Statistical mechanics and visual signal processing

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Abstract
We show how to use the language of statistical field theory to address and solve problems in which one must estimate some aspect of the environnent from the data in an array of sensors. In the field theory formulation the optimal estimator can be written as an expectation value in an ensemble where the input data act as external field. Problems at low signal-to-noise ratio can be solved in perturbation theory, while high signal-to-noise ratios are treated with a saddle-point approximation. These ideas are illustrated in detail by an example of visual motion estimation which is chosen to model a problem solved by the fly's brain. The optimal estimator bas a rich structure, adapting to various parameters of the environnent such as the mean-square contrast and the corrélation time of contrast fluctuations. This structure is in qualitative accord with existing measurements on motion sensitive neurons in the fly's brain, and the adaptive properties of the optimal estimator may help resolve conficts among different interpretations of these data. Finally we propose some crucial direct tests of the adaptive behavior
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