Derivation of a cortical normalization model from the statistics of natural images

E P Simoncelli and O Schwartz

Published in Investigative Opthalmology and Visual Science Supplement (ARVO), vol.39 pp. S-424, May 1998.

This paper has been superseded by:
Natural signal statistics and sensory gain control
O Schwartz and E P Simoncelli.
Nature Neuroscience, vol.4(8), pp. 819--825, Aug 2001 .

  • Presentation slides (331k, pdf)
    • 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.

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