Sampling in the Euclidean Motion Group and a Problem from Brain's Primary Visual Cortex

TL;DR

This paper investigates a sampling problem related to wavelet transforms on the Euclidean motion group, motivated by visual cortex behavior, providing theoretical characterization and numerical analysis of sampling parameters.

Contribution

It introduces a new sampling framework for wavelet transforms on SE(2) and analyzes the influence of parameters through dual Gramian matrices and numerical experiments.

Findings

Characterization of sampling conditions via dual Gramian matrix

Numerical relationships between sampling parameters and wavelet properties

Insights into brain-inspired sampling models

Abstract

We study a sampling problem for the abstract wavelet transform associated with the quasiregular representation of the $SE(2)$ group, for a modulated gaussian mother wavelet. This problem is motivated by the behavior of brain's primary visual cortex. We provide a characterization in terms of a dual Gramian matrix, and study numerically the relationships among the parameters defining the sampling and the mother wavelet.