Images, Statistics, and Textures

Images, statistics, and textures: a comment on "Implications of triple correlation uniqueness for texture statistics and the Julesz conjecture."

Jonathan D. Victor

J. Opt. Soc. Amer. A11, 1680-1684 (1994)


Visual textures are statistical structures, and must be considered as such to interpret the Julesz conjecture and related notions properly. Yellott presents a result concerning the statistics of individual images, but this result is of limited relevance to the study of texture perception because of the distinction between spatially-finite samples of a texture (an image) and the ensemble from which the sample is drawn. This distinction makes sense both logically and empirically. The assumptions and approximations that are required to link the mathematical notion of a texture ensemble to laboratory practice are no more severe or unnatural than those required in the application of other useful mathematical structures to biology. Interpretation of texture perception in terms of images rather than ensembles leads to an impoverished experimental and theoretical analysis.

Julesz Conjecture

The Julesz conjecture states that humans cannot distinguish between textures with identical second-order statistics. This conjecture is now known to be false. If the "
triple correlation uniqueness" theorem applied to textures as well as individual images, then extensions of the Julesz conjecture to higher orders could not be tested -- since it would be impossible to construct distinct textures with identical third-order statistics. However, the study of textures with identical third-order statistics has proven to be useful in understanding cortical processing of form.

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

Triple Correlation Uniqueness Theorem

The "triple correlation uniqueness" theorem states that a binary image is completely determined (up to translation) by its set of third-order correlations. The theorem applies only to finite individual images, and not to ensembles.

Yellott, J. (1993) Implications of triple correlation uniqueness for texture statistics and the Julesz conjecture. J. Opt. Soc. Amer. A10, 777-793.

Isodipole textures: examples and background
Publications related to texture processing
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