Julesz Ensemble: A Mathematical Definition of Texture
An input texture image
sampled image with k=0
sampled image with k=2
sampled image with k=3
sampled image with k=4
sampled image with k=7
Reference:

[1] S.C. Zhu, X.W. Liu and Y.N. Wu, "Exploring Julesz Ensembles by efficient 
     Markov Chain Monte Carlo---towards a trichromacy theory of texture", 
     IEEE Trans. PAMI, vol 22, No. 6, pp554-569, 2000.
[2] 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.
[3] 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.
[4] J. Portilla and E.P. Simoncelli, "A Parametric Texture Model Based on Joint
     Statistics of Complex Wavelet Coefficients", Int'l Journal of Computer Vision, 
     40(1), pp49-70, Sept. 2000.
[5] D.J. Heeger and J.R. Bergen, "Pyramid-based Texture Analysis and Synthesis",
     Proc. of SIGGRAPH, pp229-238, 1995. 
   To start texture study, one may ask a simple question:


                                "What is a texture?"
   The question is better understood if we think color --- the other basic visual cue.
A visible color can be identified by a single number: the wavelength of the electro-
magnetical wave. It can also be described by a decomposition of (red, green, blue).
Can we achieve such a neat definition for texture?  Julesz, a well-known psychophysicist
pioneered the texture study for two decades, motivated by the following quest:


                  “What features and statistics are characteristics of a
texture pattern, so that texture pairs that share the
same features and statistics cannot be told apart by
pre-attentive human visual perception?
” ---- Julesz 1960s-1980s Unfortunately, any sensible pursuit of a unified texture theory were easily defeated by the overwhelming diversity of texture patterns in the world. After four decades of study, an answer to Julesz's quest is emerging from multiple research streams, in particular, by three main progress. The first progress is a mathematical one: Unlike color which can be identified by a photon, a texture cannot be defined on a single pixel, or 2x2 pixels. If it cannot be defined on an nxn lattice, then a (n+1)x(n+1) lattice won't do either. So we have to define texture on an infinite lattice as a mathematical limit in the sense of van Hove. For example, a 2D plane where the statistical fluctuations of texture diminishes, thus we obtain a unique value that identifies a texture. This thought led to a definition by (Zhu, Liu, and Wu, 1999)[1,2] which identifies a texture by an equavalence class--- that they called the Julesz ensemble-- on 2D plane: The second progress is a biologic and ecologic one: A texture definition should be also consistent with human perception, and the statistics h(I) should account for image features and statistics to which our visual systems are sensitive to. This will be decided by our enviroments and the purposes of our visual system. (See another page for Human vision insights). The third progress includes connecting the math definition with existing texture models such as the Markov random field models by a minimax entropy theory (Zhu, Wu, and Mumford, 1997)[3] and an ensemble equivalence theorem borrowed from modern statistical physics (Wu, Zhu and Liu,2000)[2]. Otherwise a disconnected math definition would not be convincing. The Julesz ensemble is verified carefully by Markov chain Monte Carlo computing[1,3] and other method by Portilla and Simoncelli[4]. Generally speaking, a texture or a Julesz ensemble is an equivalent class of images on 2D plane that share identical statistics. The figure below shows an example[3]. Suppose we are given an input image (upper left corner), we can extract many statistical measures h from it, of which we select K elements, such as histograms of Gabor filter responses[5]. Matching those K statistics h yields a Julesz ensemble, from which we draw random samples. The Figure below shows results with K=2,3,4,7 histograms[3]. When we achieve the sufficient statistics, the sampled texture should appear alike to the observed one visually. The procedure is controlled by a minimax entropy principle[3]. Click here to view more result of Texture Synthesis with Julesz Ensemble