Smooth Bijective Projection in a High-order Shell
Article No.: 59, Pages 1 - 13
Abstract
We propose a new structure called a higher-order shell, which is composed of a set of triangular prisms. Each triangular prism is enveloped by three Bézier triangles (top, middle, and bottom) and three side surfaces, each of which is trimmed from a bilinear surface. Moreover, we define a continuous vector field to smoothly and bijectively transfer attributes between two surfaces inside the shell. Since the higher-order shell has several hard construction constraints, we apply an interior-point strategy to robustly and automatically construct a high-order shell for an input mesh. Specifically, the strategy starts from a valid linear shell with a small thickness. Then, the shell is optimized until the specified thickness is reached, where explicit checks ensure that the constraints are always satisfied. We extensively test our method on more than 8300 models, demonstrating its robustness and performance. Compared to state-of-the-art methods, our bijective projection is smoother, and the space between the shell and input mesh is more uniform.
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References
[1]
Noam Aigerman, Shahar Z Kovalsky, and Yaron Lipman. 2017. Spherical orbifold tu e embeddings. ACM Trans. Graph. 36, 4 (2017), 90.
[2]
R Aubry, S Dey, EL Mestreau, and BK Karamete. 2017. Boundary layer mesh generation on arbitrary geometries. Internat. J. Numer. Methods Engrg. 112, 2 (2017), 157--173.
[3]
Stéphane Calderon and Tamy Boubekeur. 2017. Bounding proxies for shape approximation. ACM Trans. Graph. 36, 4 (2017), 57--1.
[4]
Yanyun Chen, Xin Tong, Jiaping Wang, Stephen Lin, Baining Guo, and Heung-Yeung Shum. 2004. Shell texture functions. ACM Trans. Graph. 23, 3 (2004), 343--353.
[5]
Xiao-Xiang Cheng, Xiao-Ming Fu, Chi Zhang, and Shuangming Chai. 2019. Practical error-bounded remeshing by adaptive refinement. Computers & Graphics 82 (2019), 163--173.
[6]
Philippe G. Ciarlet. 1991. Basic error estimates for elliptic problems. Handbook of Numerical Analysis 2 (1991), 17--351.
[7]
Jonathan Cohen, Dinesh Manocha, and Marc Olano. 1997. Simplifying polygonal models using successive mappings. In Proceedings. Visualization'97 (Cat. No. 97CB36155). IEEE, 395--402.
[8]
Keenan Crane, Clarisse Weischedel, and Max Wardetzky. 2012. Geodesics in heat: A new approach to computing distance based on heat flow. ArXiv abs/1204.6216 (2012).
[9]
Swagatam Das and Ponnuthurai Nagaratnam Suganthan. 2011. Differential Evolution: A Survey of the State-of-the-Art. IEEE Transactions on Evolutionary Computation 15, 1 (2011), 4--31.
[10]
Saikat Dey, Robert M. O'Bara, and Mark S. Shephard. 1999. Curvilinear mesh generation in 3D. In Proceedings of the 8th International Meshing Roundtable. 407--417.
[11]
Danielle Ezuz, Justin Solomon, and Mirela Ben-Chen. 2019. Reversible harmonic maps between discrete surfaces. ACM Trans. Graph. 38, 2 (2019), 1--12.
[12]
Gerald Farin. 1986. Triangular Bernstein-Bézier patches. Comput. Aided Geom. Des. 3, 2 (1986), 83--127.
[13]
Zachary Ferguson, Pranav Jain, Denis Zorin, Teseo Schneider, and Daniele Panozzo. 2023. High-Order Incremental Potential Contact for Elastodynamic Simulation on Curved Meshes. ACM SIGGRAPH 2023 Conference Proceedings (2023).
[14]
Michael S Floater. 1997. Parametrization and smooth approximation of surface triangulations. Computer aided geometric design 14, 3 (1997), 231--250.
[15]
Rao V Garimella and Mark S Shephard. 2000. Boundary layer mesh generation for viscous flow simulations. Internat. J. Numer. Methods Engrg. 49, 1--2 (2000), 193--218.
[16]
Michael Garland and Paul S Heckbert. 1997. Surface simplification using quadric error metrics. In Proceedings of the 24th annual conference on Computer graphics and interactive techniques. 209--216.
[17]
Kai Hormann, Bruno Lévy, and Alla Sheffer. 2007. Mesh parameterization: Theory and practice. (2007).
[18]
Kai Hormann and N Sukumar. 2017. Generalized barycentric coordinates in computer graphics and computational mechanics. CRC press.
[19]
Yixin Hu, Teseo Schneider, Xifeng Gao, Qingnan Zhou, Alec Jacobson, Denis Zorin, and Daniele Panozzo. 2019. TriWild: robust triangulation with curve constraints. ACM Trans. Graph. 38, 4 (2019).
