Steffen Hinderink
Marcel Campen
Skip Abstract Section
Skip Supplemental Material SectionAbstract
A method is presented to compute volumetric maps and parametrizations of objects over 3D domains. As a key feature, continuity and bijectivity are ensured by construction. Arbitrary objects of ball topology, represented as tetrahedral meshes, are supported. Arbitrary convex as well as star-shaped domains are supported. Full control over the boundary mapping is provided. The method is based on the technique of simplicial foliations, generalized to a broader class of domain shapes and applied adaptively in a novel localized manner. This increases flexibility as well as efficiency over the state of the art, while maintaining reliability in guaranteeing map bijectivity.
Supplemental Material
- S. Mazdak Abulnaga, Oded Stein, Polina Golland, and Justin Solomon. 2023. Symmetric Volume Maps: Order-invariant Volumetric Mesh Correspondence with Free Boundary. ACM Trans. Graph. 42, 3 (2023), 25:1--25:20.Google Scholar
Digital Library
- Noam Aigerman and Yaron Lipman. 2013. Injective and Bounded Distortion Mappings in 3D. ACM Trans. Graph. 32, 4 (2013), 106:1--106:14.Google ScholarDigital Library
- Marc Alexa. 2023. Tutte Embeddings of Tetrahedral Meshes. Discrete & Computational Geometry (2023).Google Scholar
- Marc Alexa, Daniel Cohen-Or, and David Levin. 2000. As-Rigid-As-Possible Shape Interpolation. In Proc. SIGGRAPH 2000. 157--164.Google ScholarDigital Library
- Stephen Boyd and Lieven Vandenberghe. 2004. Convex Optimization. Cambridge University Press.Google ScholarDigital Library
- Hendrik Brückler, David Bommes, and Marcel Campen. 2022a. Volume Parametrization Quantization for Hexahedral Meshing. ACM Trans. Graph. 41, 4 (2022), 60:1--60:19.Google ScholarDigital Library
- Hendrik Brückler, Ojaswi Gupta, Manish Mandad, and Marcel Campen. 2022b. The 3D Motorcycle Complex for Structured Volume Decomposition. Computer Graphics Forum 41, 2 (2022), 221--235.Google ScholarCross Ref
- Marcel Campen, Ryan Capouellez, Hanxiao Shen, Leyi Zhu, Daniele Panozzo, and Denis Zorin. 2021. Efficient and Robust Discrete Conformal Equivalence with Boundary. ACM Trans. Graph. 40, 6 (2021), 261:1--261:16.Google ScholarDigital Library
- Marcel Campen, Cláudio T. Silva, and Denis Zorin. 2016. Bijective Maps from Simplicial Foliations. ACM Trans. Graph. 35, 4 (2016), 74:1--74:15.Google ScholarDigital Library
- David Cohen and Mirela Ben-Chen. 2019. Generalized volumetric foliation from inverted viscous flow. Computers & Graphics 82 (2019), 152--162.Google ScholarDigital Library
- Xingyi Du, Noam Aigerman, Qingnan Zhou, Shahar Z. Kovalsky, Yajie Yan, Danny M. Kaufman, and Tao Ju. 2020. Lifting Simplices to Find Injectivity. ACM Trans. Graph. 39, 4 (2020), 120:1--120:17.Google ScholarDigital Library
- Xingyi Du, Danny M. Kaufman, Qingnan Zhou, Shahar Z. Kovalsky, Yajie Yan, Noam Aigerman, and Tao Ju. 2022. Isometric Energies for Recovering Injectivity in Constrained Mapping. In Proc. SIGGRAPH Asia 2022. 36:1--36:9.Google ScholarDigital Library
- J. M. Escobar, E. Rodrıguez, R. Montenegro, G. Montero, and J. M. González-Yuste. 2003. Simultaneous untangling and smoothing of tetrahedral meshes. Comput. Methods Appl. Mech. Eng. 192, 25 (2003), 2775--2787.Google ScholarCross Ref
- Yu Fang, Minchen Li, Chenfanfu Jiang, and Danny M. Kaufman. 2021. Guaranteed Globally Injective 3D Deformation Processing. ACM Trans. Graph. 40, 4 (2021), 75:1--75:13.Google ScholarDigital Library
- Michael S. Floater. 1997. Parametrization and smooth approximation of surface triangulations. Computer Aided Geometric Design 14, 3 (1997), 231--250.Google ScholarDigital Library
- Michael S. Floater and Valérie Pham-Trong. 2006. Convex combination maps over triangulations, tilings, and tetrahedral meshes. Advances in Computational Mathematics 25, 4 (2006), 347--356.Google ScholarCross Ref
- Xiao-Ming Fu, Yang Liu, and Baining Guo. 2015. Computing Locally Injective Mappings by Advanced MIPS. ACM Trans. Graph. 34, 4 (2015), 71:1--71:12.Google ScholarDigital Library
