Ligang Liu
Abstract
We propose a novel approach, called Progressive Parameterizations, to compute foldover-free parameterizations with low isometric distortion on disk topology meshes. Instead of using the input mesh as a reference to define the objective function, we introduce a progressive reference that contains bounded distortion to the parameterized mesh and is as close as possible to the input mesh. After optimizing the bounded distortion energy between the progressive reference and the parameterized mesh, the parameterized mesh easily approaches the progressive reference, thereby also coming close to the input. By iteratively generating the progressive reference and optimizing the bounded distortion energy to update the parameterized mesh, our algorithm achieves high-quality parameterizations with strong practical reliability and high efficiency. We have demonstrated that our algorithm succeeds on a massive test data set containing over 20712 complex disk topology meshes. Compared to the state-of-the-art methods, our method has achieved higher computational efficiency and practical reliability.
Supplemental Material
Available for Download
We propose a novel approach, called Progressive Parameterizations, to compute foldover-free parameterizations with low isometric distortion on disk topology meshes. Instead of using the input mesh as a reference to define the objective function, we introduce a progressive reference that contains bounded distortion to the parameterized mesh and is as close as possible to the input mesh. After optimizing the bounded distortion energy between the progressive reference and the parameterized mesh, the parameterized mesh easily approaches the progressive reference, thereby also coming close to the input. By iteratively generating the progressive reference and optimizing the bounded distortion energy to update the parameterized mesh, our algorithm achieves high-quality parameterizations with strong practical reliability and high efficiency.The code is also available via GitHub at https://github.com/ChunyangYe/PP
Supplemental files.
- Noam Aigerman, Shahar Z. Kovalsky, and Yaron Lipman. 2017. Spherical Orbifold Tutte Embeddings. ACM Trans. Graph. (SIGGRAPH) 36, 4 (2017), 90:1--90:13. Google Scholar
Digital Library
- Noam Aigerman and Yaron Lipman. 2015. Orbifold Tutte Embeddings. ACM Trans. Graph. (SIGGRAPH ASIA) 34, 6 (2015), 190:1--190:12. Google ScholarDigital Library
- Noam Aigerman and Yaron Lipman. 2016. Hyperbolic Orbifold Tutte Embeddings. ACM Trans. Graph. (SIGGRAPH ASIA) 35, 6 (2016), 190:1--190:12. Google ScholarDigital Library
- Noam Aigerman, Roi Poranne, and Yaron Lipman. 2014. Lifted Bijections for Low Distortion Surface Mappings. ACM Trans. Graph. (SIGGRAPH) 33, 4 (2014), 69:1--69:12. Google ScholarDigital Library
- Marc Alexa. 2002. Linear Combination of Transformations. ACM Trans. Graph. 21, 3 (2002), 380--387. Google ScholarDigital Library
- Marc Alexa, Daniel Cohen-Or, and David Levin. 2000. As-Rigid-As-Possible Shape Interpolation. In Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH '00). 157--164. Google ScholarDigital Library
- David Bommes, Henrik Zimmer, and Leif Kobbelt. 2009. Mixed-Integer Quadrangulation. ACM Trans. Graph. (SIGGRAPH) 28, 3 (2009), 77:1--77:10. Google ScholarDigital Library
- Alon Bright, Edward Chien, and Ofir Weber. 2017. Harmonic Global Parametrization with Rational Holonomy. ACM Trans. Graph. (SIGGRAPH) 36, 4 (2017), 89:1--89:15. Google ScholarDigital Library
- Renjie Chen, Ofir Weber, Daniel Keren, and Mirela Ben-Chen. 2013. Planar Shape Interpolation with Bounded Distortion. ACM Trans. Graph. (SIGGRAPH) 32, 4 (2013), 108:1--108:12. Google ScholarDigital Library