[20]
Yixin Hu, Teseo Schneider, Bolun Wang, Denis Zorin, and Daniele Panozzo. 2020. Fast tetrahedral meshing in the wild. ACM Trans. Graph. 39, 4 (2020), 117--1.
[21]
Zhongshi Jiang, Scott Schaefer, and Daniele Panozzo. 2017. Simplicial complex augmentation framework for bijective maps. ACM Trans. Graph. 36, 6 (2017).
[22]
Zhongshi Jiang, Teseo Schneider, Denis Zorin, and Daniele Panozzo. 2020. Bijective projection in a shell. ACM Trans. Graph. 39 (2020), 1--18.
[23]
Zhongshi Jiang, Ziyi Zhang, Yixin Hu, Teseo Schneider, Denis Zorin, and Daniele Panozzo. 2021. Bijective and coarse high-order tetrahedral meshes. ACM Trans. Graph. (2021).
[24]
Xiangmin Jiao and Michael T. Heath. 2004. Overlaying Surface Meshes, Part I: Algorithms. International Journal of Computational Geometry and Applications 14, 379--402.
[25]
Miao Jin, Junho Kim, Feng Luo, and Xianfeng Gu. 2008. Discrete surface Ricci flow. IEEE Transactions on Visualization and Computer Graphics 14, 5 (2008), 1030--1043.
[26]
Yao Jin, Dan Song, Tongtong Wang, Jin Huang, Ying Song, and Lili He. 2019. A shell space constrained approach for curve design on surface meshes. Computer-Aided Design 113 (2019), 24--34.
[27]
Payam Khanteimouri and Marcel Campen. 2023. 3D Bézier Guarding: Boundary-Conforming Curved Tetrahedral Meshing. ACM Trans. Graph. 42, 6 (2023).
[28]
Liliya Kharevych, Boris Springborn, and Peter Schröder. 2006. Discrete conformal mappings via circle patterns. ACM Trans. Graph. 25, 2 (2006), 412--438.
[29]
Peter Knabner and Gerhard Summ. 2001. The invertibility of the isoparametric mapping for pyramidal and prismatic finite elements. Numer. Math. 88 (2001), 661--681.
[30]
Felix Knöppel, Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2013. Globally optimal direction fields. ACM Trans. Graph. 32, 4 (2013).
[31]
Leif Kobbelt, Swen Campagna, Jens Vorsatz, and Hans-Peter Seidel. 1998. Interactive multi-resolution modeling on arbitrary meshes. Proceedings of the 25th annual conference on Computer graphics and interactive techniques (1998).
[32]
Sebastian Koch, Albert Matveev, Zhongshi Jiang, Francis Williams, Alexey Artemov, Evgeny Burnaev, Marc Alexa, Denis Zorin, and Daniele Panozzo. 2018. ABC: A Big CAD Model Dataset for Geometric Deep Learning. 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2018), 9593--9603.
[33]
Vladislav Kraevoy and Alla Sheffer. 2004. Cross-parameterization and compatible remeshing of 3D models. ACM Trans. Graph. (2004), 861--869.
[34]
Aaron Lee, Henry Moreton, and Hugues Hoppe. 2000. Displaced subdivision surfaces. In Proceedings of the 27th annual conference on Computer graphics and interactive techniques. 85--94.
[35]
Aaron W. F. Lee, Wim Sweldens, Peter Schröder, Lawrence C. Cowsar, and David P. Dobkin. 1998. MAPS: multiresolution adaptive parameterization of surfaces. Proceedings of the 25th annual conference on Computer graphics and interactive techniques (1998).
[36]
Jerome Lengyel, Emil Praun, Adam Finkelstein, and Hugues Hoppe. 2001. Real-time fur over arbitrary surfaces. In Proceedings of the 2001 symposium on Interactive 3D graphics. 227--232.
[37]
Hsueh-Ti Derek Liu, Mark Gillespie, Benjamin Chislett, Nicholas Sharp, Alec Jacobson, and Keenan Crane. 2023. Surface Simplification using Intrinsic Error Metrics. ACM Trans. Graph. 42 (2023), 1--17.
[38]
Hsueh-Ti Derek Liu, Vladimir G. Kim, Siddhartha Chaudhuri, Noam Aigerman, and Alec Jacobson. 2020. Neural subdivision. ACM Trans. Graph. 39 (2020), 124:1--124:16.
[39]
Hsueh-Ti Derek Liu, Jiayi Eris Zhang, Mirela Ben-Chen, and Alec Jacobson. 2021b. Surface multigrid via intrinsic prolongation. ACM Trans. Graph. 40 (2021), 1--13.
[40]
Zhong-Yuan Liu, Jian-Ping Su, Hao Liu, Chunyang Ye, Ligang Liu, and Xiao-Ming Fu. 2021a. Error-bounded Edge-based Remeshing of High-order Tetrahedral Meshes. Comput. Aided Des. 139 (2021), 103080.
