Cited By
View all- Chen JSchaefer FDesbrun M(2024)Lightning-fast Method of Fundamental SolutionsACM Transactions on Graphics10.1145/365819943:4(1-16)Online publication date: 19-Jul-2024
An Adaptive Fast-Multipole-Accelerated Hybrid Boundary Integral Equation Method for Accurate Diffusion Curves
Article No.: 215, Pages 1 - 28
Abstract
In theory, diffusion curves promise complex color gradations for infinite-resolution vector graphics. In practice, existing realizations suffer from poor scaling, discretization artifacts, or insufficient support for rich boundary conditions. Previous applications of the boundary element method to diffusion curves have relied on polygonal approximations, which either forfeit the high-order smoothness of Bézier curves, or, when the polygonal approximation is extremely detailed, result in large and costly systems of equations that must be solved. In this paper, we utilize the boundary integral equation method to accurately and efficiently solve the underlying partial differential equation. Given a desired resolution and viewport, we then interpolate this solution and use the boundary element method to render it. We couple this hybrid approach with the fast multipole method on a non-uniform quadtree for efficient computation. Furthermore, we introduce an adaptive strategy to enable truly scalable infinite-resolution diffusion curves.
Supplementary Material
supplemental
- Download
- 82.12 MB
References
[1]
Ludvig af Klinteberg and Alex H Barnett. 2021. Accurate quadrature of nearly singular line integrals in two and three dimensions by singularity swapping. BIT Numerical Mathematics 61, 1 (2021), 83--118.
[2]
Gavin Barill, Neil G Dickson, Ryan Schmidt, David IW Levin, and Alec Jacobson. 2018. Fast winding numbers for soups and clouds. ACM Transactions on Graphics (TOG) 37, 4 (2018), 43.
[3]
Josh Barnes and Piet Hut. 1986. A hierarchical O (N log N) force-calculation algorithm. nature 324, 6096 (1986), 446--449.
[4]
Hedlena Bezerra, Elmar Eisemann, Doug DeCarlo, and Joëlle Thollot. 2010. Diffusion constraints for vector graphics. In Proceedings of the 8th international symposium on non-photorealistic animation and rendering. 35--42.
[5]
John C Bowers, Jonathan Leahey, and Rui Wang. 2011. A ray tracing approach to diffusion curves. In Computer Graphics Forum, Vol. 30. Wiley Online Library, 1345--1352.
[6]
Simon Boyé, Pascal Barla, and Gael Guennebaud. 2012. A vectorial solver for free-form vector gradients. ACM Transactions on Graphics (TOG) 31, 6 (2012), 1--9.
[7]
Fang Da, David Hahn, Christopher Batty, Chris Wojtan, and Eitan Grinspun. 2016. Surface-only liquids. ACM Transactions on Graphics (TOG) 35, 4 (2016), 1--12.
[8]
Mark Finch, John Snyder, and Hugues Hoppe. 2011. Freeform vector graphics with controlled thin-plate splines. ACM Transactions on Graphics (TOG) 30, 6 (2011), 1--10.
[9]
Abinand Gopal and Lloyd N Trefethen. 2019a. New Laplace and Helmholtz solvers. Proceedings of the National Academy of Sciences (2019).
[10]
Abinand Gopal and Lloyd N. Trefethen. 2019b. New Laplace and Helmholtz solvers. Proceedings of the National Academy of Sciences 116, 21 (may 2019), 10223--10225.
[11]
Leslie Greengard and Zydrunas Gimbutas. 2022. fmm2d. https://fmm2d.readthedocs.io/en/latest/
[12]
Leslie Greengard, Denis Gueyffier, Per-Gunnar Martinsson, and Vladimir Rokhlin. 2009. Fast direct solvers for integral equations in complex three-dimensional domains. Acta Numerica 18 (2009), 243--275.
[13]
Leslie Greengard and Vladimir Rokhlin. 1987. A fast algorithm for particle simulations. Journal of computational physics 73, 2 (1987), 325--348.
[14]
Johan Helsing and Rikard Ojala. 2008. On the evaluation of layer potentials close to their sources. J. Comput. Phys. 227, 5 (2008), 2899--2921.
[15]
Douglas R Hofstadter. 1979. Gödel, Escher.
[16]
Fei Hou, Qian Sun, Zheng Fang, Yong-Jin Liu, Shi-Min Hu, Hong Qin, Aimin Hao, and Ying He. 2020. Poisson Vector Graphics (PVG). IEEE Transactions on Visualization and Computer Graphics 26, 2 (2020), 1361--1371.
[17]
Yixin Hu, Teseo Schneider, Xifeng Gao, Qingnan Zhou, Alec Jacobson, Denis Zorin, and Daniele Panozzo. 2019. TriWild: robust triangulation with curve constraints. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1--15.
