Representation of local geometry in the visual system

Abstract

It is shown that a convolution with certain reasonable receptive field (RF) profiles yields the exact partial derivatives of the retinal illuminance blurred to a specified degree. Arbitrary concatenations of such RF profiles yield again similar ones of higher order and for a greater degree of blurring. By replacing the illuminance with its third order jet extension we obtain position dependent geometries. It is shown how such a representation can function as the substrate for "point processors" computing geometrical features such as edge curvature. We obtain a clear dichotomy between local and multilocal visual routines. The terms of the truncated Taylor series representing the jets are partial derivatives whose corresponding RF profiles closely mimic the well known units in the primary visual cortex. Hence this description provides a novel means to understand and classify these units. Taking the receptive field outputs as the basic input data one may devise visual routines that compute geometric features on the basis of standard differential geometry exploiting the equivalence with the local jets (partial derivatives with respect to the space coordinates).