Varifold-Based Matching of Curves via Sobolev-Type Riemannian Metrics

Bauer, Martins; Bruveris, Martins; Charon, Nicolas; Møller-Andersen, Jakob

doi:10.1007/978-3-319-67675-3_14

Cited by 3 publications

(8 citation statements)

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“…Proof of Theorem 2.6. Denote by G 1 an elastic metric with constant coefficients and by G 2 a scale-invariant elastic metric of the form (1) and (2). We first observe that both G 1 and G 2 extend to smooth Riemannian metrics on the Hilbert manifold I 2 (S 1 , R d ).…”

Section: 3mentioning

confidence: 94%

“…Using invariance properties of the metrics, we obtain the following result concerning the induced metrics on the quotient space: Theorem 2.8. Let G be an elastic metric of type (1) or (2). Then G induces a Riemannian metric on the quotient space S f (M 1 , R d ) such that the projection…”

Section: Metrics On Spaces Of Open and Closed Curvesmentioning

confidence: 99%

“…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 Kurtek and Needham [32] propose a different numerical framework for the class of G a,b -metrics based on a generalization of the transformation of Younes et.…”

Section: Introductionmentioning

confidence: 99%

“…The construction of those distance functions, unrelated to Sobolev metrics, follow the principles of geometric measure theory, which have so far been used as fidelity terms in combination with models like the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework. In the present article, building on our previous conference publication [2], we examine a fairly general class of kernel metrics on immersed open and closed curves that are induced from the representation of curves as oriented varifolds. We examine the rigorous conditions to obtain distance functions on unparametrized immersed curves.…”

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: 3mentioning

confidence: 94%

Section: Metrics On Spaces Of Open and Closed Curvesmentioning

confidence: 99%

Section: Introductionmentioning

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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“…For this reason, they have to be combined with other shape distances, either on diffeomorphism groups as in diffeomorphic matching (Charon and Trouvé, 2013) or on shape spaces as in elastic matching. The latter approach was used for Sobolev metrics on curves by Bauer, Bruveris, Charon, et al (2017 and for square root normal distances on curves in the recent conference paper by Bauer, Charon, and Harms (2019), which is a predecessor of the present work.…”

Section: Introductionmentioning

confidence: 99%

A numerical framework for elastic surface matching, comparison, and interpolation

Bauer,

Charon,

Harms

et al. 2020

Preprint

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Surface comparison and matching is a challenging problem in computer vision. While reparametrization-invariant Sobolev metrics provide meaningful elastic distances and point correspondences via the geodesic boundary value problem, solving this problem numerically tends to be difficult. Square root normal fields (SRNF) considerably simplify the computation of certain elastic distances between parametrized surfaces. Yet they leave open the issue of finding optimal reparametrizations, which induce elastic distances between unparametrized surfaces. This issue has concentrated much effort in recent years and led to the development of several numerical frameworks. In this paper, we take an alternative approach which bypasses the direct estimation of reparametrizations: we relax the geodesic boundary constraint using an auxiliary parametrization-blind varifold fidelity metric. This reformulation has several notable benefits. By avoiding altogether the need for reparametrizations, it provides the flexibility to deal with simplicial meshes of arbitrary topologies and sampling patterns. Moreover, the problem lends itself to a coarse-to-fine multiresolution implementation, which makes the algorithm scalable to large meshes. Furthermore, this approach extends readily to higher-order feature maps such as square root curvature fields and is also able to include surface textures in the matching problem. We demonstrate these advantages on several examples, synthetic and real.

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