Completeness properties of Sobolev metrics on the space of curves

Bruveris, Martins

doi:10.3934/jgm.2015.7.125

Cited by 29 publications

(70 citation statements)

References 48 publications

(81 reference statements)

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“…On the space of closed curves we additionally extend the completeness results, that were obtained first in [13,15] for Sobolev metrics with constant coefficients and then in [16] for length-weighted Sobolev metrics to this class of elastic metrics. For open curves we find a counter example showing that second order metrics with constant coefficients are not metrically complete.…”

Section: Introductionsupporting

confidence: 60%

“…Since we are working in infinite dimensions the existence of geodesics is a nontrivial question. For elastic Sobolev metrics we have the following existence results for geodesics, which are based on the results in [13,15,36]. They will serve as the theoretical foundation of the proposed numerical framework.…”

Section: 3mentioning

confidence: 99%

“…It has been recently shown in [13,15,40], that adding second derivatives to the metric allows one to obtain completeness results for the Riemannian manifolds in question: the geodesic equation is globally well-posed, the metric completion consists of all H 2 -immersions, and the metric extends to a strong Riemannian metric on the space of H 2 -immersions. Furthermore, any two curves in the same connected component can be joined by a minimizing geodesic.…”

Section: Introductionmentioning

confidence: 99%

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A relaxed approach for curve matching with elastic metrics

et al. 2019

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In this paper we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The model and resulting matching algorithm integrate within one common setting both the family of H 2 -metrics with constant coefficients and scale-invariant H 2 -metrics on both open and closed immersed curves. These families include as particular cases the class of first-order elastic metrics. An essential difference with prior approaches is the way that boundary constraints are dealt with. By leveraging varifold-based similarity metrics we propose a relaxed variational formulation for the matching problem that avoids the necessity of optimizing over the reparametrization group. Furthermore, we show that we can also quotient out finite-dimensional similarity groups such as translation, rotation and scaling groups. The different properties and advantages are illustrated through numerical examples in which we also provide a comparison with related diffeomorphic methods used in shape registration.2000 Mathematics Subject Classification. 68Q25, 68R10, 68U05. 1 Note, that the invariance of the metric is only a necessary condition for a metric on the space of parametrized curves to induce a metric on unparametrized curves. However, all the metrics considered in this article do induce Riemannian metrics on the quotient space. For details, see [36].2 This result has been later extended to the space of parametrized curves in [3]. 3 The article [37] studied the family of elastic metrics with different choices of parameters on the space of parametrized curves. This analysis was, however, not extended to the space of unparametrized curves, which is the more relevant object for applications.

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Section: Introductionsupporting

confidence: 60%

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A relaxed approach for curve matching with elastic metrics

et al. 2019

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“…The following theorem summarizes local and global well-posedness results, that are known for the class of reparametrization-invariant metrics on the space of parametrized submanifolds. [18,19], see also [57].…”

Section: The Geodesic Equationmentioning

confidence: 97%

Metric registration of curves and surfaces using optimal control

Bauer

Charon

Younes

2019

Handbook of Numerical Analysis

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This chapter presents an overview of recent developments in the analysis of shapes such as curves and surfaces through Riemannian metrics. We show that several constructions of metrics on spaces of submanifolds can be unified through the prism of Riemannian submersions, with shape space metrics being induced from metrics defined on the top spaces. Computing the resulting Riemannian distances involves solving geodesic matching problems with boundary conditions. To deal efficiently with such variational problems, one can rely on an auxiliary family of "chordal" distances to simplify the treatment of boundary conditions, which we use to come up with a relaxed inexact formulation of the matching problem. This also allows to turn shape matching into optimal control problems and give a common framework to address them in practice. We then specify our analysis to the cases of intrinsic shape metrics defined using invariant Sobolev metrics on parametrized immersions, outer shape metrics induced from metrics on diffeomorphism groups of the ambient space and finally a recent hybrid model that combines those two approaches.2000 Mathematics Subject Classification. 68Q25, 68R10, 68U05.

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“…It is not known whether this space is a smooth Banach manifold, it is however a metric length space. The structure of it is explained in more detail in the article [14], where the following completeness result is proven.…”

Section: Parametrized Curvesmentioning

confidence: 99%

A Numerical Framework for Sobolev Metrics on the Space of Curves

Bauer¹,

Bruveris²,

Harms³

et al. 2017

SIAM J. Imaging Sci.

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Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics, but their discretization is still largely missing. In this paper, we present algorithms to numerically solve the geodesic initial and boundary value problems for these metrics. The combination of these algorithms enables one to compute Karcher means in a Riemannian gradient-based optimization scheme and perform principal component analysis and clustering. Our framework is sufficiently general to be applicable to a wide class of metrics. We demonstrate the effectiveness of our approach by analyzing a collection of shapes representing HeLa cell nuclei.

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Completeness properties of Sobolev metrics on the space of curves

Cited by 29 publications

References 48 publications

A relaxed approach for curve matching with elastic metrics

A relaxed approach for curve matching with elastic metrics

Metric registration of curves and surfaces using optimal control

A Numerical Framework for Sobolev Metrics on the Space of Curves

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