Elastic Correspondence between Triangle Meshes

Ezuz, Danielle; Heeren, Behrend; Azencot, Omri; Rumpf, Martin; Ben‐Chen, Mirela

doi:10.1111/cgf.13624

Cited by 29 publications

(15 citation statements)

References 32 publications

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“…We emphasize that while our method uses the geodesic distances in the pairwise term and thus assumes an underlying isometry, it nevertheless attains good results in the quasi‐isometric case as we show for the FAUST dataset. However, if the shapes are non‐isometric as in e.g., SHREC07 [GBP07], different unary and pairwise components may be required, see e.g., [EHA*19]. We leave this direction for further investigation and future work.…”

Section: Evaluation and Resultsmentioning

confidence: 99%

Consistent Shape Matching via Coupled Optimization

Azencot

Dubrovina

Guibas

2019

Computer Graphics Forum

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We propose a new method for computing accurate point‐to‐point mappings between a pair of triangle meshes given imperfect initial correspondences. Unlike the majority of existing techniques, we optimize for a map while leveraging information from the inverse map, yielding results which are highly consistent with respect to composition of mappings. Remarkably, our method considers only a linear number of candidate points on the target shape, allowing us to work directly with high resolution meshes, and to avoid a delicate and possibly error‐prone up‐sampling procedure. Key to this dimensionality reduction is a novel candidate selection process, where the mapped points drift over the target shape, finalizing their location based on intrinsic distortion measures. Overall, we arrive at an iterative scheme where at each step we optimize for the map and its inverse by solving two relaxed Quadratic Assignment Problems using off‐the‐shelf optimization tools. We provide quantitative and qualitative comparison of our method with several existing techniques, and show that it provides a powerful matching tool when accurate and consistent correspondences are required.

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Section: Evaluation and Resultsmentioning

confidence: 99%

Consistent Shape Matching via Coupled Optimization

Azencot

Dubrovina

Guibas

2019

Computer Graphics Forum

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“…That det Dφ I > 0 almost everywhere follows directly by its definition, since |∂N r I M 1 | = 0, ϕ r I is a diffeomorphism, and the energy density in (27) is unbounded as det Dφ → 0. By its definition in (27) φ I belongs to W 1,p 0 (Ω; R d ) + Id. Moreover, since ϕ r I is a C 2 diffeomorphism and the definition of φ I in Ω in and Ω out we also have…”

Section: Existence Of Minimizers For Symmetric Matching Energiesmentioning

confidence: 91%

“…A number of works deal with shape analysis tasks using formulations based in linearized elasticity, like [31]. Shape matching using nonlinear thin shell energies has been tackled for parametric domains in [42] and for triangulated surfaces in [50,27]. Some precedents for shape analysis based on signed distance functions are [22] and [15].…”

Section: Related Workmentioning

confidence: 99%

Shape-Aware Matching of Implicit Surfaces Based on Thin Shell Energies

2017

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A shape sensitive, variational approach for the matching of surfaces considered as thin elastic shells is investigated. The elasticity functional to be minimized takes into account two different types of nonlinear energies: a membrane energy measuring the rate of tangential distortion when deforming the reference shell into the template shell, and a bending energy measuring the bending under the deformation in terms of the change of the shape operators from the undeformed into the deformed configuration. The variational method applies to surfaces described as level sets. It is mathematically well-posed, and an existence proof of an optimal matching deformation is given. The variational model is implemented using a finite element discretization combined with a narrow band approach on an efficient hierarchical grid structure. For the optimization, a regularized nonlinear conjugate gradient scheme and a cascadic multilevel strategy are used. The features of the proposed approach are studied for synthetic test cases and a collection of geometry processing examples.

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“…A number of works deal with shape analysis tasks using formulations based in linearized elasticity, like [30]. Shape matching using nonlinear thin shell energies has been tackled for parametric domains in [42] and for triangulated surfaces in [26,52]. Some precedents for shape analysis based on signed distance functions are [14,21].…”

Section: Related Workmentioning

confidence: 99%

Symmetry and scaling limits for matching of implicit surfaces based on thin shell energies

Iglesias

2021

ESAIM: M2AN

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In a recent paper by Iglesias, Rumpf and Scherzer (Found. Comput. Math. 18(4), 2018) a variational model for deformations matching a pair of shapes given as level set functions was proposed. Its main feature is the presence of anisotropic energies active only in a narrow band around the hypersurfaces that resemble the behavior of elastic shells. In this work we consider some extensions and further analysis of that model. First, we present a symmetric energy functional such that given two particular shapes, it assigns the same energy to any given deformation as to its inverse when the roles of the shapes are interchanged, and introduce the adequate parameter scaling to recover a surface problem when the width of the narrow band vanishes. Then, we obtain existence of minimizing deformations for the symmetric energy in classes of bi-Sobolev homeomorphisms for small enough widths, and prove a $\Gamma$-convergence result for the corresponding non-symmetric energies as the width tends to zero. Finally, numerical results on realistic shape matching applications demonstrating the effect of the symmetric energy are presented.

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Elastic Correspondence between Triangle Meshes

Cited by 29 publications

References 32 publications

Consistent Shape Matching via Coupled Optimization

Consistent Shape Matching via Coupled Optimization

Shape-Aware Matching of Implicit Surfaces Based on Thin Shell Energies

Symmetry and scaling limits for matching of implicit surfaces based on thin shell energies

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