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

Davide Barbieri

TL;DR

This work addresses sampling for the SE(2) wavelet transform with a modulated Gaussian mother wavelet by formulating frame conditions on a Paley–Wiener subspace and deriving a spectral criterion via a dual Gramian. The main result is a theorem that reduces the frame property to the spectrum of a finite family of matrices G(ω), enabling finite-dimensional spectral tests to decide sampling suitability for given parameters p, σ, ρ, and the orientation-set Θ. The paper also offers a practical sufficient-condition analysis under a frequency cutoff and provides numerical experiments that connect the theory to visual cortex modeling, illustrating parameter regimes that yield stable frames and highlighting the influence of lattice geometry and angle discretization. This contributes to both harmonic analysis on SE(2) and neuroscience-inspired sampling design, with potential implications for image analysis and brain-inspired computation. The findings underscore a concrete link between mathematical sampling theory and the organization of orientation maps in V1, reinforcing the relevance of coverage principles in discretized SE(2) representations.

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.

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

TL;DR

This work addresses sampling for the SE(2) wavelet transform with a modulated Gaussian mother wavelet by formulating frame conditions on a Paley–Wiener subspace and deriving a spectral criterion via a dual Gramian. The main result is a theorem that reduces the frame property to the spectrum of a finite family of matrices G(ω), enabling finite-dimensional spectral tests to decide sampling suitability for given parameters p, σ, ρ, and the orientation-set Θ. The paper also offers a practical sufficient-condition analysis under a frequency cutoff and provides numerical experiments that connect the theory to visual cortex modeling, illustrating parameter regimes that yield stable frames and highlighting the influence of lattice geometry and angle discretization. This contributes to both harmonic analysis on SE(2) and neuroscience-inspired sampling design, with potential implications for image analysis and brain-inspired computation. The findings underscore a concrete link between mathematical sampling theory and the organization of orientation maps in V1, reinforcing the relevance of coverage principles in discretized SE(2) representations.

Abstract

We study a sampling problem for the abstract wavelet transform associated with the quasiregular representation of the 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.
Paper Structure (11 sections, 2 theorems, 52 equations, 12 figures)

This paper contains 11 sections, 2 theorems, 52 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Related works
  4. Motivation from visual perception
  5. Sampling conditions
  6. A Hilbert space for a frame condition
  7. Statement of the problem
  8. Characterization in terms of a dual Gramian matrix
  9. Sufficient conditions for frequency cutoff
  10. Numerical exploration of the parameters space
  11. Conclusions

Key Result

Theorem 3.1

Let $\Gamma < \mathbb{R}^2$ be a full-rank lattice, let $\Gamma^\perp$ be its annihilator lattice, and let $\Omega \subset \mathbb{R}^2$ be a fundamental set for the quotient $\mathbb{R}^2 / \Gamma^\perp$. For $\varrho > 0$, denote by Let $p, \sigma > 0$, and let $\{(\alpha_k,\theta_k)\}_{k = 1}^N \subset \mathbb{R}^2 \times S^1$. For $\omega \in \Omega$, denote by $G(\omega) \in \mathbb{R}^{\mat

Figures (12)

  • Figure 1: Electrophysiological measurements of the representative vectors for the linear functionals that govern the activities of different V1 neurons, and fits with rotated modulated gaussians. Left, from JP1987: in cats. Right, from Ringach2002: in macaques.
  • Figure 2: Left, from Bosking1997: optical imaging measurements of OPM in V1 of a tree-shrew. Color indicates the preferred orientation of the underlying neuron. Center, from Ohki2006: preferred orientation of single neurons measured at different widths, proving the two-dimensional structure of OPM. Right, from NW1994: (A) image of OPM in macaques measured by BS1986, (B) its Discrete Fourier Transform, (D) the image that results by inverse DFT of (C) a random distribution on a thin annulus.
  • Figure 3: Artificial OPM-like distribution generated as in Figure \ref{['fig:OPM']} Right, D NW1994. In this image, colors (representing preferred orientations) are quantized into 6 channels. For each, the centers of a standard k-means clustering are marked with an X.
  • Figure 4: Top: Fourier transform, real and imaginary part of $\psi$ with $p \sigma = 0.3$. Bottom: Fourier transform, real and imaginary part of $\varphi$ with $L = 2p$ (tangent to the axes, and retaining about 80% of the mass of the original gaussian).
  • Figure 5: Numerical analysis for $p = 0.5, \sigma = 2/\pi, \varrho = 1/\sqrt{2}, N = 4$.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Theorem 3.1
  • proof
  • Proposition 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5