Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discrete Geodesic Calculus in the Space of Sobolev Curves

Sascha Beutler, Florine Hartwig, Martin Rumpf, Benedikt Wirth

TL;DR

This work develops a rigorous, convergent discretization framework for the Riemannian calculus on the space of Sobolev immersions with a second-order Sobolev metric ($m\ge 2$), including geodesics, parallel transport, covariant differentiation, and curvature. By formulating a variational time discretization and a spectral space discretization, the authors prove existence and Mosco convergence for discrete geodesics and their associated geometric operators, and they introduce three discrete path energies (linear, $\epsilon$-regularized, and rational) with provable convergence to the continuous energy. They also derive a discrete exponential map, discrete covariant derivatives, and Schild’s ladder-based parallel transport with explicit consistency and convergence rates, supported by numerical experiments on low-dimensional Sobolev curve submanifolds. The resulting framework provides a provably convergent, implementable pipeline for computing geodesics and differential geometry on Sobolev shape spaces, with potential impact on accurate shape comparison and statistical analysis of curves. Overall, the paper bridges numerical discretization and analytic convergence theory in an intrinsically infinite-dimensional, reparameterization-invariant setting, enabling reliable shape analysis via discrete Riemannian calculus.

Abstract

The Riemannian manifold of curves with a Sobolev metric is an important and frequently studied model in the theory of shape spaces. Various numerical approaches have been proposed to compute geodesics, but so far elude a rigorous convergence theory. By a slick modification of a temporal Galerkin discretization we manage to preserve coercivity and compactness properties of the continuous model and thereby are able to prove convergence for the geodesic boundary value problem. Likewise, for the numerical analysis of the geodesic initial value problem we are able to exploit the geodesic completeness of the underlying continuous model for the error control of a time-stepping approximation. In fact, we develop a convergent discretization of a comprehensive Riemannian calculus that in addition includes parallel transport, covariant differentiation, the Riemann curvature tensor, and sectional curvature, all important tools to explore the geometry of the space of curves. Selected numerical examples confirm the theoretical findings and show the qualitative behaviour. To this end, a low-dimensional submanifold of Sobolev curves with explicit formulas for ground truth covariant derivatives and curvatures are considered.

Discrete Geodesic Calculus in the Space of Sobolev Curves

TL;DR

This work develops a rigorous, convergent discretization framework for the Riemannian calculus on the space of Sobolev immersions with a second-order Sobolev metric (), including geodesics, parallel transport, covariant differentiation, and curvature. By formulating a variational time discretization and a spectral space discretization, the authors prove existence and Mosco convergence for discrete geodesics and their associated geometric operators, and they introduce three discrete path energies (linear, -regularized, and rational) with provable convergence to the continuous energy. They also derive a discrete exponential map, discrete covariant derivatives, and Schild’s ladder-based parallel transport with explicit consistency and convergence rates, supported by numerical experiments on low-dimensional Sobolev curve submanifolds. The resulting framework provides a provably convergent, implementable pipeline for computing geodesics and differential geometry on Sobolev shape spaces, with potential impact on accurate shape comparison and statistical analysis of curves. Overall, the paper bridges numerical discretization and analytic convergence theory in an intrinsically infinite-dimensional, reparameterization-invariant setting, enabling reliable shape analysis via discrete Riemannian calculus.

Abstract

The Riemannian manifold of curves with a Sobolev metric is an important and frequently studied model in the theory of shape spaces. Various numerical approaches have been proposed to compute geodesics, but so far elude a rigorous convergence theory. By a slick modification of a temporal Galerkin discretization we manage to preserve coercivity and compactness properties of the continuous model and thereby are able to prove convergence for the geodesic boundary value problem. Likewise, for the numerical analysis of the geodesic initial value problem we are able to exploit the geodesic completeness of the underlying continuous model for the error control of a time-stepping approximation. In fact, we develop a convergent discretization of a comprehensive Riemannian calculus that in addition includes parallel transport, covariant differentiation, the Riemann curvature tensor, and sectional curvature, all important tools to explore the geometry of the space of curves. Selected numerical examples confirm the theoretical findings and show the qualitative behaviour. To this end, a low-dimensional submanifold of Sobolev curves with explicit formulas for ground truth covariant derivatives and curvatures are considered.
Paper Structure (23 sections, 18 theorems, 218 equations, 11 figures)

