An unbiased implementation of regularization mechanisms

Abstract

In computer or biological vision, computation of vectorial maps of parametric quantities (e.g. feature parameters, 3D or motion cues,..) are of common use in perceptual processes. Defining them using continuous partial differential equations yields highly parallelizable regularization processes allowing to obtain well-defined estimations of these quantities. However these equations have to be sampled on real data and this step is not obvious and may introduce some bias. In order to overcome this caveat, a method, introduced by Raviat and developed by Degond and Mas-Gallic, is based on an integral approximation of the diffusion operator used in regularization mechanisms: it leads to a so-called “particle” implementation of such diffusion process. Following this formulation, the present development defines an optimal implementation of such an integral operator with the interesting property that when used on sampled data such as image pixels or 3D data voxels, it provides an unbiased implementation of the corresponding continuous operator without any other approximation. Furthermore, the method is ‘automatic’ (using symbolic computations) in the sense that given a continuous regularization mechanism, the corresponding (non-linear) discrete filter is derived automatically, as made explicit here. A step ahead, the architecture of the implementation corresponds to what is observed in cortical visual maps, leading to a certain biological plausibility. The present development is illustrated by an experiment of visual motion estimation and another experiment in image denoising.