Splines in the Space of Shells

Heeren, Behrend; Rumpf, Martin; Schröder, Peter; Wardetzky, Max; Wirth, Benedikt

doi:10.1111/cgf.12968

Cited by 23 publications

(14 citation statements)

References 33 publications

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“…For instance, physical simulation software can require solution of complex variational problems for stable implicit time‐stepping [GSS*15]. Automatic tools for interpolating between frames of an animated mesh sequence similarly must minimize objectives constructed out of nonconvex elastic terms and could benefit from preconditioned acceleration [FB11, HRWW12, HRS*16].…”

Section: Discussionmentioning

confidence: 99%

Isometry‐Aware Preconditioning for Mesh Parameterization

Bessmeltsev¹,

Schaefer

Solomon³

2017

Computer Graphics Forum

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This paper presents a new preconditioning technique for large‐scale geometric optimization problems, inspired by applications in mesh parameterization. Our positive (semi‐)definite preconditioner acts on the gradients of optimization problems whose variables are positions of the vertices of a triangle mesh in ℝ2 or of a tetrahedral mesh in ℝ3, converting localized distortion gradients into the velocity of a globally near‐rigid motion via a linear solve. We pose our preconditioning tool in terms of the Killing energy of a deformation field and provide new efficient formulas for constructing Killing operators on triangle and tetrahedral meshes. We demonstrate that our method is competitive with state‐of‐the‐art algorithms for locally injective parameterization using a variety of optimization objectives and show applications to two‐ and three‐dimensional mesh deformation.

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Section: Discussionmentioning

confidence: 99%

Isometry‐Aware Preconditioning for Mesh Parameterization

Bessmeltsev¹,

Schaefer

Solomon³

2017

Computer Graphics Forum

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“…This enables natural extrapolation of paths in shell space and the transfer of large nonlinear deformations from one shell to another. In [Heeren et al 2016], the authors extend the concept of Euclidean splines to the Riemannian manifold of discrete shells, allowing for a temporally smooth interpolation of a given set of shell keyframe poses.…”

Section: Shape Space and Flowsmentioning

confidence: 99%

The shape space of discrete orthogonal geodesic nets

Rabinovich

Hoffmann²,

Sorkine‐Hornung

2018

ACM Trans. Graph.

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Discrete orthogonal geodesic nets (DOGs) are a quad mesh analogue of developable surfaces. In this work we study continuous deformations on these discrete objects. Our main theoretical contribution is the characterization of the shape space of DOGs for a given net connectivity. We show that generally, this space is locally a manifold of a fixed dimension, apart from a set of singularities, implying that DOGs are continuously deformable. Smooth flows can be constructed by a smooth choice of vectors on the manifold's tangent spaces, selected to minimize a desired objective function under a given metric. We show how to compute such vectors by solving a linear system, and we use our findings to devise a geometrically meaningful way to handle singular points. We base our shape space metric on a novel DOG Laplacian operator, which is proved to converge under sampling of an analytical orthogonal geodesic net. We further show how to extend the shape space of DOGs by supporting creases and curved folds and apply the developed tools in an editing system for developable surfaces that supports arbitrary bending, stretching, cutting, (curved) folds, as well as smoothing and subdivision operations.

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“…Shape matching is a fundamental problem in geometry processing and computer graphics, with applications ranging from shape interpolation [HRS*16] to shape exploration [HWG14] and statistical shape analysis [BRLB14].…”

Section: Introductionmentioning

confidence: 99%

Structured Regularization of Functional Map Computations

Ren

Panine

Wonka

et al. 2019

Computer Graphics Forum

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We consider the problem of non‐rigid shape matching using the functional map framework. Specifically, we analyze a commonly used approach for regularizing functional maps, which consists in penalizing the failure of the unknown map to commute with the Laplace‐Beltrami operators on the source and target shapes. We show that this approach has certain undesirable fundamental theoretical limitations, and can be undefined even for trivial maps in the smooth setting. Instead we propose a novel, theoretically well‐justified approach for regularizing functional maps, by using the notion of the resolvent of the Laplacian operator. In addition, we provide a natural one‐parameter family of regularizers, that can be easily tuned depending on the expected approximate isometry of the input shape pair. We show on a wide range of shape correspondence scenarios that our novel regularization leads to an improvement in the quality of the estimated functional, and ultimately pointwise correspondences before and after commonly‐used refinement techniques.

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Splines in the Space of Shells

Cited by 23 publications

References 33 publications

Isometry‐Aware Preconditioning for Mesh Parameterization

Isometry‐Aware Preconditioning for Mesh Parameterization

The shape space of discrete orthogonal geodesic nets

Structured Regularization of Functional Map Computations

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