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Discrete differential operators on polygonal meshes

Authors:

Fernando De Goes,

Mathieu DesbrunAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 39, Issue 4

Article No.: 110, Pages 110:1 - 110:14

https://doi.org/10.1145/3386569.3392389

Published: 12 August 2020 Publication History

Abstract

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.

References

[1]

Ralph Abraham, Jerrold E. Marsden, and Tudor Ratiu. 1988. Manifolds, Tensor Analysis, and Applications. Applied Mathematical Sciences, Vol. 75. Springer Verlag.

[2]

Marc Alexa and Max Wardetzky. 2011. Discrete Laplacians on General Polygonal Meshes. ACM Transactions on Graphics 30, 4 (2011), Article 102.

Digital Library

[3]

Bernhard Auchmann and Stefan Kurz. 2006. A geometrically defined discrete Hodge operator on simplicial cells. IEEE Transactions on Magnetics 42, 4 (2006), 643--646.

[4]

Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, and Mirela Ben-Chen. 2015. Discrete Derivatives of Vector Fields on Surfaces - An Operator Approach. ACM Transactions on Graphics 34, 3 (2015), Article 29.

Digital Library

[5]

Lourenço Beirão da Veiga, Franco Brezzi, Andrea Cangiani, Gianmarco Manzini, Luisa Donatella Marini, and Alessandro Russo. 2013. Basic principles of Virtual Element Methods. Mathematical Models and Methods in Applied Sciences 23, 1 (2013), 199--214.

[6]

Lourenço Beirão da Veiga, Carlo Lovadina, and Alessandro Russo. 2017. Stability Analysis for the Virtual Element Method. Mathematical Models and Methods in Applied Sciences 27, 13 (2017), 2557--2594.

[7]

Franco Brezzi, Konstantin Lipnikov, and Valeria Simoncini. 2005. A Family Of Mimetic Finite Difference Methods On Polygonal And Polyhedral Meshes. Mathematical Models and Methods in Applied Sciences 15, 10 (2005), 1533--1551.

[8]

Astrid Bunge, Philipp Herholz, Misha Kazhdan, and Mario Botsch. 2020. Polygon Laplacian Made Simple. Computer Graphics Forum 39, 2 (2020).

[9]

Frédéric Cazals and Marc Pouget. 2005. Estimating differential quantities using polynomial fitting of osculating jets. Computer Aided Geometric Design 22, 2 (2005), 121--146.

[10]

Yanqing Chen, Timothy A. Davis, William W. Hager, and Sivasankaran Rajamanickam. 2008. Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate. ACM Trans. Math. Software 35, 3 (2008), 14.

Digital Library

[11]

David Cohen-Steiner and Jean-Marie Morvan. 2003. Restricted Delaunay Triangulations and Normal Cycle. In Symposium on Computational Geometry. 312--321.

[12]

Dylan M. Copeland. 2009. Boundary-Element-Based Finite Element Methods for Helmholtz and Maxwell Equations on General Polyhedral Meshes. International Journal of Mathematical and Computational Sciences 3, 9 (2009), 661--674.

[13]

Keenan Crane, Fernando de Goes, Mathieu Desbrun, and Peter Schröder. 2013a. Digital Geometry Processing with Discrete Exterior Calculus. In ACM SIGGRAPH 2013 Courses. Course #7.

[14]

Keenan Crane, Mathieu Desbrun, and Peter Schröder. 2010. Trivial Connections on Discrete Surfaces. Computer Graphics Forum 29, 5 (2010), 1525--1533.

[15]

Keenan Crane, Clarisse Weischedel, and Max Wardetzky. 2013b. Geodesics in Heat: A New Approach to Computing Distance Based on Heat Flow. ACM Transactions on Graphics 32, 5 (2013), Art. 152.

Digital Library

[16]

Ke-Yang Dai, Gui-Rong Liu, and Thoi-Trung Nguyen. 2007. An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics. Finite Elements in Analysis and Design 43, 11 (2007), 847--860.

[17]

Fernando de Goes, Mathieu Desbrun, Mark Meyer, and Tony DeRose. 2016b. Subdivision Exterior Calculus for Geometry Processing. ACM Transactions on Graphics 35, 4 (2016), Article 133.

Digital Library

[18]

Fernando de Goes, Mathieu Desbrun, and Yiying Tong. 2016a. Vector Field Processing on Triangle Meshes. In ACM SIGGRAPH Courses. Article 27, 49 pages.

[19]

Fernando de Goes, Beibei Liu, Max Budninskiy, Yiying Tong, and Mathieu Desbrun. 2014. Discrete 2-tensor Fields on Triangulations. Computer Graphics Forum 33, 5 (2014), 13--24.

Digital Library

[20]

Doug DeCarlo, Adam Finkelstein, and Szymon Rusinkiewicz. 2004. Interactive Rendering of Suggestive Contours with Temporal Coherence. In Symposium on Non-Photorealistic Animation and Rendering. 135--145.