- Edward Chien, Renjie Chen, and Ofir Weber. 2016a. Bounded Distortion Harmonic Shape Interpolation. ACM Trans. Graph. (SIGGRAPH) 35, 4 (2016), 105:1--105:15. Google ScholarDigital Library
- Edward Chien, Zohar Levi, and Ofir Weber. 2016b. Bounded Distortion Parametrization in the Space of Metrics. ACM Trans. Graph. (SIGGRAPH ASIA) 35, 6 (2016), 215:1--215:16. Google ScholarDigital Library
- S Claici, M Bessmeltsev, S Schaefer, and J Solomon. 2017. Isometry-Aware Preconditioning for Mesh Parameterization. Comput. Graph. Forum (SGP) 36, 5 (2017), 37--47. Google ScholarDigital Library
- P. Degener, J. Meseth, and R. Klein. 2003. An Adaptable Surface Parameterization Method. In Int. Meshing Roundtable. 201--213.Google Scholar
- Michael S. Floater. 2003. One-to-one Piecewise Linear Mappings over Triangulations. Math. Comput. 72 (2003), 685--696. Google ScholarDigital Library
- Michael S. Floater and Kai Hormann. 2005. Surface Parameterization: A Tutorial and Survey. In In Advances in Multiresolution for Geometric Modelling. Springer, 157--186.Google Scholar
- Xiao-Ming Fu and Yang Liu. 2016. Computing Inversion-Free Mappings by Simplex Assembly. ACM Trans. Graph. (SIGGRAPH ASIA) 35, 6 (2016). Google ScholarDigital Library
- Xiao-Ming Fu, Yang Liu, and Baining Guo. 2015. Computing Locally Injective Mappings by Advanced MIPS. ACM Trans. Graph. (SIGGRAPH) 34, 4 (2015), 71:1--71:12. Google ScholarDigital Library
- Xiao-Ming Fu, Yang Liu, John Snyder, and Baining Guo. 2014. Anisotropic Simplicial Meshing Using Local Convex Functions. ACM Trans. Graph. (SIGGRAPH ASIA) 33, 6 (2014), 182:1--182:11. Google ScholarDigital Library
- F Sebastian Grassia. 1998. Practical Parameterization of Rotations Using the Exponential Map. Journal of graphics tools 3, 3 (1998), 29--48. Google ScholarDigital Library
- Xianfeng Gu, Steven J. Gortler, and Hugues Hoppe. 2002. Geometry Images. ACM Trans. Graph. (SIGGRAPH) 21, 3 (2002), 355--361. Google ScholarDigital Library
- K. Hormann and G. Greiner. 2000. MIPS: An Efficient Global Parametrization Method. In Curve and Surface Design: Saint-Malo 1999. Vanderbilt University Press, 153--162.Google Scholar
- Kai Hormann, Bruno Lévy, and Alla Sheffer. 2007. Mesh Parameterization: Theory and Practice. In ACM SIGGRAPH 2007 Courses (SIGGRAPH '07). Google ScholarDigital Library
- Xin Hu, Xiao-Ming Fu, and Ligang Liu. 2018. Advanced Hierarchical Spherical Parameterizations. IEEE. T. Vis. Comput. Gr. 24, 6 (2018), 1930--1941.Google ScholarCross Ref
- Zhongshi Jiang, Scott Schaefer, and Daniele Panozzo. 2017. Simplicial Complex Augmentation Framework for Bijective Maps. ACM Trans. Graph. (SIGGRAPH ASIA) 36, 6 (2017), 186:1--186:9. Google ScholarDigital Library
- Shahar Z. Kovalsky, Noam Aigerman, Ronen Basri, and Yaron Lipman. 2015. Large-scale Bounded Distortion Mappings. ACM Trans. Graph. (SIGGRAPH ASIA) 34, 6 (2015), 191:1--191:10. Google ScholarDigital Library
- Shahar Z. Kovalsky, Meirav Galun, and Yaron Lipman. 2016. Accelerated Quadratic Proxy for Geometric Optimization. ACM Trans. Graph. (SIGGRAPH) 35, 4 (2016), 134:1--134:11. Google ScholarDigital Library
- Bruno Lévy, Sylvain Petitjean, Nicolas Ray, and Jérome Maillot. 2002. Least Squares Conformal Maps for Automatic Texture Atlas Generation. ACM Trans. Graph. (SIGGRAPH) 21, 3 (2002), 362--371. Google ScholarDigital Library
- Yaron Lipman. 2012. Bounded Distortion Mapping Spaces for Triangular Meshes. ACM Trans. Graph. (SIGGRAPH) 31, 4 (2012), 108:1--108:13. Google ScholarDigital Library
- Ligang Liu, Lei Zhang, Yin Xu, Craig Gotsman, and Steven J. Gortler. 2008. A Local/Global Approach to Mesh Parameterization. Comput. Graph. Forum (SGP) 27, 5 (2008), 1495--1504. Google ScholarDigital Library
- Cosmin G. Petra, Olaf Schenk, and Mihai Anitescu. 2014a. Real-time Stochastic Optimization of Complex Energy Systems on High-performance Computers. IEEE Computing in Science & Engineering 16, 5 (2014), 32--42.Google ScholarCross Ref
