Subdividing barycentric coordinates

Anisimov, Dmitry; Deng, Chongyang; Hormann, Kai

doi:10.1016/j.cagd.2016.02.005

Cited by 11 publications

(7 citation statements)

References 27 publications

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“…For higher-order problems, our method may not converge to high accuracy because our choice of linear interpolation is insufficient [Hemker 1990]. Thus, exploring high-order prolongation (e.g., subdivision barycentric coordinates [Anisimov et al 2016]) or learning-based prolongation (e.g, [Katrutsa et al 2020]) would also be valuable directions to improve the solver.…”

Section: Limitations and Future Workmentioning

confidence: 99%

Surface multigrid via intrinsic prolongation

et al. 2021

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Fig. 1. We introduce a novel geometric multigrid solver for curved surfaces. Our key ingredient is an intrinsic prolongation operator computed via parameterizing the high resolution shape via its coarsened counterpart, visualized using colored triangles. By recursively applying this self-parameterization, we obtain a hierarchy (from left to right) for our multigrid method (e.g., to solve heat geodesics [Crane et al. 2017], far left). ©model by Benoît Rogez under CC BY-NC.This paper introduces a novel geometric multigrid solver for unstructured curved surfaces. Multigrid methods are highly efficient iterative methods for solving systems of linear equations. Despite the success in solving problems defined on structured domains, generalizing multigrid to unstructured curved domains remains a challenging problem. The critical missing ingredient is a prolongation operator to transfer functions across different multigrid levels. We propose a novel method for computing the prolongation for triangulated surfaces based on intrinsic geometry, enabling an efficient geometric multigrid solver for curved surfaces. Our surface multigrid solver achieves better convergence than existing multigrid methods. Compared to direct solvers, our solver is orders of magnitude faster. We evaluate our method on many geometry processing applications and a wide variety of complex shapes with and without boundaries. By simply replacing the direct solver, we upgrade existing algorithms to interactive frame rates, and shift the computational bottleneck away from solving linear systems.

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Section: Limitations and Future Workmentioning

confidence: 99%

Surface multigrid via intrinsic prolongation

et al. 2021

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“…For further information on barycentric coordinates and its applications and generalizations, see [63][64][65][82][83][84][85][86][87][88][89][90][91][92][93][94][95].…”

Section: B22 Testing Nearby Facets For Intersectionsmentioning

confidence: 99%

Five Degree-of-Freedom Property Interpolation of Arbitrary Grain Boundaries via Voronoi Fundamental Zone Octonion Framework

Baird,

Homer,

Fullwood

et al. 2021

Preprint

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In this work we introduce the Voronoi fundamental zone octonion (VFZO) interpolation framework for grain boundary (GB) structure-property models and surrogates. The VFZO framework offers an advantage over other five degree-of-freedom (5DOF) based property interpolation methods because it is constructed as a point set in a manifold. This means that directly computed Euclidean distances approximate the original octonion distance with significantly reduced computation runtime (∼7 CPU minutes vs. 153 CPU days for a 50 000 × 50 000 pairwise-distance matrix). This increased efficiency facilitates lower interpolation error through the use of significantly more input data. We demonstrate grain boundary energy (GBE) interpolation results for a non-smooth validation function and simulated bi-crystal datasets for Fe and Ni using four interpolation methods: barycentric interpolation, Gaussian process regression (GPR) or Kriging, inverse-distance weighting (IDW), and nearest neighbor (NN) interpolation. These are evaluated for 50 000 random input GBs and 10 000 random prediction GBs. The best performance was achieved with GPR, which resulted in a reduction of the root mean square error (RMSE) by 83.0% relative to RMSE of a constant, average model. Likewise, interpolation on a large, noisy, molecular statics (MS) Fe simulation dataset improves performance by 34.4 % compared to 21.2 % in prior work. Interpolation on a small, low-noise MS Ni simulation dataset is similar to interpolation results for the original octonion metric (57.6 % vs. 56.4 %). A vectorized, parallelized, MATLAB interpolation function (interp5DOF.m) and related routines are available in our VFZO repository (gi thub.com/sgbaird-5dof/interp) which can be applied to other crystallographic point groups. The VFZO framework offers advantages for computing distances between GBs, estimating property values for arbitrary GBs, and modeling surrogates of computationally expensive 5DOF functions and simulations.

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“…For many generalized barycentric coordinates without closed-form expressions, their computation often relies on picewise approximation over a discretized domain. Anisimov et al (2016) show that subdivision schemes can be applied to refine a coarse approximation of such generalized barycentric coordinates, while retaining key properties such as smoothness and non-negativity.…”

Section: Generalized Barycentric Coordinatesmentioning

confidence: 99%

“…In Zhang et al (2014), the locality of coordinate functions are induced via the minimization of total variation which penalized the total length of level sets. The subdivision schemes proposed by Anisimov et al (2016) can be applied to refine a coarse result from Zhang et al (2014) while maintaining the locality. Recently, Anisimov et al (2017) also presented a new closed-form construction of generalized barycentric coordinates that are non-negative, smooth, and locally supported.…”

Section: Generalized Barycentric Coordinatesmentioning

confidence: 99%

A fast numerical solver for local barycentric coordinates

Tao

Deng

Zhang

2019

Computer Aided Geometric Design

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The local barycentric coordinates (LBC), proposed in Zhang et al. (2014), demonstrate good locality and can be used for local control on function value interpolation and shape deformation. However, it has no closedform expression and must be computed by solving an optimization problem, which can be time-consuming especially for high-resolution models. In this paper, we propose a new technique to compute LBC efficiently. The new solver is developed based on two key insights. First, we prove that the non-negativity constraints in the original LBC formulation is not necessary, and can be removed without affecting the solution of the optimization problem. Furthermore, the removal of this constraint allows us to reformulate the computation of LBC as a convex constrained optimization for its gradients, followed by a fast integration to recover the coordinate values. The reformulated gradient optimization problem can be solved using ADMM, where each step is trivially parallelizable and does not involve global linear system solving, making it much more scalable and efficient than the original LBC solver. Numerical experiments verify the effectiveness of our technique on a large variety of models.

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Subdividing barycentric coordinates

Cited by 11 publications

References 27 publications

Surface multigrid via intrinsic prolongation

Surface multigrid via intrinsic prolongation

Five Degree-of-Freedom Property Interpolation of Arbitrary Grain Boundaries via Voronoi Fundamental Zone Octonion Framework

A fast numerical solver for local barycentric coordinates

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