[Paper Review] Covariant un-reduction for curve matching

Alexis Arnaudon, Marco Castrillón López+1 more|arXiv (Cornell University)|Aug 21, 2015

Morphological variations and asymmetry9 references3 citations

TL;DR

This paper introduces a covariant un-reduction framework for curve matching in field theories, extending prior work by formulating un-reduction in a space-time covariant setting using Sobolev metrics. It enables flexible, parametrization-invariant shape comparisons by treating spatial and temporal variables on equal footing, with numerical results showing convergence of explicit Euler schemes to expected solutions.

ABSTRACT

The process of un-reduction, a sort of reversal of reduction by the Lie group symmetries of a variational problem, is explored in the setting of field theories. This process is applied to the problem of curve matching in the plane, when the curves depend on more than one independent variable. This situation occurs in a variety of instances such as matching of surfaces or comparison of evolution between species. A discussion of the appropriate Lagrangian involved in the variational principle is given, as well as some initial numerical investigations.

Motivation & Objective

• To address the parametrization dependence in geodesic distance computations for curve matching by reformulating un-reduction in an inner metric setting.
• To generalize the un-reduction process from classical mechanics to a covariant field theory framework, allowing multiple independent variables for shape comparison.
• To replace curvature-weighted metrics with Sobolev metrics to avoid arbitrarily small geodesic distances and simplify un-reduced equations.
• To validate the framework through numerical experiments, demonstrating convergence of explicit schemes in initial value problems.
• To enable new applications such as matching cylindrical surfaces and spatio-temporal analysis by treating space and time symmetrically.

Proposed method

• Formulate Lagrange-Poincaré reduction for the space of planar curves under the action of the diffeomorphism group $\mathrm{Diff}(S^1)$, yielding equations on the quotient space of shapes.
• Implement covariant un-reduction by introducing an independent parametrization field, allowing the un-reduced dynamics to project onto reduced solutions.
• Use Sobolev inner metrics (e.g., $H^1$) instead of curvature-weighted metrics to ensure well-posed geodesic distances and simplify the un-reduction equations.
• Apply the un-reduction equations in a two-dimensional space-time field theory setting, where both spatial and temporal variables are treated as independent parameters.
• Use a forward Euler scheme for numerical integration of the initial value problem, with the un-reduced equations governing the coupled dynamics of shape and parametrization.
• Compute the distance between curves using the method of currents, which is parametrization-independent, to validate the numerical results.

Experimental results

Research questions

• RQ1 How can un-reduction be generalized from classical mechanics to a covariant field theory framework for curve matching?
• RQ2 What are the advantages of using Sobolev metrics over curvature-weighted metrics in the context of un-reduction and shape matching?
• RQ3 Can the un-reduction framework ensure parametrization-invariant geodesic distances between curves?
• RQ4 How does the inclusion of additional independent variables (e.g., spatial or temporal) improve the flexibility of shape comparison compared to time-warping approaches?
• RQ5 To what extent can explicit numerical schemes like forward Euler converge to correct solutions in the un-reduced system without additional stabilization?

Key findings

• The use of Sobolev metrics in the un-reduction framework eliminates the issue of arbitrarily small geodesic distances that plagues curvature-weighted metrics.
• The un-reduced equations take a simpler form with Sobolev metrics compared to the more complex curvature-weighted formulations used in prior work.
• Numerical simulations using the forward Euler method converged to the expected solution, even without adaptive time-stepping or stabilization.
• The coupling between vertical (shape) and horizontal (parametrization) dynamics decreased with finer time resolution, improving deformation quality.
• The distance between curves, computed via the method of currents, remained consistent across different parametrizations, confirming parametrization independence.
• The framework enables new applications such as matching cylindrical surfaces and spatio-temporal analysis by treating space and time as symmetric variables in the action functional.