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Derivation
of a Cortical Normalization Model from the Statistics of Natural Images
E P Simoncelli and O Schwartz. ARVO, vol 39, page S-424, May 1998.
Purpose:
Several successful models of cortical visual processing are based on
linear transformation followed by rectification and normalization (in
which each neuron's output is divided by the pooled activity of other
neurons). We show that this form of nonlinear decomposition is optimally
matched to the statistics of natural images, in that it can produce
neural responses that are nearly statistically independent. Methods:
We examine the statistics of monochromatic natural images. One can always
find a linear transformation (i.e., principal component analysis) that
eliminates second-order dependencies (correlations). This transform
is, however, not unique. Several authors (e.g., Bell & Sejnowski,
Olshausen & Field) have used higher-order measurements to further
constrain the choice of transform. The resulting basis functions are
localized in spatial position, orientation and scale, and the associated
coefficients are decorrelated and generally more independent than principal
components. Results: We find that the coefficients of such transforms
exhibit important higher-order statistical dependencies that cannot
be eliminated with linear processing. Specifically, rectified coefficients
corresponding to coefficients at neighboring spatial positions, orientations
and scales are highly correlated, even when the underlying linear coefficients
are decorrelated. The optimal method of removing these dependencies
is to divide each coefficient by a weighted combination of its rectified
neighbors. Conclusions: Our analysis provides a theoretical justification
for divisive normalization models of cortical processing. Perhaps more
importantly, the statistical measurements explicitly specify the weights
that should be used in computing the normalization signal, and thus
offer the opportunity to test directly (through physiological measurements)
the ecological hypothesis that visual neural computations are optimally
matched to the statistics of images.