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), pp247-265, 2000.
[2] S.C. Zhu, Y.N. Wu, and D.B.Mumford, "Minimax Entropy Principle and Its Applications to 
    Texture Modeling", Neural Computation, 9, pp1627-1660, nov. 1997.
[3] S.C. Zhu, Y.N. Wu, and D.B.Mumford, "FRAME : Filters, Random fields And Maximum 
    Entropy---towards a unified theory for texture modeling", Int'l Journal of 
    Computer Vision, 27(2), pp1-20, 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 
                   micro-canonically 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.