Skip to main content
SpringerLink
Log in

Survey on Discrete Surface Ricci Flow

  • Survey
  • Published:
Journal of Computer Science and Technology Aims and scope Submit manuscript

Abstract

Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a nonlinear heat diffusion process, and becomes constant eventually. Ricci flow is a powerful computational tool to design Riemannian metrics by prescribed curvatures. Surface Ricci flow has been generalized to the discrete setting. This work surveys the theory of discrete surface Ricci flow, its computational algorithms, and the applications for surface registration and shape analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Zeng W, Gu X. Ricci Flow for Shape Analysis and Surface Registration. Springer New York, 2013.

  2. Thurston W. The Geometry and Topology of 3-Manifolds. Princeton University Press, 1997.

  3. Andreev E M. Complex polyhedra in Lobačhevskiî spaces. Mat. Sb. (N.S.), 1970, 81(123): 445–478. (in Russian)

  4. Andreev E M. Convex polyhedra of finite volume in Lobačhevskiî space. Mat. Sb. (N.S.), 1970, 83(125): 256–260. (in Russian)

  5. Koebe P. Kontaktprobleme der konformen abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math. Phys. Kl., 1936, 88: 141–164.

  6. Rodin B, Sullivan D. The convergence of circle packings to the Riemann mapping. Journal of Differential Geometry, 1987, 26(2): 349–360.

    MATH  MathSciNet  Google Scholar 

  7. Chow B, Luo F. Combinatorial Ricci flows on surfaces. Journal of Differential Geometry, 2003, 63(1): 97–129.

    MATH  MathSciNet  Google Scholar 

  8. Marden A, Rodin B. On Thurston’s formulation and proof of Andreev’s theorem. In Lecture Notes in Math. 1435, Ruscheweyh S, Satt E, Salinas L et al. (eds.), Springer Berlin, 1990, pp.103–116.

  9. Colin de Verdiére Y. Un principe variationnel pour les empilements de cercles. Invent. Math., 1991, 104: 655–669.

  10. Stephenson K. Introduction to Circle Packing: The Theory of Discrete Analytic Functions. Cambridge University Press, 2005.

  11. He Z X, Schramm O. On the convergence of circle packings to the Riemann map. Invent. Math., 1996, 125(2): 285–305.

    Article  MATH  MathSciNet  Google Scholar 

  12. Bowers P L, Stephenson K. Uniformizing Dessins and Belyi Maps via Circle Packing. Amer. Math. Soc., 2004.

  13. Guo R. Local rigidity of inversive distance circle packing. Trans. Amer. Math. Soc., 2011, 363: 4757–4776.

    Article  MATH  MathSciNet  Google Scholar 

  14. Luo F. Combinatorial Yamabe flow on surfaces. Contemp. Math., 2004, 6(5): 765–780.

    Article  MATH  MathSciNet  Google Scholar 

  15. Springborn B, Schröder P, Pinkall U. Conformal equivalence of triangle meshes. ACM Trans. Graph., 2008, 27(3): Article No. 77.

  16. Glickenstein D. A combinatorial Yamabe flow in three dimensions. Topology, 2005, 44(4): 791–808.

    Article  MATH  MathSciNet  Google Scholar 

  17. Glickenstein D. A maximum principle for combinatorial Yamabe flow. Topology, 2005, 44(4): 809–825.

    Article  MATH  MathSciNet  Google Scholar 

  18. Glickenstein D. Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds. Journal of Differential Geometry, 2011, 87(2): 201–238.

    MATH  MathSciNet  Google Scholar 

  19. Guo R. Combinatorial Yamabe flow on hyperbolic surfaces with boundary. Communications in Contemporary Mathematics, 2011, 13(5): 827–842.

    Article  MATH  MathSciNet  Google Scholar 

  20. Zhang M, Guo R, Zeng W, Luo F, Yau S T, Gu X. The unified surface Ricci flow. Graphic Models, 2014, 76(5): 321–339.

    Article  Google Scholar 

  21. Brägger W. Kreispackungen und triangulierugen. Enseign. Math., 1992, 38: 201–217.

    MATH  MathSciNet  Google Scholar 

  22. Rivin I. Euclidean structures of simplicial surfaces and hyperbolic volume. Ann. Math., 1994, 139: 553–580.

    Article  MATH  MathSciNet  Google Scholar 

  23. Leibon G. Characterizing the Delaunay decompositions of compact hyperbolic surface. Geom. & Topol., 2002, 6: 361–391.

    Article  MATH  MathSciNet  Google Scholar 

  24. Bobenko A I, Springborn B A. Variational principles for circle patterns and Koebe’s theorem. Trans. Amer. Math. Soc., 2004, 356(2): 659–689.

    Article  MATH  MathSciNet  Google Scholar 

  25. Guo R, Luo F. Rigidity of polyhedral surface, II. Geom. & Topol., 2009, 13: 1265–1312.

  26. Springborn B. A variational principle for weighted Delaunay triangulation and hyperideal polyhedra. Journal of Differential Geometry, 2008, 78(2): 333–367.

    MATH  MathSciNet  Google Scholar 

  27. Luo F. Rigidity of polyhedral surfaces. arXiv:math.GT/0612714, 2006.

  28. Dai J, Gu X, Luo F. Variational Principles for Discrete Surfaces (Advanced Lectures in Mathematics). High Education Press and International Press, 2007.

  29. Chow B, Lu P, Ni L. Hamilton’s Ricci Flow (Graduate Studies in Mathematics). American Mathematical Society, 2006.

  30. Schoen R, Yau S T. Lecture on Differential Geometry, Volume 1. International Press Incorporated, Boston, 1994.

  31. Gu X, Yau S T. Computational Conformal Geometry. International Press, 2008.

  32. Hamilton R. Ricci flow on surfaces. Mathematics and General Relativity, Contemporary Mathematics AMS, Providence, RI, 1988, 71: 237–262.

