.Mumford, "Minimax Entropy Principle and Its Applications to Texture Modeling", Neural Computation, 9, pp16271660, nov. 1997. [3] S.C. Zhu, Y.N. Wu, and D.B.Mumford, "FRAME : Filters, Random fields And Maximum Entropytowards a unified theory for texture modeling", Int'l Journal of Computer Vision, 27(2), pp120, 1998.
a). A finite texture image is considered
as viewing an infinite texture through windows (lattice lambda).

b). As we move from infinity 2D plane to finite lattice, disjoint Julesz ensembles began to overlap due to statistical fluctuations. A texture is described by a prob dist. (model). 
The Julesz ensemble definition is a mathematical idealization and provides an important concept. Associated with each Julesz ensemble is a distribution q(I; h) which is uniform inside the ensemble and zero outside. In real applications, textures are observed and computed on finite lattices. This is, in fact, equivalent to viewing an infinite texture image through some small windows, and thus any finite image is characterized by a conditional distribution of the Julesz ensemble q(I, h) given some boundary conditions. It is proven in (Wu, Zhu and Liu, 2000)[1] that this conditional distribution is exactly the Gibbs (FRAME) models studied in [2] and [3] by Zhu, Wu, and Mumford 1997, 1998. The FRAME model is learned using a minimax entropy principle, and it generalized traditional Markov random field models. Thus the Julesz ensemble is a limit distribution of the Markov random field models.
In fact, the equivalence theorem [1] was observed a century ago by Gibbs in statistical mechanics:
“If a system of a great number of degrees of freedom is microcanonically distributed in phase, any very small part of it may be regarded as canonically distributed.”Gibbs 1902 This equivalence theorem proves the consistence between Julesz ensemble (Conceptualization) and Gibbs and FRAME models (modeling). This also unifies two main research streams in vision research: One is MRF modeling, and the other is matching statistics. This makes the framework elegantly unified.