Perfect Laplacians for Polygon Meshes

Herholz, Philipp; Kyprianidis, Jan Eric; Alexa, Marc

doi:10.1111/cgf.12709

Cited by 13 publications

(12 citation statements)

References 28 publications

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“…In [Alexa and Wardetzky 2011], a geometric approach was described to construct a discrete Laplacian for polygonal surfaces that exploits the gradient of the magnitude of the polygonal vector area [Sullivan 2008] combined with a MFDbased inner product stabilization [Brezzi et al 2005]. The work of Herholz et al [2015] later characterized polygonal meshes that admit a discrete Laplacian with only non-negative weights, while Sharp et al [2019] enriched this polygonal operator with vertex-to-vertex rotations in order to assemble a discrete vector Laplacian. Recently, Bunge et al [2020] adapted the virtual node method to non-flat polygons by refining each polygon with a triangle fan, emanating not from the face centroid but from the position that minimizes the sum of squared triangle areas.…”

Section: Vertex Residualmentioning

confidence: 99%

Discrete differential operators on polygonal meshes

Goes¹,

Butts²,

Desbrun

2020

ACM Trans. Graph.

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Geometry processing of surface meshes relies heavily on the discretization of differential operators such as gradient, Laplacian, and covariant derivative. While a variety of discrete operators over triangulated meshes have been developed and used for decades, a similar construction over polygonal meshes remains far less explored despite the prevalence of non-simplicial surfaces in geometric design and engineering applications. This paper introduces a principled construction of discrete differential operators on surface meshes formed by (possibly non-flat and non-convex) polygonal faces. Our approach is based on a novel mimetic discretization of the gradient operator that is linear-precise on arbitrary polygons. Equipped with this discrete gradient, we draw upon ideas from the Virtual Element Method in order to derive a series of discrete operators commonly used in graphics that are now valid over polygonal surfaces. We demonstrate the accuracy and robustness of our resulting operators through various numerical examples, before incorporating them into existing geometry processing algorithms.

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Section: Vertex Residualmentioning

confidence: 99%

Discrete differential operators on polygonal meshes

Goes¹,

Butts²,

Desbrun

2020

ACM Trans. Graph.

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show abstract

“…, x n ] is the eigenvectors' matrix and Λ is the diagonal matrix of the eigenvalues (λ i ) n i=1 . Analogous discretisations apply to polygonal [3,22] and tetrahedral [27,51] meshes, or point sets [29].…”

Section: Discretisation Of Spectral Kernels and Distancesmentioning

confidence: 99%

A unified definition and computation of Laplacian spectral distances

Patanè

2019

Pattern Recognition

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Laplacian spectral kernels and distances (e.g., biharmonic, heat diffusion, wave kernel distances) are easily defined through a filtering of the Laplacian eigenpairs. They play a central role in several applications, such as dimensionality reduction with spectral embeddings, diffusion geometry, image smoothing, geometric characterisations and embeddings of graphs. Extending the results recently derived in the discrete setting [38,39] to the continuous case, we propose a novel definition of the Laplacian spectral kernels and distances, whose approximation requires the solution of a set of inhomogeneous Laplace equations. Their discrete counterparts are equivalent to a set of sparse, symmetric, and well-conditioned linear systems, which are efficiently solved with iterative methods. Finally, we discuss the optimality of the Laplacian spectrum for the approximation of the spectral kernels, the relation between the spectral and Green kernels, and the stability of the spectral distances with respect to the evaluation of the Laplacian spectrum and to multiple Laplacian eigenvalues.

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“…Frequency-based ideas have been widely used for mesh analysis and processing in a broad range of applications -see [LZ10] and papers therein for a recent survey. The majority of these works utilize the spectrum (eigenfunctions / eigenvalues) of the Laplace-Beltrami operator on discrete manifolds [DRW10] -mostly triangular meshes, although extensions to polygonal cases exist [HKA15]. Applications include, among others, mesh smoothing, compression, shape segmentation, matching, and parameterization.…”

Section: Related Workmentioning

confidence: 99%

A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

Diamanti

Tang

et al. 2018

Computer Graphics Forum

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Spectral mesh analysis and processing methods, namely ones that utilize eigenvalues and eigenfunctions of linear operators on meshes, have been applied to numerous geometric processing applications. The operator used predominantly in these methods is the Laplace‐Beltrami operator, which has the often‐cited property that it is intrinsic, namely invariant to isometric deformation of the underlying geometry, including rigid transformations. Depending on the application, this can be either an advantage or a drawback. Recent work has proposed the alternative of using the Dirac operator on surfaces for spectral processing. The available versions of the Dirac operator either only focus on the extrinsic version, or introduce a range of mixed operators on a spectrum between fully extrinsic Dirac operator and intrinsic Laplace operator. In this work, we introduce a unified discretization scheme that describes both an extrinsic and intrinsic Dirac operator on meshes, based on their continuous counterparts on smooth manifolds. In this discretization, both operators are very closely related, and preserve their key properties from the smooth case. We showcase various applications of our operators, with improved numerics over prior work.

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Perfect Laplacians for Polygon Meshes

Cited by 13 publications

References 28 publications

Discrete differential operators on polygonal meshes

Discrete differential operators on polygonal meshes

A unified definition and computation of Laplacian spectral distances

A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

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