Isodipole Textures

Isodipole Textures

Matlab code
unsolved problems
Home Page

Introduction: Isodipole Textures

Bela Julesz is well-known for his many contributions to understanding visual perception. In 1962, he conjectured that "effortless" visual processing cannot discriminate textures which have the same power spectrum, or, equivalently, identical second-order correlations. Such textures are known as "isodipole" textures. More generally, textures whose statistics are identical up to order N-1, but not order N, are known as N-th order textures.

This conjecture turned out to be false, but established the idea that texture discrimination experiments are a useful probe of the computational processes underlying early vision. One reason for this is that natural scenes are distinguished from Gaussian noise fields by virtue of their high-order correlations, and isodipole textures isolate individual high-order correlations in as pure a manner as possible.

To implement this program, one must be able to generate a diverse set of isodipole textures. This page presents a family of such algorithms to construct isodipole textures of order N. The isodipole textures generated by these algorithms are also maximum-entropy textures, subject to the constraints of the recursion rule used to build the texture. Download a VSS 2005 poster on this.

Formally, a "texture" is a statistical ensemble of images, not an individual image. Statements about the statistics of the texture relate to the complete ensemble, not to individual images. To be rigorous, experiments based on isodipole textures must therefore explore a number of examples of each texture, to be sure that findings are not due to the idiosyncracies of a particular example. This issue is discussed more fully elsewhere.

top introduction algorithm generalizations references


Isodipole textures may be constructed by a recursive scheme, which might be called a plane Markov process. These constructions lead to texture ensembles which have a number of nice theoretical properties, including ergodicity and maximum entropy. See for example
E. N. Gilbert's work.

A texture example consists of an assignment of pixel values 0 or 1 to a lattice ai,j of points, and a rendering of these pixel values as an image (typically 0 as black, 1 as white). Here, i represents the row and j represents the column of the pixel in the lattice.

A set of "integer recursion parameters" (pk,qk) (k = 1, 2, ..., N-1) is chosen. The values pk must be non-negative (say, in the range 0 to P, P arbitrary). For values of k for which pk=0, then qk must be positive (say, in the range 1 to Q, Q arbitrary). This is to ensure that the recursion rule (below) can be implemented. Even within these limits, the recursion parameters (pk,qk) cannot be chosen arbitrarily; certain choices do not lead to Nth-order textures because they imply correlations of lower order. See E. N. Gilbert's work.

The texture sample is initiated by assigning the pixels ai,j (0 <= i < P or 0 <=j < Q) to values 0 or 1, randomly and with equal probability. For all other values of i and j, the value of ai,j is determined by the recursion rule
ai,j = ai-p1,j-q1 + ai-p2,j-q2 + ... + ai-pN-1,j-qN-1,
where addition is interpreted mod 2. The restrictions on the recursion parameters (pk,qk) ensure that this recursion can be carried out at each step.

Recursion parameters for some sample textures:
even texture: N=4, (p1,q1)=(0,1), (p2,q2)=(1,0), (p3,q3)=(1,1).
odd texture: N=3, (p1,q1)=(0,1), (p2,q2)=(1,0).
cross texture: N=4, (p1,q1)=(1,1), (p2,q2)=(1,-1), (p3,q3)=(2,0).
tee texture: N=4, (p1,q1)=(1,1), (p2,q2)=(1,0), (p3,q3)=(1,-1).

top introduction algorithm generalizations references


top introduction algorithm generalizations references


The Julesz conjecture

Julesz, B. (1962) Visual pattern discrimination. IRE Trans. Inf. Theory IT-8, 84-92

Julesz, B., Gilbert, E.N., Shepp, L.A., & Frisch, H.L. (1973) Inability of humans to discriminate between visual textures that agree in second-order statistics -- revisited. Perception 2, 391-405 (1973).

Original counterexamples to the Julesz conjecture

Caelli, T., & Julesz, B. (1978) On perceptual analyzers underlying visual texture discrimination. Part I. Biological Cybernetics 28, 167-175.

Caelli, T., Julesz, B., & Gilbert E.N. (1978) On perceptual analyzers underlying visual texture discrimination. Part II. Biological Cybernetics 29, 201-214.

Julesz, B., Gilbert, E., and Victor, J.D. (1978) Visual discrimination of textures with identical third-order statistics. Biological Cybernetics 31, 137-140.

Victor, J.D. and Brodie, S. (1978) Discriminable textures with identical Buffon needle statistics. Biological Cybernetics 31, 231-234.

Theory of planar Markov processes

Gilbert, E.N. (1980) Random colorings of a lattice on squares in the plane. SIAM J. Alg. Disc. Meth. 1, 152-159.

Work from the Maddess lab Link to downloads

Seamons, J.W., Barbosa, M.S., Bubna-Litic, A., & Maddess, T. (2015) A lower bound on the number of mechanisms for discriminating fourth and higher order spatial correlations. Vision Res. 108, 41-48.

Maddess, T. & Nagai Y. (2001) Discriminating isotrigon textures. Vision Res. 41, 3837-3860.

Maddess, T., Davey, M. & Yang, E. (1999) Discrimination of complex textures by bees. J. Comp. Physiol. A, 184, 107-177.

Work from our lab

Review article

Other references from our lab related to visual texture and form processing

Swedish translation created by Anna Chekovsky
Bulgarian translation of this page created by Stoil Dragomirov
Indonesian translation of this page created by Jordan Silaen at ChameleonJohn

top introduction algorithm generalizations