  33. Chow B. The Ricci flow on the 2-sphere. Journal of Differential Geometry, 1991, 33(2): 325–334.

    MATH  MathSciNet  Google Scholar 

  34. Bobenko A, Pinkall U, Springborn B. Discrete conformal maps and ideal hyperbolic polyhedra. arXiv:1005.2698, 2013.

  35. He Z. Rigidity of infinite disk patterns. Ann. Math., 1999 149(1): 1–33.

    Article  MATH  Google Scholar 

  36. Jin M, Kim J, Luo F, Gu X. Discrete surface Ricci flow. IEEE Transactions on Visualization and Computer Graphic, 2008, 14(5): 1030–1043.

    Article  Google Scholar 

  37. Zeng W, Yin X, Zhang M, Luo F, Gu X. Generalized Koebe’s method for conformal mapping multiply connected domains. In Proc. SIAM/ACM Joint Conference on Geometric and Physical Modeling, Oct. 2009, pp.89–100.

  38. Hernandez C, Vogiatzis G, Brostow G J, Stenger B, Cipolla R. Non-rigid photometric stereo with colored lights. In Proc. the 11th IEEE International Conference on Computer Vision, Oct. 2007.

  39. Wang Y, Gupta M, Zhang S, Wang S, Gu X, Samaras D, Huang P. High resolution tracking of non-rigid motion of densely sampled 3D data using harmonic maps. International Journal of Computer Vision, 2008, 76(3): 283–300.

    Article  Google Scholar 

  40. Zeng W, Samaras D, Gu X. Ricci flow for 3D shape analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2010, 32(4): 662–677.

    Article  Google Scholar 

  41. Zeng W, Gu X. Registration for 3D surfaces with large deformations using quasi-conformal curvature flow. In Proc. IEEE Conference on Computer Vision and Pattern Recognition, June 2011, pp.2457–2464.

  42. Lui L M, Wong T, Zeng W, Gu X, Thompson P, Chan T, Yau S T. Optimization of surface registrations using Beltrami holomorphic flow. Journal of Scientific Computing, 2012, 50(3): 557–585.

    Article  MATH  MathSciNet  Google Scholar 

  43. Horner M, Ries L, Krapcho M, Neyman N, Aminou R, Howlader N, Altekruse S, Feuer E, Huang L, Mariotto A, Miller B, Lewis D, Eisner M, Stinchcomb D, Edwards B. SEER cancer statistics review, 1975–2006. http://seer.cancer.gov/csr/1975 2006/, Apr. 2015.

  44. Center M, Jemal A, Smith R A, Ward E. Worldwide variations in colorectal cancer. CA: A Cancer Journal for Clinicians, 2009, 59(6): 366–378.

    Google Scholar 

  45. Hong L, Muraki S, Kaufman A, Bartz D, He T. Virtual voyage: Interactive navigation in the human colon. In Proc. the 24th ACM SIGGRAPH, Aug. 1997, pp.27–34.

  46. Johnson C, Dachman A. CT colography: The next colon screening examination. Radiology, 2000, 216(2): 331–341.

    Article  Google Scholar 

  47. Qiu F, Fan Z, Yin X, Kauffman A, Gu X. Colon flattening with discrete Ricci flow. In Proc. International Conference on Medical Image Computing and Computer Assisted Intervention, Sept. 2009.

  48. Zeng W, Marino J, Chaitanya Gurijala K, Gu X, Kaufman A. Supine and prone colon registration using quasiconformal mapping. IEEE Transactions on Visualization and Computer Graphics, 2010, 16(6): 1348–1357.

    Article  Google Scholar 

  49. Gu X, Wang Y, Chan T, Thompson P M, Yau S T. Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Transactions on Medical Imaging, 2004, 23(8): 949–958.

    Article  Google Scholar 

  50. Wang Y, Shi J, Yin X, Gu X, Chan T, Yau S T, Toga A W, Thompson P M. Brain surface conformal parameterization with the Ricci flow. IEEE Transactions on Medical Imaging, 2012, 31(2): 251–264.

    Article  Google Scholar 

  51. Zeng W, Gu X. 3D dynamics analysis in Teichm¨uller space. In Proc. ICCV Workshops, Nov. 2011, pp. 1610–1617.

  52. Sharon E, Mumford D. 2D-shape analysis using conformal mapping. International Journal of Computer Vision, 2006, 70(1): 55–75.

    Article  Google Scholar 

  53. Lui L M, Zeng W, Chan T F, Yau S T, Gu X. Shape representation of planar objects with arbitrary topologies using conformal geometry. In Proc. the 11th European Conference on Computer Vision, Sept. 2010.

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, State University of New York at Stony Brook, Stony Brook, NY, 11794-4400, U.S.A.

    Min Zhang & Xianfeng David Gu

  2. School of Computing and Information Sciences, Florida International University, Miami, Florida, 33199, U.S.A.

    Wei Zeng

  3. Department of Mathematics, Oregon State University, Corvallis, OR, 97331-4605, U.S.A.

    Ren Guo

  4. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, U.S.A.

    Feng Luo

Authors
  1. Min Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ren Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Feng Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xianfeng David Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Min Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, M., Zeng, W., Guo, R. et al. Survey on Discrete Surface Ricci Flow. J. Comput. Sci. Technol. 30, 598–613 (2015). https://doi.org/10.1007/s11390-015-1548-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11390-015-1548-8

Keywords

Search

Navigation