[18]
Libo Huang, Torsten Hädrich, and Dominik L Michels. 2019. On the accurate large-scale simulation of ferrofluids. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1--15.
[19]
Peter Ilbery, Luke Kendall, Cyril Concolato, and Michael McCosker. 2013. Biharmonic diffusion curve images from boundary elements. ACM Transactions on Graphics (TOG) 32, 6 (2013), 1--12.
[20]
Alec Jacobson et al. 2021. gptoolbox: Geometry Processing Toolbox. http://github.com/alecjacobson/gptoolbox.
[21]
Alec Jacobson, Daniele Panozzo, et al. 2018. libigl: A simple C++ geometry processing library. https://libigl.github.io/.
[22]
Alec Jacobson, Tino Weinkauf, and Olga Sorkine. 2012. Smooth Shape-Aware Functions with Controlled Extrema. Comput. Graph. Forum 31, 5 (2012), 1577--1586.
[23]
Doug L James and Dinesh K Pai. 1999. Artdefo: accurate real time deformable objects. In Proceedings of the 26th annual conference on Computer graphics and interactive techniques. 65--72.
[24]
Stefan Jeschke. 2016. Generalized Diffusion Curves: An Improved Vector Representation for Smooth-Shaded Images. In Computer Graphics Forum, Vol. 35. Wiley Online Library, 71--79.
[25]
Stefan Jeschke, David Cline, and Peter Wonka. 2009. A GPU Laplacian solver for diffusion curves and Poisson image editing. In ACM SIGGRAPH Asia 2009 papers. 1--8.
[26]
Stefan Jeschke, David Cline, and Peter Wonka. 2011. Estimating color and texture parameters for vector graphics. In Computer Graphics Forum, Vol. 30. Wiley Online Library, 523--532.
[27]
Todd Keeler and Robert Bridson. 2014. Ocean waves animation using boundary integral equations and explicit mesh tracking. In ACM SIGGRAPH 2014 Posters. 1--1.
[28]
P Kolm and V Rokhlin. 2001. Numerical quadratures for singular and hypersingular integrals. Computers & Mathematics with Applications 41, 3--4 (2001), 327--352.
[29]
Yijun Liu. 2009. Fast multipole boundary element method: theory and applications in engineering. Cambridge university press.
[30]
Manish Mandad and Marcel Campen. 2020. Bézier guarding: precise higher-order meshing of curved 2D domains. ACM Trans. Graph. 39, 4 (2020), 103.
[31]
Per-Gunnar Martinsson. 2019. Fast direct solvers for elliptic PDEs. SIAM.
[32]
Bailey Miller, Rohan Sawhney, Keenan Crane, and Ioannis Gkioulekas. 2023. Boundary Value Caching for Walk on Spheres. arXiv preprint arXiv:2302.11825 (2023).
[33]
Alexandrina Orzan, Adrien Bousseau, Holger Winnemöller, Pascal Barla, Joëlle Thollot, and David Salesin. 2008. Diffusion curves: a vector representation for smooth-shaded images. ACM Transactions on Graphics (TOG) 27, 3 (2008), 1--8.
[34]
Wai-Man Pang, Jing Qin, Michael Cohen, Pheng-Ann Heng, and Kup-Sze Choi. 2011. Fast rendering of diffusion curves with triangles. IEEE Computer Graphics and Applications 32, 4 (2011), 68--78.
[35]
Susanne Pfalzner and Paul Gibbon. 1997. Many-body tree methods in physics.
[36]
Romain Prévost, Wojciech Jarosz, and Olga Sorkine-Hornung. 2015. A vectorial framework for ray traced diffusion curves. In Computer Graphics Forum, Vol. 34. Wiley Online Library, 253--264.
[37]
Ante Qu and Doug L James. 2021. Fast linking numbers for topology verification of loopy structures. ACM Trans. Graph. 40 (2021), 106.
[38]
Rohan Sawhney and Keenan Crane. 2020. Monte Carlo geometry processing: A grid-free approach to PDE-based methods on volumetric domains. ACM Transactions on Graphics 39, 4 (2020).
[39]
Rohan Sawhney, Bailey Miller, Ioannis Gkioulekas, and Keenan Crane. 2023. Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions. arXiv preprint arXiv:2302.11815 (2023).
[40]
Rohan Sawhney, Dario Seyb, Wojciech Jarosz, and Keenan Crane. 2022. Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficients. arXiv preprint arXiv:2201.13240 (2022).
