Spectral coarsening of geometric operators

Liu, Hsueh-Ti Derek; Jacobson, Alec; Ovsjanikov, Maks

doi:10.1145/3306346.3322953

Cited by 19 publications

(45 citation statements)

References 69 publications

(74 reference statements)

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“…The goal of spectral coarsening is to reduce the size of a discrete operator, derived from a 3D shape, while preserving its spectral properties. Liu et al [2019] show that it is possible to have a significant reduction without affecting the low-frequency eigenvectors and eigenvalues. They visualize the preservation of spectral properties with the inner product matrix between eigenvectors (see Fig.…”

Section: Methodsmentioning

confidence: 93%

“…However, their method only supports matrices with a chordal sparsity pattern already, which is not applicable to our problem because most discrete operators are not chordal. In contrast, we utilize the ideas from [Sun and Vandenberghe 2015] to handle any sparsity pattern of choice, and the strategies in [Zheng et al 2017b[Zheng et al , 2020 to develop a chordal ADMM solver for the spectral coarsening energy [Liu et al 2019]. We exploit the fact that many discrete operators are sparse and symmetric to perform a change of variables to significantly reduce the computational cost.…”

Section: Chordal Graphs In Sparse Matrix Optimizationmentioning

confidence: 99%

“…Nasikun et al [2018] use a combination of Poisson disk sampling and local polynomial bases to efficiently solve an approximate Laplacian eigenvalue problem of a mesh. Beyond the Laplace operator, Liu et al [2019] propose an algebraic approach to coarsen common geometric operators while preserving spectral properties. Lescoat et al [2020] extend the formulation to achieve spectral-preserving mesh simplification.…”

Section: Geometry Coarseningmentioning

confidence: 99%

“…Our approach is purely algebraic. Our convex formulation leads us to have better spectral preservation compared to the similar algebraic approach [Liu et al 2019] in spectral coarsening. Our flexibility in controlling the sparsity allows us to post-process the results of spectral simplification [Lescoat et al 2020] and further improve its quality.…”

Section: Geometry Coarseningmentioning

confidence: 99%

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Chordal decomposition for spectral coarsening

et al. 2020

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We introduce a novel solver to significantly reduce the size of a geometric operator while preserving its spectral properties at the lowest frequencies. We use chordal decomposition to formulate a convex optimization problem which allows the user to control the operator sparsity pattern. This allows for a trade-off between the spectral accuracy of the operator and the cost of its application. We efficiently minimize the energy with a change of variables and achieve state-of-the-art results on spectral coarsening. Our solver further enables novel applications including volume-to-surface approximation and detaching the operator from the mesh, i.e., one can produce a mesh tailor-made for visualization and optimize an operator separately for computation.

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Section: Methodsmentioning

confidence: 93%

Section: Chordal Graphs In Sparse Matrix Optimizationmentioning

confidence: 99%

Section: Geometry Coarseningmentioning

confidence: 99%

Section: Geometry Coarseningmentioning

confidence: 99%

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Chordal decomposition for spectral coarsening

et al. 2020

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“…Schemes for the approximate solution of eigenproblems are static condensation [Bathe 2014] in engineering and the Nyström method [Williams and Seeger 2001] and random projections [Halko et al 2011] in machine learning. Approximation schemes for the Laplace-Beltrami eigenproblem on surfaces have been introduced in Chuang et al [2009], Lescoat et al [2020], Liu et al [2019], and Nasikun et al [2018]. In contrast to the eigensolvers we consider in this work, these schemes do not provide any guarantee on the approximation quality of the eigenpairs.…”

Section: Related Workmentioning

confidence: 99%

The Hierarchical Subspace Iteration Method for Laplace–Beltrami Eigenproblems

Nasikun

Hildebrandt

2022

ACM Trans. Graph.

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Sparse eigenproblems are important for various applications in computer graphics. The spectrum and eigenfunctions of the Laplace–Beltrami operator, for example, are fundamental for methods in shape analysis and mesh processing. The Subspace Iteration Method is a robust solver for these problems. In practice, however, Lanczos schemes are often faster. In this article, we introduce the Hierarchical Subspace Iteration Method (HSIM) , a novel solver for sparse eigenproblems that operates on a hierarchy of nested vector spaces. The hierarchy is constructed such that on the coarsest space all eigenpairs can be computed with a dense eigensolver. HSIM uses these eigenpairs as initialization and iterates from coarse to fine over the hierarchy. On each level, subspace iterations, initialized with the solution from the previous level, are used to approximate the eigenpairs. This approach substantially reduces the number of iterations needed on the finest grid compared to the non-hierarchical Subspace Iteration Method. Our experiments show that HSIM can solve Laplace–Beltrami eigenproblems on meshes faster than state-of-the-art methods based on Lanczos iterations, preconditioned conjugate gradients, and subspace iterations.

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