- Xiao-Ming Fu, Jian-Ping Su, Zheng-Yu Zhao, Qing Fang, Chunyang Ye, and Ligang Liu. 2021. Inversion-free geometric mapping construction: A survey. Computational Visual Media 7, 3 (2021), 289--318.Google ScholarCross Ref
- Robert Furch. 1924. Zur Grundlegung der kombinatorischen Topologie. Abhandlungen aus dem mathematischen Seminar der Universität Hamburg 3, 1 (1924), 69--88.Google Scholar
- Vladimir Garanzha, Igor Kaporin, Liudmila Kudryavtseva, François Protais, Nicolas Ray, and Dmitry Sokolov. 2021. Foldover-free maps in 50 lines of code. ACM Trans. Graph. 40, 4 (2021), 102:1--102:16.Google ScholarDigital Library
- Mark Gillespie, Boris Springborn, and Keenan Crane. 2021. Discrete Conformal Equivalence of Polyhedral Surfaces. ACM Trans. Graph. 40, 4 (2021), 103:1--103:20.Google ScholarDigital Library
- Yixin Hu, Qingnan Zhou, Xifeng Gao, Alec Jacobson, Denis Zorin, and Daniele Panozzo. 2018. Tetrahedral Meshing in the Wild. ACM Trans. Graph. 37, 4 (2018), 60:1--60:14.Google ScholarDigital Library
- Lisa Huynh and Yotam Gingold. 2015. Bijective Deformations in ℝn via Integral Curve Coordinates. arXiv:1505.00073Google Scholar
- S. S. Iyengar, Xin Li, Huanhuan Xu, Supratik Mukhopadhyay, N. Balakrishnan, Amit Sawant, and Puneeth Iyengar. 2012. Toward More Precise Radiotherapy Treatment of Lung Tumors. Computer 45, 1 (2012), 59--65.Google ScholarDigital Library
- Zhongshi Jiang, Scott Schaefer, and Daniele Panozzo. 2017. Simplicial Complex Augmentation Framework for Bijective Maps. ACM Trans. Graph. 36, 6 (2017), 186:1--186:9.Google ScholarDigital Library
- Bert Jüttler, Sofia Maroscheck, Myung-Soo Kim, and Q Youn Hong. 2019. Arc fibrations of planar domains. Computer Aided Geometric Design 71 (2019), 105--118.Google ScholarDigital Library
- Shahar Z. Kovalsky, Noam Aigerman, Ronen Basri, and Yaron Lipman. 2014. Controlling Singular Values with Semidefinite Programming. ACM Trans. Graph. 33, 4 (2014), 68:1--68:13.Google ScholarDigital Library
- Xin Li, Xiaohu Guo, Hongyu Wang, Ying He, Xianfeng Gu, and Hong Qin. 2007. Harmonic Volumetric Mapping for Solid Modeling Applications. In Proc. SPM 2007. 109--120.Google ScholarDigital Library
- Juncong Lin, Jiazhi Xia, Xing Gao, Minghong Liao, Ying He, and Xianfeng Gu. 2015. Interior structure transfer via harmonic 1-forms. Multimedia Tools and Applications 74, 1 (2015), 139--158.Google ScholarDigital Library
- Manish Mandad, Ruizhi Chen, David Bommes, and Marcel Campen. 2022. Intrinsic mixed-integer polycubes for hexahedral meshing. Computer Aided Geometric Design 94 (2022), 102078.Google ScholarCross Ref
- Tobias Martin, Guoning Chen, Suraj Musuvathy, Elaine Cohen, and Charles Hansen. 2012. Generalized Swept Mid-structure for Polygonal Models. Computer Graphics Forum 31, 4 (2012), 805--814.Google ScholarDigital Library
- Tobias Martin, Elaine Cohen, and Robert M. Kirby. 2008. Volumetric Parameterization and Trivariate B-spline Fitting using Harmonic Functions. In Proc. SPM 2008. 269--280.Google ScholarDigital Library
- I. Moerdijk and J. Mrčun. 2003. Introduction to Foliations and Lie Groupoids. Cambridge University Press.Google Scholar
- Alexander Naitsat, Yufeng Zhu, and Yehoshua Y. Zeevi. 2020. Adaptive Block Coordinate Descent for Distortion Optimization. Computer Graphics Forum 39, 6 (2020), 360--376.Google ScholarCross Ref
- M. Nieser, U. Reitebuch, and K. Polthier. 2011. CubeCover - Parameterization of 3D Volumes. Computer Graphics Forum 30, 5 (2011), 1397--1406.Google ScholarCross Ref
- Valentin Z. Nigolian, Marcel Campen, and David Bommes. 2023. Expansion Cones: A Progressive Volumetric Mapping Framework. ACM Trans. Graph. 42, 4 (2023).Google ScholarDigital Library
- Matthew Overby, Danny Kaufman, and Rahul Narain. 2021. Globally Injective Geometry Optimization with Non-Injective Steps. Computer Graphics Forum 40, 5 (2021), 111--123.Google ScholarCross Ref
- Nico Pietroni, Marcel Campen, Alla Sheffer, Gianmarco Cherchi, David Bommes, Xifeng Gao, Riccardo Scateni, Franck Ledoux, Jean Remacle, and Marco Livesu. 2022. Hex-Mesh Generation and Processing: A Survey. ACM Trans. Graph. 42, 2 (2022), 16:1--16:44.Google Scholar
- Roman Poya, Rogelio Ortigosa, and Theodore Kim. 2023. Geometric Optimisation Via Spectral Shifting. ACM Trans. Graph. 42, 3 (2023), 29:1--29:15.Google ScholarDigital Library
