“Bounded distortion parametrization in the space of metrics” by Chien, Levi and Weber
Conference:
Type(s):
Title:
- Bounded distortion parametrization in the space of metrics
Session/Category Title: Parameterization & Remeshing
Presenter(s)/Author(s):
Abstract:
We present a framework for global parametrization that utilizes the edge lengths (squared) of the mesh as variables. Given a mesh with arbitrary topology and prescribed cone singularities, we flatten the original metric of the surface under strict bounds on the metric distortion (various types of conformal and isometric measures are supported). Our key observation is that the space of bounded distortion metrics (given any particular bounds) is convex, and a broad range of useful and well-known distortion energies are convex as well. With the addition of nonlinear Gaussian curvature constraints, the parametrization problem is formulated as a constrained optimization problem, and a solution gives a locally injective map. Our method is easy to implement. Sequential convex programming (SCP) is utilized to solve this problem effectively. We demonstrate the flexibility of the method and its uncompromised robustness and compare it to state-of-the-art methods.
References:
1. Aigerman, N., and Lipman, Y. 2013. Injective and bounded distortion mappings in 3D. ACM Transactions on Graphics (TOG) 32, 4, 106.
2. Aigerman, N., Poranne, R., and Lipman, Y. 2014. Lifted bijections for low distortion surface mappings. ACM Transactions on Graphics (TOG) 33, 4, 69.
3. ApS, M. 2015. The MOSEK optimization toolbox for MATLAB manual. Version 7.1 (Revision 28).
4. Ben-Chen, M., Gotsman, C., and Bunin, G. 2008. Conformal flattening by curvature prescription and metric scaling. Computer Graphics Forum 27, 2, 449–458. Cross Ref
5. Bobenko, A. I., Pinkall, U., and Springborn, B. A. 2015. Discrete conformal maps and ideal hyperbolic polyhedra. Geometry & Topology 19, 4, 2155–2215. Cross Ref
6. Bommes, D., Zimmer, H., and Kobbelt, L. 2009. Mixed-integer quadrangulation. ACM Trans. Graph. 28, 3, 77.
7. Bommes, D., Campen, M., Ebke, H.-C., Alliez, P., and Kobbelt, L. 2013. Integer-grid maps for reliable quad meshing. ACM Transactions on Graphics (TOG) 32, 4, 98.
8. 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.
9. Botsch, M., Kobbelt, L., Pauly, M., Alliez, P., and Lévy, B. 2010. Polygon mesh processing. CRC press.
10. Boyd, S., and Vandenberghe, L. 2004. Convex optimization. Cambridge university press.
11. Campen, M., Bommes, D., and Kobbelt, L. 2015. Quantized global parametrization. ACM Transactions on Graphics (TOG) 34, 6, 192.
12. Chen, R., and Weber, O. 2015. Bounded distortion harmonic mappings in the plane. ACM Transactions on Graphics (TOG) 34, 4, 73.
13. Chen, R., Weber, O., Keren, D., and Ben-Chen, M. 2013. Planar shape interpolation with bounded distortion. ACM Transactions on Graphics (TOG) 32, 4, 108.
14. Chien, E., Chen, R., and Weber, O. 2016. Bounded distortion harmonic shape interpolation. ACM TOG 35, 4.
15. Dinh, Q. T., and Diehl, M. 2010. Local convergence of sequential convex programming for nonconvex optimization. In Recent Advances in Optimization and its Applications in Engineering. Springer, 93–102.
16. Eppstein, D. 2005. Quasiconvex programming. Combinatorial and Computational Geometry 52, 287–331.
17. Floater, M., and Hormann, K. 2005. Surface Parameterization: a Tutorial and Survey. Advances In Multiresolution For Geometric Modelling.
18. Grant, M., Boyd, S., and Ye, Y., 2008. Cvx: Matlab software for disciplined convex programming.
19. Hormann, K., Lévy, B., and Sheffer, A. 2007. Mesh parameterization: Theory and practice. SIGGRAPH Course Notes.
20. Kälberer, F., Nieser, M., and Polthier, K. 2007. Quad-Cover: Surface Parameterization using Branched Coverings. Computer Graphics Forum 26, 3, 375–384. Cross Ref
21. Kharevych, L., Springborn, B., and Schröder, P. 2006. Discrete conformal mappings via circle patterns. ACM Trans. Graph. 25, 2, 412–438.
