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AN OPTIMAL CONTROL APPROACH TO CONFORMAL FLATTENING OF TRIANGULATED SURFACES

  • PARK, YESOM (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY) ;
  • LEE, BYUNGJOON (DEPARTMENT OF MATHEMATICS, THE CATHOLIC UNIVERSITY OF KOREA) ;
  • MIN, CHOHONG (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY)
  • Received : 2019.09.10
  • Accepted : 2019.12.04
  • Published : 2019.12.25

Abstract

This article presents a new approach for conformal flattening with optimal cone singularity. The algorithm here takes an optimal control for selecting optimal cones and uses the Ricci flow to force the flattening. This work is considered as a modification to the work of Soliman et al. [1] in the sense that they make use of the Yamabe equation for the flattening, which is an approximation of the Ricci flow. We present a numerical algorithm based on the optimal control with the mathematical background. Several numerical results validate that our method is optimal in total cone angle and usage of the Ricci flow ensures the conformal flattening while selecting optimal cones.

Acknowledgement

Supported by : NRF, POSCO TJ Park Foundation

References

  1. Yousuf Soliman, Dejan Slepcev, and Keenan Crane. Optimal cone singularities for conformal flattening. ACM Transactions on Graphics (TOG), 37(4):105, 2018.
  2. Cindy Grimm and Denis Zorin. Surface modeling and parameterization with manifolds: Siggraph 2006 course notes. ACM, 2006.
  3. Rhaleb Zayer, Bruno Levy, and Hans-Peter Seidel. Linear angle based parameterization. 2007.
  4. Zhengyu Su, Wei Zeng, Rui Shi, Yalin Wang, Jian Sun, and Xianfeng Gu. Area preserving brain mapping. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2235-2242, 2013.
  5. Bennett Chow and Feng Luo. Combinatorial ricci flows on surfaces. Journal of Differential Geometry, 63(1):97-129, 2003. https://doi.org/10.4310/jdg/1080835659
  6. Miao Jin, Junho Kim, Feng Luo, and Xianfeng Gu. Discrete surface ricci flow. IEEE Transactions on Visualization and Computer Graphics, 14(5):1030-1043, 2008. https://doi.org/10.1109/TVCG.2008.57
  7. Liliya Kharevych, Boris Springborn, and Peter Schroder. Discrete conformal mappings via circle patterns. ACM Transactions on Graphics (TOG), 25(2):412-438, 2006. https://doi.org/10.1145/1138450.1138461
  8. Ashish Myles and Denis Zorin. Global parametrization by incremental flattening. ACM Transactions on Graphics (TOG), 31(4):109, 2012.
  9. RS Hamilton. The ricci flow on surfaces. mathematics and general relativity, 237-262. American Mathematical Society, 1988.
  10. Boris Springborn, Peter Schroder, and Ulrich Pinkall. Conformal equivalence of triangle meshes. ACM Transactions on Graphics (TOG), 27(3):77, 2008.
  11. Charles A Micchelli, Lixin Shen, Yuesheng Xu, and Xueying Zeng. Proximity algorithms for the l1/tv image denoising model. Advances in Computational Mathematics, 38(2):401-426, 2013. https://doi.org/10.1007/s10444-011-9243-y
  12. Emmanuel J Candes, Justin K Romberg, and Terence Tao. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 59(8):1207-1223, 2006.
  13. Richard Baraniuk and Philippe Steeghs. Compressive radar imaging. In 2007 IEEE radar conference, pages 128-133. IEEE, 2007.
  14. Yong-Liang Yang, Ren Guo, Feng Luo, Shi-Min Hu, and Xianfeng Gu. Generalized discrete ricci flow. In Computer Graphics Forum, volume 28, pages 2005-2014. Wiley Online Library, 2009.
  15. Feng Luo. Combinatorial yamabe flow on surfaces. Communications in Contemporary Mathematics, 6(05):765-780, 2004. https://doi.org/10.1142/S0219199704001501