[21]

Mathieu Desbrun, Eva Kanso, and Yiying Tong. 2008. Discrete Differential Forms for Computational Modeling. In Discrete Differential Geometry, A.I. Bobenko et al. (Ed.). Birkhäuser Basel, Chapter 16, 287--324.

[22]

Qiang. Du, Vance. Faber, and Max. Gunzburger. 1999. Centroidal Voronoi Tessellations: Applications and Algorithms. SIAM Rev. 41, 4 (1999), 637--676.

[23]

Michael Floater. 2003. Mean value coordinates. Computer Aided Geometric Design 20, 1 (2003), 19--27.

Digital Library

[24]

Arun L. Gain, Cameron Talischi, and Glaucio H. Paulino. 2014. On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Computer Methods in Applied Mechanics and Engineering 282 (2014), 132--160.

[25]

Andrew Gillette, Alexander Rand, and Chandrajit Bajaj. 2016. Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes. Computational Methods in Applied Mathematics 16, 4 (2016), 667--683.

[26]

Eitan Grinspun, Yotam Gingold, Jason Reisman, and Denis Zorin. 2006. Computing discrete shape operators on general meshes. Computer Graphics Forum 25, 3 (2006), 547--556.

[27]

Eric Heitz and Fabrice Neyret. 2018. High-Performance By-Example Noise Using a Histogram-Preserving Blending Operator. Symposium on High-Performance Graphics 1, 2 (2018), Article 31.

Digital Library

[28]

Philipp Herholz, Jan Eric Kyprianidis, and Marc Alexa. 2015. Perfect Laplacians for Polygon Meshes. Computer Graphics Forum 34, 5 (2015), 211--218.

Digital Library

[29]

Klaus Hildebrandt and Konrad Polthier. 2011. Generalized shape operators on polyhedral surfaces. Computer Aided Geometric Design 28, 5 (2011), 321--343.

Digital Library

[30]

Anil N. Hirani. 2003. Discrete Exterior Calculus. Ph.D. Dissertation. Caltech.

Digital Library

[31]

Michael Holst and Ari Stern. 2012. Geometric Variational Crimes: Hilbert Complexes, Finite Element Exterior Calculus, and Problems on Hypersurfaces. Foundations of Computational Mathematics 12, 3 (2012), 263--293.

Digital Library

[32]

Evangelos Kalogerakis, Patricio Simari, Derek Nowrouzezahrai, and Karan Singh. 2007. Robust Statistical Estimation of Curvature on Discretized Surfaces. In Symposium on Geometry Processing. 13--22.

[33]

Felix Knöppel, Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2013. Globally Optimal Direction Fields. ACM Transactions on Graphics 32, 4 (2013), Article 59.

Digital Library

[34]

Konstantin Lipnikov, Gianmarco Manzini, and Mikhail Shashkov. 2014. Mimetic Finite Difference Method. J. Comput. Phys. 257 (2014), 1163--1227.

Digital Library

[35]

Beibei Liu, Yiying Tong, Fernando de Goes, and Mathieu Desbrun. 2016. Discrete Connection and Covariant Derivative for Vector Field Analysis and Design. ACM Transactions on Graphics 35, 3 (2016), Article 23.

Digital Library

[36]

Ligang Liu, Lei Zhang, Yin Xu, Craig Gotsman, and Steven J. Gortler. 2008. A Local/Global Approach to Mesh Parameterization. Computer Graphics Forum 27, 5 (2008), 1495--1504.

Digital Library

[37]

Michael Mengolini, Matías F. Benedetto, and Alejandro M. Aragón. 2019. An engineering perspective to the virtual element method and its interplay with the standard finite element method. Computer Methods in Applied Mechanics and Engineering 350 (2019), 995--1023.

[38]

Mark Meyer, Mathieu Desbrun, Peter Schröder, and Alan H. Barr. 2003. Discrete Differential-Geometry Operators for Triangulated 2-Manifolds. In Visualization and Mathematics III. 35--57.

[39]

Patrick Mullen, Yiying Tong, Pierre Alliez, and Mathieu Desbrun. 2008. Spectral Conformal Parameterization. Computer Graphics Forum 27, 5 (2008), 1487--1494.

Digital Library

[40]

Konrad Polthier and Eike Preuss. 2003. Identifying Vector Field Singularities Using a Discrete Hodge Decomposition. Visualization and Mathematics 3 (2003).

[41]

Pierre-Arnaud Raviart and Jean-Marie Thomas. 1977. A Mixed Finite Element Method for Second Order Elliptic Problems. 66 (1977), 292--315.

[42]

Sergej Rjasanow and Steffen Weißer. 2012. Higher Order BEM-Based FEM on Polygonal Meshes. SIAM J. Numer. Anal. 50, 5 (2012), 2357--2378.