22. Kovalsky, S. Z., Aigerman, N., Basri, R., and Lipman, Y. 2014. Controlling singular values with semidefinite programming. ACM Transactions on Graphics (TOG) 33, 4, Article 68.
23. Kovalsky, S. Z., Aigerman, N., Basri, R., and Lipman, Y. 2015. Large-scale bounded distortion mappings. ACM Transactions on Graphics (TOG) 34, 6, 191.
24. Lai, Y.-K., Jin, M., Xie, X., He, Y., Palacios, J., Zhang, E., Hu, S.-M., and Gu, X. 2010. Metric-driven rosy field design and remeshing. Visualization and Computer Graphics, IEEE Transactions on 16, 1, 95–108.
25. Lee, J. M. 1997. Riemannian Manifolds: An Introduction to Curvature. Springer. Cross Ref
26. Levi, Z., and Zorin, D. 2014. Strict minimizers for geometric optimization. ACM Transactions on Graphics (TOG) 33, 6, 185.
27. Lévy, B., Petitjean, S., Ray, N., and Maillot, J. 2002. Least squares conformal maps for automatic texture atlas generation. ACM Transactions on Graphics (TOG) 21, 3, 362–371.
28. Lipman, Y. 2012. Bounded distortion mapping spaces for triangular meshes. ACM Transactions on Graphics (TOG) 31, 4, 108.
29. Liu, L., Zhang, L., Xu, Y., Gotsman, C., and Gortler, S. 2008. A local/global approach to mesh parameterization. Computer Graphics Forum 27, 5, 1495–1504.
30. Liu, T., Gao, M., Zhu, L., Sifakis, E., and Kavan, L. 2016. Fast and robust inversion-free shape manipulation. Computer Graphics Forum 35, 2. Cross Ref
31. Maillot, J., Yahia, H., and Verroust, A. 1993. Interactive texture mapping. Proceedings of SIGGRAPH 1993, 27–34.
32. Myles, A., and Zorin, D. 2012. Global parametrization by incremental flattening. ACM Trans. Graph. 31, 4 (July), 109:1–109:11.
33. Myles, A., and Zorin, D. 2013. Controlled-distortion constrained global parametrization. ACM Trans. Graph. 32, 4 (July), 105:1–105:14.
34. Myles, A., Pietroni, N., and Zorin, D. 2014. Robust field-aligned global parametrization. ACM Trans. Graph. 33, 4 (July), 135:1–135:14.
35. Poranne, R., and Lipman, Y. 2014. Provably good planar mappings. ACM Transactions on Graphics (TOG) 33, 4, 76.
36. Schreiner, J., Asirvatham, A., Praun, E., and Hoppe, H. 2004. Inter-surface mapping. ACM Transactions on Graphics (TOG) 23, 3, 870–877.
37. Schüller, C., Kavan, L., Panozzo, D., and Sorkine-Hornung, O. 2013. Locally injective mappings. In Computer Graphics Forum, vol. 32, Wiley Online Library, 125–135.
38. Sheffer, A., and de Sturler, E. 2001. Parameterization of faceted surfaces for meshing using angle-based flattening. Engineering with Computers 17, 3, 326–337. Cross Ref
39. Sorkine, O., Cohen-Or, D., Goldenthal, R., and Lischinski, D. 2002. Bounded-distortion piecewise mesh parameterization. In Proceedings of IEEE Visualization, IEEE Computer Society, 355–362.
40. Springborn, B., Schröder, P., and Pinkall, U. 2008. Conformal equivalence of triangle meshes. ACM Transactions on Graphics (TOG) 27, 3, 77.
41. Vaxman, A., Campen, M., Diamanti, O., Panozzo, D., Bommes, D., Hildebrandt, K., and Ben-Chen, M. 2016. Directional field synthesis, design, and processing – state of the art report. Computer Graphics Forum 35, 2. Cross Ref
42. Weber, O., and Zorin, D. 2014. Locally inective parametrization with arbitrary fixed boundaries. ACM Transactions on Graphics (TOG) 33, 4, 75.
43. Weber, O., Myles, A., and Zorin, D. 2012. Computing extremal quasiconformal maps. Computer Graphics Forum 31, 5, 1679–1689.
