research-article

Integrable PolyVector fields

Authors:

Daniele Panozzo,

Olga Sorkine-HornungAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 34, Issue 4

Article No.: 38, Pages 1 - 12

https://doi.org/10.1145/2766906

Published: 27 July 2015 Publication History

Abstract

We present a framework for designing curl-free tangent vector fields on discrete surfaces. Such vector fields are gradients of locally-defined scalar functions, and this property is beneficial for creating surface parameterizations, since the gradients of the parameterization coordinate functions are then exactly aligned with the designed fields. We introduce a novel definition for discrete curl between unordered sets of vectors (PolyVectors), and devise a curl-eliminating continuous optimization that is independent of the matchings between them. Our algorithm naturally places the singularities required to satisfy the user-provided alignment constraints, and our fields are the gradients of an inversion-free parameterization by design.

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References

[1]

Alliez, P., Cohen-Steiner, D., Devillers, O., Lévy, B., and Desbrun, M. 2003. Anisotropic polygonal remeshing. ACM Trans. Graph. 22, 3, 485--493.

Digital Library

[2]

Azencot, O., Ben-Chen, M., Chazal, F., and Ovsjanikov, M. 2013. An operator approach to tangent vector field processing. Comput. Graph. Forum 32, 5, 73--82.

Digital Library

[3]

Bommes, D., Zimmer, H., and Kobbelt, L. 2009. Mixed-integer quadrangulation. ACM Trans. Graph. 28, 3, 77:1--77:10.

Digital Library

[4]

Bommes, D., Campen, M., Ebke, H.-C., Alliez, P., and Kobbelt, L. 2013. Integer-grid maps for reliable quad meshing. ACM Trans. Graph. 32, 4, 98:1--98:12.

Digital Library

[5]

Bommes, D., Lévy, B., Pietroni, N., Puppo, E., Silva, C., Tarini, M., and Zorin, D. 2013. Quad-mesh generation and processing: A survey. Computer Graphics Forum 32, 6, 51--76.

Digital Library

[6]

Botsch, M., Kobbelt, L., Pauly, M., Alliez, P., and Lévy, B. 2010. Polygon Mesh Processing. AK Peters.

[7]

Campen, M., and Kobbelt, L. 2014. Dual strip weaving: Interactive design of quad layouts using elastica strips. ACM Trans. Graph. 33, 6, 183:1--183:10.

Digital Library

[8]

Crane, K., Desbrun, M., and Schröder, P. 2010. Trivial connections on discrete surfaces. Comput. Graph. Forum 29, 5.

[9]

Diamanti, O., Vaxman, A., Panozzo, D., and Sorkine-Hornung, O. 2014. Designing N-PolyVector fields with complex polynomials. Computer Graphics Forum 33, 5, 1--11.

Digital Library

[10]

Dong, S., Bremer, P.-T., Garland, M., Pascucci, V., and Hart, J. C. 2006. Spectral surface quadrangulation. ACM Trans. Graph. 25, 3 (July), 1057--1066.

Digital Library

[11]

Ebke, H.-C., Bommes, D., Campen, M., and Kobbelt, L. 2013. QEx: Robust quad mesh extraction. ACM Trans. Graph. 32, 6, 168:1--168:10.

Digital Library

[12]

Ebke, H.-C., Campen, M., Bommes, D., and Kobbelt, L. 2014. Level-of-detail quad meshing. ACM Trans. Graph. 33, 6, 184:1--184:11.

Digital Library

[13]

Fisher, M., Schröder, P., Desbrun, M., and Hoppe, H. 2007. Design of tangent vector fields. ACM Trans. Graph. 26, 3.

Digital Library

[14]

Hertzmann, A., and Zorin, D. 2000. Illustrating smooth surfaces. In Proc. ACM SIGGRAPH, 517--526.

Digital Library

[15]

Kälberer, F., Nieser, M., and Polthier, K. 2007. QuadCover -- surface parameterization using branched coverings. Computer Graphics Forum 26, 3, 375--384.