- Michael Rabinovich, Roi Poranne, Daniele Panozzo, and Olga Sorkine-Hornung. 2017. Scalable Locally Injective Mappings. ACM Trans. Graph. 36, 4 (2017), 16:1--16:16.Google ScholarDigital Library
- Patrick Schmidt, Janis Born, Marcel Campen, and Leif Kobbelt. 2019. Distortion-Minimizing Injective Maps Between Surfaces. ACM Trans. Graph. 38, 6 (2019), 156:1--156:15.Google ScholarDigital Library
- Christian Schüller, Ladislav Kavan, Daniele Panozzo, and Olga Sorkine-Hornung. 2013. Locally Injective Mappings. Computer Graphics Forum 32, 5 (2013), 125--135.Google ScholarDigital Library
- Hang Si. 2015. TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator. ACM Trans. Math. Softw. 41, 2 (2015), 11:1--11:36.Google ScholarDigital Library
- Breannan Smith, Fernando De Goes, and Theodore Kim. 2019. Analytic Eigensystems for Isotropic Distortion Energies. ACM Trans. Graph. 38, 1 (2019), 3:1--3:15.Google ScholarDigital Library
- Jason Smith and Scott Schaefer. 2015. Bijective Parameterization with Free Boundaries. ACM Trans. Graph. 34, 4 (2015), 70:1--70:9.Google ScholarDigital Library
- Tommaso Sorgente, Silvia Biasotti, and Michela Spagnuolo. 2022. Polyhedron kernel computation using a geometric approach. Computers & Graphics 105 (2022), 94--104.Google ScholarDigital Library
- Jian-Ping Su, Xiao-Ming Fu, and Ligang Liu. 2019. Practical Foldover-Free Volumetric Mapping Construction. Computer Graphics Forum 38, 7 (2019), 287--297.Google ScholarCross Ref
- Jian-Ping Su, Chunyang Ye, Ligang Liu, and Xiao-Ming Fu. 2020. Efficient Bijective Parameterizations. ACM Trans. Graph. 39, 4 (2020), 111:1--111:8.Google ScholarDigital Library
- Thomas Toulorge, Christophe Geuzaine, Jean-François Remacle, and Jonathan Lambrechts. 2013. Robust untangling of curvilinear meshes. J. Comput. Phys. 254 (2013), 8--26.Google ScholarDigital Library
- Sofia Trautner, Bert Jüttler, and Myung-Soo Kim. 2021. Representing planar domains by polar parameterizations with parabolic parameter lines. Computer Aided Geometric Design 85 (2021), 101966.Google ScholarDigital Library
- W. T. Tutte. 1963. How to draw a graph. Proc. Lond. Math. Soc. 13 (1963), 743--767.Google ScholarCross Ref
- Yalin Wang, Xianfeng Gu, Tony F. Chan, Paul M. Thompson, and Shing-Tung Yau. 2004. Volumetric harmonic brain mapping. In Proc. ISBI 2004. 1275--1278.Google Scholar
- Ofir Weber and Denis Zorin. 2014. Locally Injective Parametrization with Arbitrary Fixed Boundaries. ACM Trans. Graph. 33, 4 (2014), 75:1--75:12.Google ScholarDigital Library
- Jiazhi Xia, Ying He, Shuchu Han, Chi-Wing Fu, Feng Luo, and Xianfeng Gu. 2010. Parameterization of Star-Shaped Volumes Using Green's Functions. In Proc. GMP 2010. 219--235.Google ScholarDigital Library
- Yongjie Zhang, Wenyan Wang, and Thomas J. R. Hughes. 2012. Solid T-spline construction from boundary representations for genus-zero geometry. Comput. Methods Appl. Mech. Eng. 249--252 (2012), 185--197.Google Scholar
- Qingnan Zhou and Alec Jacobson. 2016. Thingi10K: A Dataset of 10,000 3D-Printing Models. arXiv:1605.04797Google Scholar
Index Terms
- Galaxy Maps: Localized Foliations for Bijective Volumetric Mapping
Recommendations
Expansion Cones: A Progressive Volumetric Mapping Framework
Volumetric mapping is a ubiquitous and difficult problem in Geometry Processing and has been the subject of research in numerous and various directions. While several methods show encouraging results, the field still lacks a general approach with ...
Bijective maps from simplicial foliations
This paper presents a method for bijective parametrization of 2D and 3D objects over canonical domains. While a range of solutions for the two-dimensional case are well-known, our method guarantees bijectivity of mappings also for a large, ...
Constructing hexahedral shell meshes via volumetric polycube maps
Shells are three-dimensional structures. One dimension, the thickness, is much smaller than the other two dimensions. Shell structures can be widely found in many real-world objects. This paper presents a method to construct a layered hexahedral mesh ...
Comments