research-article

Subdivision Directional Fields

Authors:

Amir VaxmanAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 39, Issue 2

Article No.: 11, Pages 1 - 20

https://doi.org/10.1145/3375659

Published: 07 February 2020 Publication History

Abstract

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the mixed finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure preserving: it reproduces curl-free fields precisely and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover’s distance computation, for efficient and robust computation.

References

[1]

Omri Azencot, Mirela Ben-Chen, Frédéric Chazal, and Maks Ovsjanikov. 2013. An operator approach to tangent vector field processing.Computer Graphics Forum 32, 5 (2013), 73--82.

[2]

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 (May 2015), Article 29.

Digital Library

[3]

Ivo Babuška and Manil Suri. 1994. The P and H-P versions of the finite element method, basic principles and properties. SIAM Review 36, 4 (Dec. 1994), 578--632.

Digital Library

[4]

Mirela Ben-Chen, Adrian Butscher, Justin Solomon, and Leonidas Guibas. 2010. On discrete killing vector fields and patterns on surfaces. Computer Graphics Forum 29, 5 (2010), 1701--1711.

[5]

M. Bertram. 2004. Biorthogonal loop-subdivision wavelets. Computing 72, 1--2 (April 2004), 29--39.

Digital Library

[6]

Henning Biermann, Adi Levin, and Denis Zorin. 2000. Piecewise smooth subdivision surfaces with normal control. In Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’00). ACM, New York, NY, 113--120.

Digital Library

[7]

David Bommes, Henrik Zimmer, and Leif Kobbelt. 2009. Mixed-integer quadrangulation. ACM Transactions on Graphics 28, 3 (2009), Article 77.

Digital Library

[8]

Alain Bossavit. 1998. Introduction: Maxwell equations. In Computational Electromagnetism, A. Bossavit (Ed.). Academic Press, San Diego, CA, 1--30.

[9]

Achi Brandt. 1977. Multi-level adaptive solutions to boundary-value problems. Mathematics of Computation 31, 138 (1977), 333--390.

[10]

Christopher Brandt, Leonardo Scandolo, Elmar Eisemann, and Klaus Hildebrandt. 2017. Spectral processing of tangential vector fields. Computer Graphics Forum 35, 6 (2016), 338--353.

Digital Library

[11]

Marcel Campen, David Bommes, and Leif Kobbelt. 2015. Quantized global parametrization. ACM Transactions on Graphics 34, 6 (2015), Article 192.

Digital Library

[12]

Fehmi Cirak, Michael J. Scott, Erik K. Antonsson, Michael Ortiz, and Peter Schröder. 2002. Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision. Computer-Aided Design 34, 2 (2002), 137--148.

[13]

Keenan Crane, Fernando de Goes, Mathieu Desbrun, and Peter Schröder. 2013. Digital geometry processing with discrete exterior calculus. In Proceedings of ACM SIGGRAPH 2013 Courses (SIGGRAPH’13).

Digital Library

[14]

Keenan Crane, Mathieu Desbrun, and Peter Schröder. 2010. Trivial connections on discrete surfaces. Computer Graphics Forum 29, 5 (2010), 1525--1533.

[15]

Michel Crouzeix and Pierre-Arnaud Raviart. 1973. Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. ESAIM: Mathematical Modelling and Numerical Analysis—Modélisation Mathématique et Analyse Numérique 7, R3 (1973), 33--75. http://eudml.org/doc/193250.

[16]

Wolfgang Dahmen. 1986. Subdivision algorithms converge quadratically. Journal of Computational and Applied Mathematics 16, 2 (1986), 145--158.

Digital Library

[17]

Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. 2008. Computational Geometry: Algorithms and Applications (3rd ed.). Springer-Verlag TELOS, Santa Clara, CA.