[41]
Teseo Schneider, Yixin Hu, Jérémie Dumas, Xifeng Gao, Daniele Panozzo, and Denis Zorin. 2018. Decoupling simulation accuracy from mesh quality. ACM Trans. Graph. 37, 6 (2018), 280:1--280:14.
[42]
Camille Schreck, Christian Hafner, and Chris Wojtan. 2019. Fundamental solutions for water wave animation. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1--14.
[43]
Jonathan Richard Shewchuk. 2005. Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator. In Applied Computational Geometry Towards Geometric Engineering: FCRC'96 Workshop, WACG'96 Philadelphia, PA, May 27--28, 1996 Selected Papers. Springer, 203--222.
[44]
Timothy Sun, Papoj Thamjaroenporn, and Changxi Zheng. 2014. Fast multipole representation of diffusion curves and points. ACM Trans. Graph. 33, 4 (2014), 53--1.
[45]
Xin Sun, Guofu Xie, Yue Dong, Stephen Lin, Weiwei Xu, Wencheng Wang, Xin Tong, and Baining Guo. 2012. Diffusion curve textures for resolution independent texture mapping. ACM Transactions on Graphics (TOG) 31, 4 (2012), 1--9.
[46]
Kenshi Takayama, Olga Sorkine, Andrew Nealen, and Takeo Igarashi. 2010. Volumetric modeling with diffusion surfaces. In ACM SIGGRAPH Asia 2010 papers. 1--8.
[47]
JJ van de Gronde. 2010. A high quality solver for diffusion curves. Ph.D. Dissertation. Faculty of Science and Engineering.
[48]
Tian Xia, Binbin Liao, and Yizhou Yu. 2009. Patch-based image vectorization with automatic curvilinear feature alignment. ACM Transactions on Graphics (TOG) 28, 5 (2009), 1--10.
[49]
Guofu Xie, Xin Sun, Xin Tong, and Derek Nowrouzezahrai. 2014. Hierarchical diffusion curves for accurate automatic image vectorization. ACM Transactions on Graphics (TOG) 33, 6 (2014), 1--11.
[50]
Chris Yu, Henrik Schumacher, and Keenan Crane. 2021. Repulsive curves. ACM Transactions on Graphics (TOG) 40, 2 (2021), 1--21.
[51]
Xinxin Zhang and Robert Bridson. 2014. A PPPM Fast Summation Method for Fluids and Beyond. ACM Trans. Graph. 33, 6 (2014).
[52]
Shuang Zhao, Frédo Durand, and Changxi Zheng. 2017. Inverse diffusion curves using shape optimization. IEEE Transactions on Visualization and Computer Graphics 24, 7 (2017), 2153--2166.
Index Terms
- An Adaptive Fast-Multipole-Accelerated Hybrid Boundary Integral Equation Method for Accurate Diffusion Curves
Recommendations
An interpolation-based fast-multipole accelerated boundary integral equation method for the three-dimensional wave equation
A new fast multipole method (FMM) is proposed to accelerate the time-domain boundary integral equation method (TDBIEM) for the three-dimensional wave equation. The proposed algorithm is an enhancement of the interpolation-based FMM for the time-domain ...
A Hybrid Boundary Element and Boundary Integral Equation Method for Accurate Diffusion Curves
SA '22: SIGGRAPH Asia 2022 Technical CommunicationsIn theory, diffusion curves promise complex color gradations for infinite-resolution vector graphics. In practice, existing realizations suffer from poor scaling, discretization artifacts, or insufficient support for rich boundary conditions. In this ...
Fast multipole accelerated singular boundary method for the 3D Helmholtz equation in low frequency regime
This paper presents a fast multipole accelerated singular boundary method (SBM) to the solution of the large-scale three-dimensional Helmholtz equation at low frequency. By using a desingularization strategy to directly compute singular kernels in the ...
Comments
Information & Contributors
Information
Published In
Copyright © 2023 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: 05 December 2023
Published in TOG Volume 42, Issue 6
Check for updates
Author Tags
Qualifiers
- Research-article
Funding Sources
- Swiss National Science Foundation's Early Postdoc.Mobility fellowship
- NSERC Discovery Grants
- National Research Foundation, Korea
- Canada Research Chairs Program
- Sloan Research Fellowship
- Ontario Early Research Award program
- NSERC Discovery Grant
- DSI Catalyst Grant program
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- View Citations1Total Citations
- 80Total Downloads
- Downloads (Last 12 months)80
- Downloads (Last 6 weeks)12
Reflects downloads up to 03 Sep 2024
Other Metrics
Citations
Cited By
View all- Chen JSchaefer FDesbrun M(2024)Lightning-fast Method of Fundamental SolutionsACM Transactions on Graphics10.1145/365819943:4(1-16)Online publication date: 19-Jul-2024
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.
Sign in