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

TL;DR

This paper explores how contact geometry influences Gabor frame construction in models of the primary visual cortex, revealing geometric conditions for frame existence and analyzing extensions involving frequency and scale.

Contribution

It establishes a geometric framework linking contact geometry to Gabor frames in visual cortex models and demonstrates how lattice scaling ensures frame conditions.

Findings

A specific lattice and Gabor system are determined by the Legendrian circle bundle structure.

A natural modification of the profile function generates Gabor frames respecting the contact geometry.

A proposed profile window function does not produce frames in the extended model.

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.