Digital Library

[18]

Fernando de Goes, Mathieu Desbrun, and Yiying Tong. 2016a. Vector field processing on triangle meshes. In Proceedings of ACM SIGGRAPH 2016 Courses (SIGGRAPH’16). ACM, New York, NY, Article 27, 49 pages.

Digital Library

[19]

Fernando de Goes, Mathieu Desbrun, Mark Meyer, and Tony DeRose. 2016b. Subdivision exterior calculus for geometry processing. ACM Transactions on Graphics 35, 4 (July 2016), Article 133, 11 pages.

Digital Library

[20]

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), 1--12.

Digital Library

[21]

M. Desbrun, A. N. Hirani, M. Leok, and J. E. Marsden. 2005. Discrete exterior calculus. arXiv:math/0508341.

[22]

Olga Diamanti, Amir Vaxman, Daniele Panozzo, and Olga Sorkine-Hornung. 2014. Designing -PolyVector fields with complex polynomials. Computer Graphics Forum 33, 5 (2014), 1--11.

Digital Library

[23]

Olga Diamanti, Amir Vaxman, Daniele Panozzo, and Olga Sorkine-Hornung. 2015. Integrable PolyVector fields. ACM Transactions on Graphics 34, 4 (2015), Article 38.

Digital Library

[24]

Michael Garland and Paul S. Heckbert. 1997. Surface simplification using quadric error metrics. In Proceedings of of the 24th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’97). 209--216.

[25]

Eitan Grinspun, Petr Krysl, and Peter Schröder. 2002. CHARMS: A simple framework for adaptive simulation. ACM Transactions on Graphics 21, 3 (July 2002), 281--290.

Digital Library

[26]

Aaron Hertzmann and Denis Zorin. 2000. Illustrating smooth surfaces. In Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’00). 517--526.

Digital Library

[27]

Anil N. Hirani. 2003. Discrete Exterior Calculus. Ph.D. Dissertation. California Institute of Technology.

[28]

Jinghao Huang and Peter Schröder. 2010. sqrt(3)-based 1-form subdivision. In Curves and Surfaces: Proceedings of the 7th International Conference, Revised Selected Papers. 351--368.

Digital Library

[29]

T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. 2005. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194, 39 (2005), 4135--4195.

[30]

Bert Jüttler, Angelos Mantzaflaris, Ricardo Perl, and Martin Rumpf. 2016. On numerical integration in isogeometric subdivision methods for PDEs on surfaces. Computer Methods in Applied Mechanics and Engineering 302 (2016), 131--146.

[31]

Felix Kälberer, Matthias Nieser, and Konrad Polthier. 2007. QuadCover—Surface parameterization using branched coverings. Computer Graphics Forum 26, 3 (2007), 375--384.

[32]

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

[33]

Bei-Bei 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

[34]

Songrun Liu, Alec Jacobson, and Yotam Gingold. 2014. Skinning cubic Bézier splines and Catmull-Clark subdivision surfaces. ACM Transactions on Graphics 33, 6 (Nov. 2014), Article 190, 9 pages.

Digital Library

[35]

Yang Liu, Helmut Pottmann, Johannes Wallner, Yong-Liang Yang, and Wenping Wang. 2006. Geometric modeling with conical meshes and developable surfaces. ACM Transactions on Graphics 25, 3 (2006), 1--9.

Digital Library

[36]

Charles Teorell Loop. 1987. Smooth Subdivision Surfaces Based on Triangles. Master’s Thesis. Department of Mathematics, University of Utah.

[37]

Michael Lounsbery, Tony D. DeRose, and Joe Warren. 1997. Multiresolution analysis for surfaces of arbitrary topological type. ACM Transactions on Graphics 16, 1 (Jan. 1997), 34--73.

Digital Library

[38]

P. Mullen, A. McKenzie, D. Pavlov, L. Durant, Y. Tong, E. Kanso, J. E. Marsden, and M. Desbrun. 2011. Discrete Lie advection of differential forms. Foundations of Computational Mathematics 11, 2 (2011).