Digital Library

[43]

Szymon Rusinkiewicz. 2004. Estimating Curvatures and Their Derivatives on Triangle Meshes. In Symposium on 3D Data Processing, Visualization, and Transmission. 486--493.

[44]

Nicholas Sharp, Yousuf Soliman, and Keenan Crane. 2019. The Vector Heat Method. ACM Transactions on Graphics 38, 3 (2019), Article 24.

Digital Library

[45]

Breannan Smith, Fernando de Goes, and Theodore Kim. 2019. Analytic Eigensystems for Isotropic Distortion Energies. ACM Transactions on Graphics 38, 1, Article 3 (2019), 15 pages.

Digital Library

[46]

Jason Smith and Scott Schaefer. 2015. Bijective Parameterization with Free Boundaries. ACM Transactions on Graphics 34, 4 (2015).

Digital Library

[47]

Justin Solomon, Mirela Ben-Chen, Adrian Butscher, and Leonidas Guibas. 2011. As-Killing-As-Possible Vector Fields for Planar Deformation. Computer Graphics Forum 30, 5 (2011), 1543--1552.

[48]

Natarajan Sukumar and Alireza Tabarraei. 2004. Conforming polygonal finite elements. Internat. J. Numer. Methods Engrg. 61, 12 (2004), 2045--2066.

[49]

John M. Sullivan. 2008. Curvatures of Smooth and Discrete Surfaces. Birkhäuser Basel, Chapter 9, 175--188.

[50]

Robert W. Sumner and Jovan Popoviç. 2004. Deformation Transfer for Triangle Meshes. ACM Transactions on Graphics 23, 3 (2004).

Digital Library

[51]

Xu-Hai Tang, Sheng-Chuan Wu, Chao Zheng, and Jian-Hai Zhang. 2009. A novel virtual node method for polygonal elements. Appl. Math. Mech. 30 (2009), Article 1233.

[52]

Gabriel Taubin. 1995. Estimating the tensor of curvature of a surface from a polyhedral approximation. In IEEE International Conference on Computer Vision. 902--907.

[53]

Yiying Tong, Santiago Lombeyda, Anil N. Hirani, and Mathieu Desbrun. 2003. Discrete Multiscale Vector Field Decomposition. ACM Transactions on Graphics 22, 3 (2003), 445--452.

Digital Library

[54]

Amir Vaxman, Marcel Campen, Olga Diamanti, David Bommes, Klaus Hildebrandt, Mirela Ben-Chen Technion, and Daniele Panozzo. 2017. Directional Field Synthesis, Design, and Processing. In ACM SIGGRAPH Courses. Article 12, 30 pages.

[55]

Eugene L. Wachspress. 1975. A Rational Finite Element Basis. Mathematics in Science and Engineering, Vol. 114. Academic Press.

[56]

Max Wardetzky. 2006. Discrete Differential Operators on Polyhedral Surfaces: Convergence and Approximation. Ph.D. Dissertation. Freie Universit at Berlin.

[57]

Yong-Liang Yang, Yu-Kun Lai, Shi-Min Hu, and Helmut Pottmann. 2006. Robust Principal Curvatures on Multiple Scales. In Symposium on Geometry Processing. 223--226.

[58]

Yizhou Yu, Kun Zhou, Dong Xu, Xiaohan Shi, Hujun Bao, Baining Guo, and Heung-Yeung Shum. 2004. Mesh Editing with Poisson-Based Gradient Field Manipulation. In ACM Conference on Computer Graphics and Interactive Techniques (SIGGRAPH). 644--651.

[59]

Eugene Zhang, Konstantin Mischaikow, and Greg Turk. 2006. Vector Field Design on Surfaces. ACM Transactions on Graphics 25, 4 (2006), 1294--1326.

Digital Library

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Wu JLi HLi BZheng XZhang D(2024)Optimization of robotic path planning and navigation point configuration based on convolutional neural networksFrontiers in Neurorobotics10.3389/fnbot.2024.140665818Online publication date: 4-Jun-2024
https://doi.org/10.3389/fnbot.2024.1406658
de Goes FDesbrun M(2024)Stochastic Computation of Barycentric CoordinatesACM Transactions on Graphics10.1145/365813143:4(1-13)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658131
He ZZhuo PLin HDai J(2024)3D shape descriptor design based on HKS and persistent homology with stability analysisComputer Aided Geometric Design10.1016/j.cagd.2024.102326111:COnline publication date: 18-Jul-2024
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Index Terms

Discrete differential operators on polygonal meshes
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Mesh models
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Discretization

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Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 39, Issue 4

August 2020

1732 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3386569

Editor:
Szymon Rusinkiewicz
Princeton University

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Copyright © 2020 Owner/Author.

Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the Owner/Author.

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Published: 12 August 2020

Published in TOG Volume 39, Issue 4

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