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)

#### Summary

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.

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#### 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.

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