[39]

Ashish Myles and Denis Zorin. 2012. Global parametrization by incremental flattening. ACM Transactions on Graphics 31, 4 (2012), 131--149.

Digital Library

[40]

Thien Nguyen, Keçstutis Karc̆iauskas, and Jörg Peters. 2014. A comparative study of several classical, discrete differential and isogeometric methods for solving Poisson’s equation on the disk. Axioms 3, 2 (2014), 280--299.

[41]

Konstantin Poelke and Konrad Polthier. 2016. Boundary-aware Hodge decompositions for piecewise constant vector fields. Computer-Aided Design 78, C (Sept. 2016), 126--136.

[42]

Konrad Polthier and Eike Preuß. 2003. Identifying vector field singularities using a discrete hodge decomposition. In Visualization and Mathematics III. Mathematics and Visualization. Springer, 113--134.

[43]

Hartmut Prautzsch, Wolfgang Boehm, and Marco Paluszny. 2002. Bezier and B-Spline Techniques. Springer-Verlag, Berlin, Germany.

[44]

Nicolas Ray, Bruno Vallet, Wan Chiu Li, and Bruno Lévy. 2008. N-symmetry direction field design. ACM Transactions on Graphics 27, 2 (2008), Article 10.

Digital Library

[45]

Lawrence Roy, Prashant Kumar, Sanaz Golbabaei, Yue Zhang, and Eugene Zhang. 2018. Interactive design and visualization of branched covering spaces. IEEE Transactions on Visualization and Computer Graphics 24, 1 (Jan. 2018), 843--852.

[46]

Andrew Sageman-Furnas, Albert Chern, Mirela Ben-Chen, and Amir Vaxman. 2019. Chebyshev nets from commuting PolyVector fields. ACM Transactions on Graphics 38, 6 (Nov. 2019), Article 172.

Digital Library

[47]

Marc Schober and Manfred Kasper. 2007. Comparison of hp-adaptive methods in finite element electromagnetic wave propagation. COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 26, 2 (2007), 431--446.

[48]

Nicholas Sharp, Yousuf Soliman, and Keenan Crane. 2019. The vector heat method. ACM Transactions on Graphics 38, 3 (2019), Article 24.

Digital Library

[49]

Meged Shoham, Amir Vaxman, and Mirela Ben-Chen. 2019. Hierarchical functional maps between subdivision surfaces. Computer Graphics Forum 38, 5 (2019), 55--73.

[50]

Justin Solomon, Raif Rustamov, Leonidas Guibas, and Adrian Butscher. 2014. Earth mover’s distances on discrete surfaces. ACM Transactions on Graphics 33, 4 (July 2014), Article 67, 12 pages.

Digital Library

[51]

Jos Stam. 2003. Flows on surfaces of arbitrary topology. ACM Transactions on Graphics 22, 3 (July 2003), 724--731.

Digital Library

[52]

Gilbert Strang and George Fix. 2008. An Analysis of the Finite Element Method. Wellesley-Cambridge Press.

[53]

Bernhard Thomaszewski, Markus Wacker, and Wolfgang Strasser. 2006. A consistent bending model for cloth simulation with corotational subdivision finite elements. In Proceedings of the 2006 ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA’06). 107--116. http://dl.acm.org/citation.cfm?id=1218064.1218079.

Digital Library

[54]

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

[55]

Amir Vaxman2019. Directional: A Library for Directional Field Synthesis, Design, and Processing. Retrieved January 6, 2020 from https://github.com/avaxman/Directional.

[56]

Amir Vaxman, Marcel Campen, Olga Diamanti, Daniele Panozzo, David Bommes, Klaus Hildebrandt, and Mirela Ben-Chen. 2016. Directional field synthesis, design, and processing. Computer Graphics Forum 35, 2 (2016), 545--572.

