Gabor frames from contact geometry in models of the primary visual cortex

Abstract

We analyze the interplay between contact geometry and Gabor filters signal analysis in geometric models of the primary visual cortex. We show in particular that a specific framed lattice and an associated Gabor system is determined by the Legendrian circle bundle structure of the $3$-manifold of contact elements on a surface (which models the V1-cortex), together with the presence of an almost-complex structure on the tangent bundle of the surface (which models the retinal surface). We identify a scaling of the lattice, also dictated by the manifold geometry, that ensures the frame condition is satisfied. We then consider a $5$-dimensional model where receptor profiles also involve a dependence on frequency and scale variables, in addition to the dependence on position and orientation. In this case we show that a proposed profile window function does not give rise to frames (even in a distributional sense), while a natural modification of the same generates Gabor frames with respect to the appropriate lattice determined by the contact geometry.