A fast numerical solver for local barycentric coordinates

Tao, Jiong; Deng, Bailin; Zhang, Jie

doi:10.1016/j.cagd.2019.04.006

Cited by 12 publications

(8 citation statements)

References 43 publications

(56 reference statements)

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“…Coordinates, only cage vertex positions are considered. problem dealing with dense mesh models, as pointed out by [190]. So they propose a new efficient solver for the optimization of LBC.…”

Section: Tions In Previous Methods Like Mvc and Harmonicmentioning

confidence: 99%

A Revisit of Shape Editing Techniques: from the Geometric to the Neural Viewpoint

Yuan

Lai

et al. 2021

Preprint

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3D shape editing is widely used in a range of applications such as movie production, computer games and computer aided design. It is also a popular research topic in computer graphics and computer vision. In past decades, researchers have developed a series of editing methods to make the editing process faster, more robust, and more reliable. Traditionally, the deformed shape is determined by the optimal transformation and weights for an energy term. With increasing availability of 3D shapes on the Internet, data-driven methods were proposed to improve the editing results. More recently as the deep neural networks became popular, many deep learning based editing methods have been developed in this field, which is naturally data-driven. We mainly survey recent research works from the geometric viewpoint to those emerging neural deformation techniques and categorize them into organic shape editing methods and man-made model editing methods. Both traditional methods and recent neural network based methods are reviewed.

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Section: Tions In Previous Methods Like Mvc and Harmonicmentioning

confidence: 99%

A Revisit of Shape Editing Techniques: from the Geometric to the Neural Viewpoint

Yuan

Lai

et al. 2021

Preprint

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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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“…The resulting coordinates have all desired properties, but must be treated numerically. This includes harmonic [JMD*07] and local barycentric coordinates [ZDL*14, TDZ19], which are usually approximated by piecewise linear functions over a dense triangulation T of Ω and require to solve a global problem to determine the values of the functions at the vertices of T . Another construction from this category are maximum entropy coordinates [HS08], which can be evaluated at any v ∈ Ω by solving a local convex optimization problem, but this approach relies on the choice of certain prior functions, and it is not clear how to choose the latter for a given polygon, so that shape artefacts in the coordinates are avoided.…”

Section: Introductionmentioning

confidence: 99%

Maximum Likelihood Coordinates

Chang

Deng

Hormann

2023

Computer Graphics Forum

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Any point inside a d‐dimensional simplex can be expressed in a unique way as a convex combination of the simplex's vertices, and the coefficients of this combination are called the barycentric coordinates of the point. The idea of barycentric coordinates extends to general polytopes with n vertices, but they are no longer unique if n > d+1. Several constructions of such generalized barycentric coordinates have been proposed, in particular for polygons and polyhedra, but most approaches cannot guarantee the non‐negativity of the coordinates, which is important for applications like image warping and mesh deformation. We present a novel construction of non‐negative and smooth generalized barycentric coordinates for arbitrary simple polygons, which extends to higher dimensions and can include isolated interior points. Our approach is inspired by maximum entropy coordinates, as it also uses a statistical model to define coordinates for convex polygons, but our generalization to non‐convex shapes is different and based instead on the project‐and‐smooth idea of iterative coordinates. We show that our coordinates and their gradients can be evaluated efficiently and provide several examples that illustrate their advantages over previous constructions.

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A fast numerical solver for local barycentric coordinates

Cited by 12 publications

References 43 publications

A Revisit of Shape Editing Techniques: from the Geometric to the Neural Viewpoint

A Revisit of Shape Editing Techniques: from the Geometric to the Neural Viewpoint

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

Maximum Likelihood Coordinates

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