Chordal decomposition for spectral coarsening

Chen, Honglin; Liu, Hsueh-Ti Derek; Jacobson, Alec; Levin, David

doi:10.1145/3414685.3417789

Cited by 5 publications

(6 citation statements)

References 51 publications

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“…We demonstrate our coarsening method using meshes and complexes and compare it against a baseline. Spectral approximations are evaluated using a generalization of functional maps [Chen et al 2020;Lescoat et al 2020;Liu et al 2019] 𝐶 = 𝑈 𝑇 𝑐 𝑃𝑈 . that are applicable to all Laplacian operators, Ideally 𝐶 should resemble a diagonal matrix.…”

Section: Discussionmentioning

confidence: 99%

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Generalized Spectral Coarsening

Keros¹,

Subr²

2022

Preprint

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Reducing the size of triangle meshes and their higher-dimensional counterparts, called simplicial complexes, while preserving important geometric or topological properties is an important problem in computer graphics and geometry processing. Such salient properties are captured by local shape descriptors via linear differential operators -often variants of Laplacian matrices. The eigenfunctions of Laplacians yield a convenient and useful set of bases that define a spectral domain for geometry processing (akin to the famous Fourier spectrum which uses eigenfunctions of the derivative operator). Existing methods for spectrum-preserving coarsening focus on 0-dimensional Laplacian operators that are defined on vertices (0-dimensional simplices).We propose a generalized spectral coarsening method that considers multiple Laplacian operators of possibly different dimensionalities in tandem. Our simple algorithm greedily decides the order of contractions of simplices based on a quality function per simplex. The quality function quantifies the error due to removal of that simplex on a chosen band within the spectrum of the coarsened geometry. We demonstrate that our method is useful to achieve band-pass filtering on both meshes as well as general simplicial complexes. CCS CONCEPTS• Computing methodologies → Shape analysis.

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Section: Discussionmentioning

confidence: 99%

“…A tilde above the respective notations denotes their coarsened versions. They formulate coarsening as an optimization problem subject to various sparsity conditions [Liu et al 2019], by detaching the mesh from the operator [Chen et al 2020], and localizing error computation in order to form a parallelizable strategy [Lescoat et al 2020].…”

Section: Introductionmentioning

confidence: 99%

Generalized Spectral Coarsening

Keros¹,

Subr²

2022

Preprint

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“…The latter set of methods have been shown to lead to excellent performance and scalability on tasks involving individual shapes, such as computing their Shape‐DNA [RWP06] descriptors, or performing mesh filtering. Similarly, there exist several spectral coarsening and simplification approaches [LJO19, LLT * 20, CLJL20] that explicitly aim to coarsen operators, such as the Laplacian while preserving their low frequency eigenapairs. Unfortunately, these methods typically rely on the eigenfunctions on the dense shapes, while the utility of the former approaches in the context of functional maps has not yet been fully analyzed and exploited, in part, since, as we show below, this requires local approximation bounds.…”

Section: Related Workmentioning

confidence: 99%

Scalable and Efficient Functional Map Computations on Dense Meshes

Magnet

Ovsjanikov

2023

Computer Graphics Forum

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We propose a new scalable version of the functional map pipeline that allows to efficiently compute correspondences between potentially very dense meshes. Unlike existing approaches that process dense meshes by relying on ad‐hoc mesh simplification, we establish an integrated end‐to‐end pipeline with theoretical approximation analysis. In particular, our method overcomes the computational burden of both computing the basis, as well the functional and pointwise correspondence computation by approximating the functional spaces and the functional map itself. Errors in the approximations are controlled by theoretical upper bounds assessing the range of applicability of our pipeline. With this construction in hand, we propose a scalable practical algorithm and demonstrate results on dense meshes, which approximate those obtained by standard functional map algorithms at the fraction of the computation time. Moreover, our approach outperforms the standard acceleration procedures by a large margin, leading to accurate results even in challenging cases.

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“…Similar methods have also been applied to tetrahedral mesh simplification, based on volume, quadric-based, and isosurfacepreserving criteria [Chiang and Lu 2003;Chopra and Meyer 2002;Vo et al 2007]. Recent methods formulate coarsening as an optimization problem subject to various sparsity conditions [Liu et al 2019], by detaching the mesh from the operator [Chen et al 2020] and localizing error computation to form a parallelizable strategy [Lescoat et al 2020]. The cotan Laplacian is a popular choice, and is used, via its functional maps [Ovsjanikov et al 2016], to identify correspondences between partial meshes [Rodolà et al 2017] .…”

Section: Related Workmentioning

confidence: 99%

Spectral Coarsening with Hodge Laplacians

Keros

Subr

2023

Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Proceedings

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generalized spectral domain example workflowFigure 1: We present a method to simplify geometry while preserving targeted bandwidths in a customizable spectral domain.The input to our algorithm is a generalized spectral domain, a combination of spectral bands from multiple discrete Laplacian operators. Our method includes a simple łliftingž operator to enable the workflow shown on the right. The example shows diffusion distances over a tetrahedral mesh of an engine part with 20 small holes that are preserved. Removing 80% of the input simplices, and seeking to preserve topology (lower portion of the spectrum), leads to some visible differences with respect to the ground truth but the quantitative comparisons indicate low error (≈ 3.6𝑒 − 7, shown in the supplemental material).

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Chordal decomposition for spectral coarsening

Cited by 5 publications

References 51 publications

Generalized Spectral Coarsening

Generalized Spectral Coarsening

Scalable and Efficient Functional Map Computations on Dense Meshes

Spectral Coarsening with Hodge Laplacians

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