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An inexact matching approach for the comparison of plane curves with general elastic metrics

Yashil Sukurdeep, Martin Bauer, Nicolas Charon

TL;DR

The paper tackles robust comparison of planar curves under elastic metrics by an inexact matching formulation that blends the $F_{a,b}$ transform with varifold fidelity terms. This approach reduces the geodesic problem to a tractable end-curve optimization, enabling effective handling of noise, inconsistencies, and minor topological variations. Key contributions include the explicit $F_{a,b}$ transform with an isometric property, a reparametrization-invariant varifold discrepancy, and a relaxed matching framework that supports partial and topology-robust matching. The method yields efficient discretization, rotational invariance, and potential extensions to surfaces and weighted partial-data matching, with promising preliminary numerical results.

Abstract

This paper introduces a new mathematical formulation and numerical approach for the computation of distances and geodesics between immersed planar curves. Our approach combines the general simplifying transform for first-order elastic metrics that was recently introduced by Kurtek and Needham, together with a relaxation of the matching constraint using parametrization-invariant fidelity metrics. The main advantages of this formulation are that it leads to a simple optimization problem for discretized curves, and that it provides a flexible approach to deal with noisy, inconsistent or corrupted data. These benefits are illustrated via a few preliminary numerical results.

An inexact matching approach for the comparison of plane curves with general elastic metrics

TL;DR

The paper tackles robust comparison of planar curves under elastic metrics by an inexact matching formulation that blends the transform with varifold fidelity terms. This approach reduces the geodesic problem to a tractable end-curve optimization, enabling effective handling of noise, inconsistencies, and minor topological variations. Key contributions include the explicit transform with an isometric property, a reparametrization-invariant varifold discrepancy, and a relaxed matching framework that supports partial and topology-robust matching. The method yields efficient discretization, rotational invariance, and potential extensions to surfaces and weighted partial-data matching, with promising preliminary numerical results.

Abstract

This paper introduces a new mathematical formulation and numerical approach for the computation of distances and geodesics between immersed planar curves. Our approach combines the general simplifying transform for first-order elastic metrics that was recently introduced by Kurtek and Needham, together with a relaxation of the matching constraint using parametrization-invariant fidelity metrics. The main advantages of this formulation are that it leads to a simple optimization problem for discretized curves, and that it provides a flexible approach to deal with noisy, inconsistent or corrupted data. These benefits are illustrated via a few preliminary numerical results.

Paper Structure

This paper contains 8 sections, 13 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Proposed method
  3. Elastic metrics on the space of curves
  4. The $F_{a,b}$ transform
  5. Varifold fidelity terms
  6. Relaxed formulation of the geodesic problem
  7. Experimental data and results
  8. Conclusion and further extensions

Figures (5)

  • Figure 1: Geodesic between a circle and the red target curve obtained for $a=1$, $b=0.8$ and $\lambda =1000$.
  • Figure 2: Effect of $\lambda$ on the geodesic path and estimated distance. The last column shows the result obtained with the exact matching algorithm of KuNe2018.
  • Figure 3: Left: source (blue) and noisy target curve (red). Right: estimated geodesic with our proposed approach for $a=1$, $b=0.5$ and $\lambda=40$.
  • Figure 4: Multidimensional scaling plots obtained with the proposed relaxed approach (left) and the exact matching approach (right).
  • Figure 5: Left: example of two curves that have similar geometric images, but have different topologies. Right: estimated matching by our algorithm.