Gauge Invariance of Degenerate Riemannian Metrics

Tumpach, Alice Barbara

doi:10.1090/noti1350

Cited by 8 publications

(3 citation statements)

References 2 publications

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“…Algorithmically, the initial value problem can be solved by a time-discretization of the geodesic equation (Bauer, Harms, and Michor, 2011) or using discrete geodesic calculus (Rumpf and Wirth, 2015). The boundary value problem on function spaces can be solved by the direct method of variational calculus (Su, Bauer, Preston, et al, 2019;Tumpach, 2016;Tumpach et al, 2015) or, for certain elastic metrics, by exploiting isometries to simpler spaces. For the boundary value problem on shape spaces, one has to search additionally for an optimal reparametrization.…”

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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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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“…Shape analysis of surfaces in R 3 has been motivated by many applications in bioinformatics, computer graphics and medical imaging, see e.g., [16,18,2,11,13,28]. In most applications the actual parametrization of the surfaces under consideration is unknown and one is only able to observe the "shape" of the object, i.e., a priori the point correspondences between the surfaces are unknown and should be an output of the performed analysis.…”

Section: Introductionmentioning

confidence: 99%

“…In the past years several metrics and frameworks have been proposed as potential approaches to this goal, see e.g., [15,21,20,4,25,28]. In particular, a class of elastic metrics has been proposed in [14], which is defined as a weighted sum of three components that measure the differences in shearing, stretching and bending of the surface.…”

Section: Introductionmentioning

confidence: 99%

Shape Analysis of Surfaces Using General Elastic Metrics

Su¹,

Bauer²,

Preston³

et al. 2019

Preprint

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In this article we introduce a family of elastic metrics on the space of parametrized surfaces in 3D space using a corresponding family of metrics on the space of vector valued one-forms. We provide a numerical framework for the computation of geodesics with respect to these metrics. The family of metrics is invariant under rigid motions and reparametrizations; hence it induces a metric on the "shape space" of surfaces. This new class of metrics generalizes a previously studied family of elastic metrics and includes in particular the M.

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Three Methods to Put a Riemannian Metric on Shape Space

Tumpach

Preston

2023

Lecture Notes in Computer Science

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Gauge Invariance of Degenerate Riemannian Metrics

Cited by 8 publications

References 2 publications

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

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

Shape Analysis of Surfaces Using General Elastic Metrics

Three Methods to Put a Riemannian Metric on Shape Space

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