[16]

Knöppel, F., Crane, K., Pinkall, U., and Schröder, P. 2013. Globally optimal direction fields. ACM Trans. Graph. 32, 4.

Digital Library

[17]

Kuzmin, A., Luisier, M., and Schenk, O. 2013. Fast methods for computing selected elements of the Green's function in massively parallel nanoelectronic device simulations. In Proc. Euro-Par, 533--544.

Digital Library

[18]

Lefebvre, S., and Hoppe, H. 2006. Appearance-space texture synthesis. ACM Trans. Graph. 25, 3, 541--548.

Digital Library

[19]

Li, Y., Bao, F., Zhang, E., Kobayashi, Y., and Wonka, P. 2011. Geometry synthesis on surfaces using field-guided shape grammars. IEEE Trans. Vis. Comput. Graph. 17, 2, 231--243.

Digital Library

[20]

Ling, R., Huang, J., Jüttler, B., Sun, F., Bao, H., and Wang, W. 2014. Spectral quadrangulation with feature curve alignment and element size control. ACM Trans. Graph. 34, 1.

Digital Library

[21]

Lipman, Y. 2012. Bounded distortion mapping spaces for triangular meshes. ACM Trans. Graph. 31, 4, 108:1--108:13.

Digital Library

[22]

Liu, Y., Xu, W., Wang, J., Zhu, L., Guo, B., Chen, F., and Wang, G. 2011. General planar quadrilateral mesh design using conjugate direction field. ACM Trans. Graph. 30, 6.

Digital Library

[23]

Marinov, M., and Kobbelt, L. 2004. Direct anisotropic quad-dominant remeshing. In Proc. Pacific Graphics, 207--216.

Digital Library

[24]

Myles, A., and Zorin, D. 2012. Global parametrization by incremental flattening. ACM Trans. Graph. 31, 4.

Digital Library

[25]

Myles, A., and Zorin, D. 2013. Controlled-distortion constrained global parametrization. ACM Trans. Graph. 32, 4.

Digital Library

[26]

Myles, A., Pietroni, N., and Zorin, D. 2014. Robust field-aligned global parametrization. ACM Trans. Graph. 33, 4.

Digital Library

[27]

Palacios, J., and Zhang, E. 2007. Rotational symmetry field design on surfaces. ACM Trans. Graph. 26, 3.

Digital Library

[28]

Panozzo, D., Lipman, Y., Puppo, E., and Zorin, D. 2012. Fields on symmetric surfaces. ACM Trans. Graph. 31, 4.

Digital Library

[29]

Panozzo, D., Puppo, E., Tarini, M., and Sorkine-Hornung, O. 2014. Frame fields: Anisotropic and non-orthogonal cross fields. ACM Trans. Graph. 33, 4, 134:1--134:11.

Digital Library

[30]

Polthier, K., and Preuss, E. 2003. Identifying vector field singularities using a discrete Hodge decomposition. In Visualization and Mathematics III, 113--134.

[31]

Ray, N., Li, W. C., Lévy, B., Sheffer, A., and Alliez, P. 2006. Periodic global parameterization. ACM Trans. Graph. 25, 4, 1460--1485.

Digital Library

[32]

Ray, N., Vallet, B., Li, W. C., and Lévy, B. 2008. N-symmetry direction field design. ACM Trans. Graph. 27, 2.

Digital Library

[33]

Ray, N., Vallet, B., Alonso, L., and Lévy, B. 2009. Geometry-aware direction field processing. ACM Trans. Graph. 29, 1, 1:1--1:11.

Digital Library

[34]

Schüller, C., Kavan, L., Panozzo, D., and Sorkine-Hornung, O. 2013. Locally injective mappings. Computer Graphics Forum 32, 5, 125--135.

Digital Library

[35]

Takayama, K., Panozzo, D., Sorkine-Hornung, A., and Sorkine-Hornung, O. 2013. Sketch-based generation and editing of quad meshes. ACM Trans. Graph. 32, 4, 97:1--97:8.

Digital Library

[36]

Weber, O., and Zorin, D. 2014. Locally injective parametrization with arbitrary fixed boundaries. ACM Trans. Graph. 33, 4 (July), 75:1--75:12.