- Cosmin G. Petra, Olaf Schenk, Miles Lubin, and Klaus Gartner. 2014b. An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization. SIAM Journal on Scientific Computing 36, 2 (2014), C139--C162.Google ScholarCross Ref
- Roi Poranne, Marco Tarini, Sandro Huber, Daniele Panozzo, and Olga Sorkine-Hornung. 2017. Autocuts: Simultaneous Distortion and Cut Optimization for UV Mapping. ACM Trans. Graph. (SIGGRAPH ASIA) 36, 6 (2017), 215:1--215:11. Google ScholarDigital Library
- Michael Rabinovich, Roi Poranne, Daniele Panozzo, and Olga Sorkine-Hornung. 2017. Scalable Locally Injective Maps. ACM Trans. Graph. 36, 2 (2017). Google ScholarDigital Library
- Jarek Rossignac and Álvar Vinacua. 2011. Steady Affine Motions and Morphs. ACM Trans. Graph. 30, 5 (2011), 116. Google ScholarDigital Library
- Olaf Schenk, Andreas Wächter, and Michael Hagemann. 2007. Mtching-based Preprocessing Algorithms to the Solution of Saddle-point Problems in Large-scale Nonconvex Interior-point Optimization. Computational Optimization and Applications 36, 2 (2007), 321--341. Google ScholarDigital Library
- Christian Schüller, Ladislav Kavan, Daniele Panozzo, and Olga Sorkine-Hornung. 2013. Locally Injective Mappings. Comput. Graph. Forum (SGP) 32, 5 (2013), 125--135. Google ScholarDigital Library
- Alla Sheffer. 2002. Spanning Tree Seams for Reducing Parameterization Distortion of Triangulated Surfaces. In Shape Modeling International. 61--66. Google ScholarDigital Library
- Alla Sheffer and John C Hart. 2002. Seamster: Inconspicuous Low-distortion Texture Seam Layout. In Proceedings of the conference on Visualization'02. 291--298. Google ScholarDigital Library
- Alla Sheffer, Bruno Levy, Maxim Mogilnitsky, and Alexander Bogomyakov. 2005. ABF++: Fast and Robust Angle Based Flattening. ACM Trans. Graph. 24, 2 (2005), 311--330. Google ScholarDigital Library
- Alla Sheffer, Emil Praun, and Kenneth Rose. 2006. Mesh Parameterization Methods and Their Applications. Found. Trends. Comput. Graph. Vis. 2, 2 (2006), 105--171. Google ScholarDigital Library
- Anna Shtengel, Roi Poranne, Olga Sorkine-Hornung, Shahar Z. Kovalsky, and Yaron Lipman. 2017. Geometric Optimization via Composite Majorization. ACM Trans. Graph. (SIGGRAPH) 36, 4 (2017), 38:1--38:11. Google ScholarDigital Library
- Jason Smith and Scott Schaefer. 2015. Bijective Parameterization with Free Boundaries. ACM Trans. Graph. (SIGGRAPH) 34, 4 (2015), 70:1--70:9. Google ScholarDigital Library
- Olga Sorkine and Marc Alexa. 2007. As-Rigid-As-Possible Surface Modeling. In Symposium on Geometry Processing. 109--116. Google ScholarDigital Library
- Olga Sorkine, Daniel Cohen-Or, Rony Goldenthal, and Dani Lischinski. 2002. Bounded-distortion Piecewise Mesh Parameterization. In Proceedings of the Conference on Visualization '02. 355--362. Google ScholarDigital Library
- W. T. Tutte. 1963. How to Draw A Graph. In Proceedings of the London Mathematical Society, Vol. 13. 747--767.Google Scholar
- Ofir Weber and Craig Gotsman. 2010. Controllable Conformal Maps for Shape Deformation and Interpolation. ACM Trans. Graph. (SIGGRAPH) 29, 4 (2010), 78:1--78:11. Google ScholarDigital Library
- Ofir Weber and Denis Zorin. 2014. Locally Injective Parametrization with Arbitrary Fixed Boundaries. ACM Trans. Graph. (SIGGRAPH) 33, 4 (2014), 75:1--75:12. Google ScholarDigital Library
- Kun Zhou, John Synder, Baining Guo, and Heung-Yeung Shum. 2004. Iso-charts: Stretch-driven Mesh Parameterization Using Spectral Analysis. In Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing. 45--54. Google ScholarDigital Library
Index Terms
Progressive parameterizations
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