[41]
Andrea Maggiordomo, Yury Uralsky, Henry P. Moreton, and Marco Tarini. 2023. The inverse barycentric displacement problem. The Visual Computer (2023).
[42]
Manish Mandad and Marcel Campen. 2020. Bézier guarding: precise higher-order meshing of curved 2D domains. ACM Trans. Graph. 39, 4 (2020), 103:1--103:15.
[43]
Joseph SB Mitchell, David M Mount, and Christos H Papadimitriou. 1987. The discrete geodesic problem. SIAM J. Comput. 16, 4 (1987), 647--668.
[44]
Ferrante Neri and Ville Tirronen. 2010. Recent advances in differential evolution: a survey and experimental analysis. Artif. Intell. Rev. 33 (2010), 61--106.
[45]
Maks Ovsjanikov, Mirela Ben-Chen, Justin Solomon, Adrian Butscher, and Leonidas Guibas. 2012. Functional maps: a flexible representation of maps between shapes. ACM Trans. Graph. (2012), 1--11.
[46]
Daniele Panozzo, Ilya Baran, Olga Diamanti, and Olga Sorkine-Hornung. 2013. Weighted averages on surfaces. ACM Trans. Graph. 32, 4 (2013), 1--12.
[47]
Jianbo Peng, Daniel Kristjansson, and Denis Zorin. 2004. Interactive modeling of topologically complex geometric detail. In ACM SIGGRAPH 2004 Papers. 635--643.
[48]
Nico Pietroni, Marco Tarini, and Paolo Cignoni. 2009. Almost isometric mesh parameterization through abstract domains. IEEE Transactions on Visualization and Computer Graphics 16, 4 (2009), 621--635.
[49]
Serban D Porumbescu, Brian Budge, Louis Feng, and Kenneth I Joy. 2005. Shell maps. ACM Trans. Graph. 24, 3 (2005), 626--633.
[50]
Emil Praun, Wim Sweldens, and Peter Schröder. 2001. Consistent mesh parameterizations. In Proceedings of the 28th annual conference on Computer graphics and interactive techniques. 179--184.
[51]
Eloi Ruiz-Gironés, Josep Sarrate, and Xevi Roca. 2021. Measuring and improving the geometric accuracy of piece-wise polynomial boundary meshes. J. Comput. Phys. (oct 2021), 22 pages.
[52]
Johnathan Richard Shewchuk. 1996. Robust Adaptive Floating-Point Geometric Predicates. Association for Computing Machinery, 141--150.
[53]
Jonathan Richard Shewchuk. 1997. Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates. Discrete & Computational Geometry 18, 3 (1997), 305--363.
[54]
Meged Shoham, Amir Vaxman, and Mirela Ben-Chen. 2019. Hierarchical functional maps between subdivision surfaces. In Computer Graphics Forum, Vol. 38. Wiley Online Library, 55--73.
[55]
Vitaly Surazhsky and Craig Gotsman. 2001. Morphing stick figures using optimized compatible triangulations. In Proceedings Ninth Pacific Conference on Computer Graphics and Applications. Pacific Graphics 2001. IEEE, 40--49.
[56]
Jean-Marc Thiery, Julien Tierny, and Tamy Boubekeur. 2012. CageR: cage-based reverse engineering of animated 3D shapes. In Computer Graphics Forum, Vol. 31. Wiley Online Library, 2303--2316.
[57]
William Thomas Tutte. 1963. How to draw a graph. Proceedings of the London Mathematical Society 3, 1 (1963), 743--767.
[58]
Lifeng Wang, Xi Wang, Xin Tong, Stephen Lin, Shimin Hu, Baining Guo, and Heung-Yeung Shum. 2003. View-dependent displacement mapping. ACM Trans. Graph. 22, 3 (2003), 334--339.
[59]
Xi Wang, Xin Tong, Stephen Lin, Shimin Hu, Baining Guo, and Heung-Yeung Shum. 2004. Generalized displacement maps. In Eurographics Symposium on Rendering (EGSR). 227--233.
[60]
Jing Xu and Andrey N. Chernikov. 2014. Automatic curvilinear quality mesh generation driven by smooth boundary and guaranteed fidelity. Procedia Engineering 82 (2014), 200--212.
[61]
Wen-Xiang Zhang, Qi Wang, Jianzhi Guo, Shuangming Chai, Ligang Liu, and Xiao-Ming Fu. 2022. Constrained Remeshing Using Evolutionary Vertex Optimization. Comput. Graph. Forum 41 (2022).
[62]
Qingnan Zhou and Alec Jacobson. 2016. Thingi10K: A Dataset of 10,000 3D-Printing Models. ArXiv abs/1605.04797 (2016).
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Publication History
Published: 19 July 2024
Published in TOG Volume 43, Issue 4
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- National Key R&D Program of China
- National Natural Science Foundation of China
- Major Project of Science and Technology of Anhui Province
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