Digital Library

[37]

Zhang, E., Mischaikow, K., and Turk, G. 2006. Vector field design on surfaces. ACM Trans. Graph. 25, 4, 1294--1326.

Digital Library

[38]

Zhang, M., Huang, J., Liu, X., and Bao, H. 2010. A wave-based anisotropic quadrangulation method. ACM Trans. Graph. 29, 4, 118:1--118:8.

Digital Library

Cited By

Mitropoulou IVaxman ADiamanti ODillenburger B(2024)Fabrication-aware strip-decomposable quadrilateral meshesComputer-Aided Design10.1016/j.cad.2023.103666168:COnline publication date: 1-Mar-2024
https://dl.acm.org/doi/10.1016/j.cad.2023.103666
Shepherd KHiemstra RGu XHughes T(2024)Extraction of surface quad layouts from quad layout immersions: application to an isogeometric model of car crashEngineering with Computers10.1007/s00366-024-02007-wOnline publication date: 14-Aug-2024
https://doi.org/10.1007/s00366-024-02007-w
Coiffier GCorman E(2023)The Method of Moving Frames for Surface Global ParametrizationACM Transactions on Graphics10.1145/360428242:5(1-18)Online publication date: 10-Jun-2023
https://dl.acm.org/doi/10.1145/3604282
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Index Terms

Integrable PolyVector fields
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Theory of computation
  1. Randomness, geometry and discrete structures

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Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 34, Issue 4

August 2015

1307 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/2809654

Issue’s Table of Contents

Copyright © 2015 ACM.

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Association for Computing Machinery

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Publication History

Published: 27 July 2015

Published in TOG Volume 34, Issue 4

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Cited By

Mitropoulou IVaxman ADiamanti ODillenburger B(2024)Fabrication-aware strip-decomposable quadrilateral meshesComputer-Aided Design10.1016/j.cad.2023.103666168:COnline publication date: 1-Mar-2024
https://dl.acm.org/doi/10.1016/j.cad.2023.103666
Shepherd KHiemstra RGu XHughes T(2024)Extraction of surface quad layouts from quad layout immersions: application to an isogeometric model of car crashEngineering with Computers10.1007/s00366-024-02007-wOnline publication date: 14-Aug-2024
https://doi.org/10.1007/s00366-024-02007-w
Coiffier GCorman E(2023)The Method of Moving Frames for Surface Global ParametrizationACM Transactions on Graphics10.1145/360428242:5(1-18)Online publication date: 10-Jun-2023
https://dl.acm.org/doi/10.1145/3604282
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
Guţan OHegde SBerumen EBessmeltsev MChien E(2023)Singularity‐Free Frame Fields for Line Drawing VectorizationComputer Graphics Forum10.1111/cgf.1490142:5Online publication date: 10-Aug-2023
https://doi.org/10.1111/cgf.14901
Deng ZLiu YPan HJabi WZhang JDeng B(2023)Sketch2PQ: Freeform Planar Quadrilateral Mesh Design via a Single SketchIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2022.317085329:9(3826-3839)Online publication date: 1-Sep-2023
https://doi.org/10.1109/TVCG.2022.3170853
Shepherd KHiemstra RHughes T(2023)The quad layout immersion: A mathematically equivalent representation of a surface quadrilateral layoutComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2023.116445417(116445)Online publication date: Dec-2023
https://doi.org/10.1016/j.cma.2023.116445
Zhu TXu ZLiu LFu X(2023)Modeling with discrete equivalence classes of planar quadsComputers & Graphics10.1016/j.cag.2023.07.032115(404-411)Online publication date: Oct-2023
https://doi.org/10.1016/j.cag.2023.07.032
Noma YUmetani NKawahara Y(2022)Fast Editing of Singularities in Field-Aligned Stripe PatternsSIGGRAPH Asia 2022 Conference Papers10.1145/3550469.3555387(1-8)Online publication date: 29-Nov-2022
https://dl.acm.org/doi/10.1145/3550469.3555387
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
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