[57]

Ke Wang, Weiwei, Yiying Tong, Mathieu Desbrun, and Peter Schroder. 2006. Edge subdivision schemes and the construction of smooth vector fields. ACM Transactions on Graphics 25, 3 (2006), 1--8.

Digital Library

[58]

Max Wardetzky. 2006. Discrete Differential Operators on Polyhedral Surfaces--Convergence and Approximation. Ph.D. Dissertation. Freie Universität Berlin.

[59]

Mirko Zadravec, Alexander Schiftner, and Johannes Wallner. 2010. Designing quad-dominant meshes with planar faces. Computer Graphics Forum 29, 5 (2010), 1671--1679.

[60]

Eugene Zhang, Konstantin Mischaikow, and Greg Turk. 2006. Vector field design on surfaces. ACM Transactions on Graphics 25, 4 (2006), 1--33.

Digital Library

Cited By

Capouellez RZorin D(2023)Metric Optimization in Penner CoordinatesACM Transactions on Graphics10.1145/361839442:6(1-19)Online publication date: 5-Dec-2023
https://dl.acm.org/doi/10.1145/3618394
Chen ZKaufman DSkouras MVouga E(2023)Complex Wrinkle Field EvolutionACM Transactions on Graphics10.1145/359239742:4(1-19)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592397
Boksebeld IVaxman A(2022)High-Order Directional FieldsACM Transactions on Graphics10.1145/3550454.355545541:6(1-17)Online publication date: 30-Nov-2022
https://dl.acm.org/doi/10.1145/3550454.3555455
Show More Cited By

Index Terms

Subdivision Directional Fields
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling

Recommendations

General triangular midpoint subdivision

Smoothness of general triangular midpoint surfaces for arbitrary control meshes. C 1 analysis tools for infinite classes of triangular subdivision schemes.New spectral properties of subdivision matrices.Smoothness analysis of Loop subdivision surfaces. ...
Edge subdivision schemes and the construction of smooth vector fields
SIGGRAPH '06: ACM SIGGRAPH 2006 Papers

Vertex- and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete ...
Edge subdivision schemes and the construction of smooth vector fields

Vertex- and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 39, Issue 2

April 2020

136 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3381407

Editor:
Marc Alexa
TU Berlin, Germany

Issue’s Table of Contents

Copyright © 2020 ACM.

Permission to make digital or hard copies of all or part 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 components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 07 February 2020

Accepted: 01 November 2019

Revised: 01 October 2019

Received: 01 October 2018

Published in TOG Volume 39, Issue 2

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

5
Total Citations
View Citations
495
Total Downloads

Downloads (Last 12 months)28
Downloads (Last 6 weeks)1

Reflects downloads up to 10 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Capouellez RZorin D(2023)Metric Optimization in Penner CoordinatesACM Transactions on Graphics10.1145/361839442:6(1-19)Online publication date: 5-Dec-2023
https://dl.acm.org/doi/10.1145/3618394
Chen ZKaufman DSkouras MVouga E(2023)Complex Wrinkle Field EvolutionACM Transactions on Graphics10.1145/359239742:4(1-19)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592397
Boksebeld IVaxman A(2022)High-Order Directional FieldsACM Transactions on Graphics10.1145/3550454.355545541:6(1-17)Online publication date: 30-Nov-2022
https://dl.acm.org/doi/10.1145/3550454.3555455
Meekes MVaxman A(2021)Unconventional patterns on surfacesACM Transactions on Graphics10.1145/3450626.345993340:4(1-16)Online publication date: 19-Jul-2021
https://dl.acm.org/doi/10.1145/3450626.3459933
Soliman YChern ADiamanti OKnöppel FPinkall USchröder P(2021)Constrained willmore surfacesACM Transactions on Graphics10.1145/3450626.345975940:4(1-17)Online publication date: 19-Jul-2021
https://dl.acm.org/doi/10.1145/3450626.3459759

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

HTML Format

View this article in HTML Format.

Media

Figures

Other

Tables

View Issue’s Table of Contents