This paper contains 23 sections, 18 theorems, 218 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Related Work
  4. Review of the space of Sobolev immersions
  5. A priori estimates on metric balls
  6. Existence of shortest geodesics
  7. Variational time discretization and fully discrete geodesics
  8. Upper and lower bounds on length element and resulting $\epsilon$-regularized path energy
  9. Time discretization for $m=2$ based on $\epsilon$-free upper bounds of the length element
  10. Existence and Mosco convergence of the time discretization
  11. Space discretization
  12. Discrete exponential and differentials
  13. Discrete exponential map
  14. Discrete parallel transport and covariant derivative
  15. Consistency of the discrete Riemann curvature tensor
  16. ...and 8 more sections

Key Result

Proposition 2.3

Let $g$ be the Sobolev metric of order $m\geq 2$ and ${\mathrm{dist}}(\cdot,\cdot)$ the induced distance and let $l_c \coloneqq \int_{{\mathrm S^1}} \,{\mathrm{d}} s$ denote the length of the curve $c$. Then the maps with $p=2$ for $k=m$ and $p=\infty$ for $1\leq k\leq m-1$ are Lipschitz continuous on every metric ball $B_r(c_0)\subset\operatorname{Imm}^m$. In particular, $\Vert c'\Vert_{L^\infty

Figures (11)

  • Figure 1: Illustration of the lower bound $|q|$ for $\vert \mathbf{c}_{\hat{c},\check c}'(t)\vert$.
  • Figure 2: Error between a discrete geodesic $\mathbf{c}^K$ (${\mathcal{W}}={\mathcal{W}}_{\mathrm{rat}}$ on the left, ${\mathcal{W}}={{\mathcal{W}}_{\mathrm{reg}}^{\epsilon}}$ on the right) computed for increasing $K$ and a reference geodesic $\mathbf{c}$ (computed with $N = 80, M = 640, K = 2^{11}$). We use $\mathrm{err}=(\frac{1}{K+1} \sum_{k=0}^K \| \mathbf{c}^K(\tfrac{k}{K}) - \mathbf{c}(\tfrac{k}{K})\|_{W_\theta^r}^2)^{1/2}$ as a time-discrete $L_t^2W_\theta^r$-norm of the error with $r=0$ (dotted), $r=1$ (dashed), and $r=2$ (solid).
  • Figure 3: Discrete geodesics (orange) between the two grey curves, extrapolated beyond the second curve via the discrete exponential map (blue), computed with the weights $(a_0,a_1,a_2)=(10^{-4}, 1, 10^{-2})$, $(1, 1, 10^{-2})$, and $(10^{-4}, 1, 1)$ in \ref{['def:SobolevMetricCurves']} (top to bottom).
  • Figure 4: Discrete geodesics (orange) between the two grey curves, extrapolated beyond the second curve via the discrete exponential map (blue), using $(a_0,a_1,a_2)=(10^{-4}, 1, 10^{-2})$ in \ref{['def:SobolevMetricCurves']}.
  • Figure 5: Discrete geodesics (orange) between the two grey curves, extrapolated beyond the second curve via the discrete exponential map (blue) using the Sobolev metric of order $m=3$ with weights $(a_0,a_1,a_2,a_3)=(10^{-4}, 1, 10^{-2}, 10^{-4})$, $(10^{-4}, 1, 10^{-2}, 10^{-3})$, and $(10^{-4}, 1, 10^{-2}, 10^{-2})$ in \ref{['def:SobolevMetricCurves']} (top to bottom).
  • ...and 6 more figures

Theorems & Definitions (42)

  • Definition 2.1: Sobolev metric on $\operatorname{Imm}^m$
  • Remark 2.2: Sobolev metric is smooth and strong
  • Proposition 2.3: A priori estimates Br15
  • Remark 2.4: Logarithm of length element
  • Remark 2.5: Extension of estimates to $\operatorname{Imm}^m$
  • Lemma 2.6: Coercivity and boundedness of the metric Br15, BrMiMu14
  • Lemma 2.7: Weak convergence of arclength derivatives Br15
  • Theorem 2.8: Existence of geodesic paths Br15
  • proof
  • Remark 2.9: Reachable curves
  • ...and 32 more