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Sobolev Metrics on Spaces of Discrete Regular Curves

Jonathan Cerqueira, Emmanuel Hartman, Eric Klassen, Martin Bauer

Abstract

Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and considers the induced Riemannian metric on a finite dimensional approximation space. Surprisingly, the theoretical properties of the corresponding finite dimensional Riemannian manifolds have not yet been studied in detail, which is the content of the present article. Our main theorem concerns metric and geodesic completeness and mirrors the results of the infinite dimensional setting as obtained by Bruveris, Michor and Mumford.

Sobolev Metrics on Spaces of Discrete Regular Curves

Abstract

Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and considers the induced Riemannian metric on a finite dimensional approximation space. Surprisingly, the theoretical properties of the corresponding finite dimensional Riemannian manifolds have not yet been studied in detail, which is the content of the present article. Our main theorem concerns metric and geodesic completeness and mirrors the results of the infinite dimensional setting as obtained by Bruveris, Michor and Mumford.
Paper Structure (15 sections, 12 theorems, 67 equations, 6 figures)

This paper contains 15 sections, 12 theorems, 67 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivation and Background:
  3. Main Contributions:
  4. Conclusions and future work:
  5. Acknowledgements:
  6. Reparametrization invariant Sobolev metrics on the space of smooth curves
  7. Discrete Sobolev Metrics on Discrete Regular Curves
  8. Completeness Results
  9. Geodesics, curvature and a comparison to Kendall's shape space
  10. The geodesic initial value problem:
  11. The geodesic boundary value problem:
  12. Gaussian Curvature of the Space of Triangles and Kendall's shape metric:
  13. Constant Coefficient Sobolev Metrics
  14. Approximating derivatives in $L^{\infty}$
  15. Proof of Lemma \ref{['metricequivalence']}

Key Result

Lemma 1

Let $m\geq 0$ and let ${G}^m$ be the Riemannian metric as defined in smoothmetricdef. Let $c\in \operatorname{Imm}(S^1,\mathbb{R}^d)$ and $h,k\in T_c\operatorname{Imm}(S^1,\mathbb{R}^d)$. Then we have:

Figures (6)

  • Figure 1: A pictorial representation of the discrete derivative. The derivative $D_s^j h$ can be seen here as a scaled first difference of $D_s^{j-1}h$. Both the even case and the odd case are shown. Note on the right for odd $j$ the derivative $D^{j+1}_sh_i$ is associated with the vertex $v_i$ while for even $j$ the derivative $D^{j+1}_sh_i$ is associated with the edge $e_i$.
  • Figure 2: Riemannian exponential map w.r.t. to the metric ${g}^2$.
  • Figure 3: Examples of solutions to the geodesic boundary value problem w.r.t. to the metric ${g}^2$. First line: from a hexagon to a spiked figure. Second line: from the spiked figure to a blunt one.
  • Figure 4: Geodesics in the space of triangles w.r.t. to four different Riemannian metrics. We display geodesics with respect to ${g}^0$ (first line), ${g}^1$ (second line), ${g}^2$ (third line), and the Kendall metric (last line). To the left of each is a plot of the three edge lengths of the triangles along each geodesic.
  • Figure 5: The Gaussian curvatures of the space of triangles triangles under the Kendall metric and the discretized scale invariant Sobolev metric ${g}^m$ for $m=0,1,2$). From left to right we have ${g}^m$ for $m=0,1,$ and $2$ and the Kendall metric (with the triangles drawn in). The scale is a symmetric log scale given by $\operatorname{sign}(x)\log(|x+\operatorname{sign}(x)|)$.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 1: Reparametrization invariant Sobolev Metrics
  • Remark 1: Scale invariant vs. non-scale invariant metrics
  • Lemma 1
  • proof
  • Lemma 2: michor2005vanishingbauer2024completeness
  • Theorem 1: bauer2022sobolevbruveris2017completeness
  • proof
  • Remark 2: Fractional Order Metrics
  • Remark 3
  • Definition 2
  • ...and 24 more