Association Fields via Cuspless Sub-Riemannian Geodesics in SE(2)

Area of Science:

• Computational geometry
• Computer vision
• Mathematical psychophysics

Background:

• Modeling perceptual organization (gestalt principles) in psychophysics requires understanding underlying association fields.
• Previous work established conditions for connecting initial and final states using globally minimizing geodesics in SE(2).
• Analyzing sub-Riemannian geodesics and their exponential map is crucial for applied imaging.

Purpose of the Study:

• To analyze the properties of sub-Riemannian geodesics and the exponential map in the context of minimizing curve curvature.
• To investigate the geometry of vision models proposed by Petitot and Citti & Sarti.
• To develop an efficient algorithm for solving the associated boundary value problem.

Main Methods:

• Minimization of curvature for planar curves with fixed start/end positions and directions.
• Analysis of the exponential map on the sub-Riemannian manifold SE(2) using both arc-length (t) and spatial arc-length (s) parametrizations.
• Development and application of a novel algorithm for solving the boundary value problem.

Main Results:

• The reachable set [Formula: see text] is contained within the half-space x≥0, with specific endpoints reachable at angle π.
• The boundary [Formula: see text] comprises endpoints of minimizers associated with cusps.
• Spatial arc-length parametrization (s) simplifies analysis of the exponential map, curvature, cusp-surface, and boundary value problem.
• Sub-Riemannian geodesics align with Petitot's circle bundle model and exhibit similarity to association field lines.

Conclusions:

• The study provides a detailed analysis of sub-Riemannian geodesics relevant to visual perception models.
• A simplified approach using spatial arc-length parametrization enhances understanding and computation.
• The findings offer a computational framework for modeling visual association fields and perceptual organization.