Advertisement

Journal of Scientific Computing

, Volume 58, Issue 3, pp 705–725 | Cite as

Folding-Free Global Conformal Mapping for Genus-0 Surfaces by Harmonic Energy Minimization

  • Rongjie LaiEmail author
  • Zaiwen Wen
  • Wotao Yin
  • Xianfeng Gu
  • Lok Ming Lui
Article

Abstract

Surface conformal maps between genus-0 surfaces play important roles in applied mathematics and engineering, with applications in medical image analysis and computer graphics. Previous work (Gu and Yau in Commun Inf Syst 2(2):121–146, 2002) introduces a variational approach, where global conformal parameterization of genus-0 surfaces was addressed through minimizing the harmonic energy, with two weaknesses: its gradient descent iteration is slow, and its solutions contain undesired parameterization foldings when the underlying surface has long sharp features. In this paper, we propose an algorithm that significantly accelerates the harmonic energy minimization and a method that iteratively removes foldings by taking advantages of the weighted Laplace–Beltrami eigen-projection. Experimental results show that the proposed approaches compute genus-0 surface harmonic maps much faster than the existing algorithm in Gu and Yau (Commun Inf Syst 2(2):121–146, 2002) and the new results contain no foldings.

Keywords

Harmonic energy minimization Conformal map Optimization with orthogonality constraints Weighted Laplace–Beltrami eigenfunctions 

Notes

Acknowledgments

Rongjie Lai’s work is supported by Zumberge Individual Award from USC’s James H. Zumberge Faculty Research and Innovation Fund. Zaiwen Wen’s work is supported in part by NSFC Grant 11101274 and Research Fund (20110073120069) for the Doctoral Program of Higher Education of China. Wotao Yin’s work is supported in part by NSF Grant DMS-0748839 and ONR Grant N00014-08-1-1101. Xianfeng Gu’s work is is supported in part of NSF Nets 1016286, NSF IIS 0916286, NSF CCF 1081424 and ONR N000140910228. Lok Ming Lui’s work is supported in part of the CUHK Direct Grant (Project ID: 2060413) and HKRGC GRF (Project ID: 2130271).

References

  1. 1.
    Gu, X., Yau, S.: Computing conformal structures of surfaces. Commun. Inf. Syst. 2(2), 121–146 (2002)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Gu, X., Wang, Y., Chan, T.F., Thompson, P., Yau, S.T.: Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Trans. Med. Imaging 23, 949–958 (2004)CrossRefGoogle Scholar
  3. 3.
    Lui, L.M., Wang, Y., Thompson, P.M., Chan, T.F.: Landmark constrained genus zero surface conformal mapping and its application to brain mapping research. Appl. Numer. Math. 57, 847–858 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Lui, L.M., Gu, X., Chan, T.F., Yau, S.-T.: Variational method on riemann surfaces using conformal parameterization and its applications to image processing. Methods Appl. Anal. 15(4), 513–538 (2008)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Levy, B., Petitjean, S., Ray, N., Maillot, J.: Least squares conformal maps for automatic texture atlas generation. Proceeding of ACM SIGGRAPH (2002)Google Scholar
  6. 6.
    Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., Stuetzle, W.: Multiresolution analysis of arbitrary meshes. Proceeding of ACM SIGGRAPH (1995)Google Scholar
  7. 7.
    Schoen, R., Yau, S.-T.: Lectures on Differential Geometry, vol. 2. International Press, Cambridge (1994)Google Scholar
  8. 8.
    Alliez, P., Meyer, M., Desbrun, M.: Interactive geometry remeshing. Proceeding of ACM SIGGRAPH (2002)Google Scholar
  9. 9.
    Kanai, T., Suzuki, H., Kimura, F.: Three-dimensional geometric metamorphosis based on harmonic maps. Vis. Comput. 14(4), 166–176 (1998)CrossRefGoogle Scholar
  10. 10.
    Hurdal, M.K., Stephenson, K., Bowers, P.L., Sumners, D.W.L., Rottenberg, D.A.: Coordinate systems for conformal cerebellar flat maps. NeuroImage 11, S467 (2000)CrossRefGoogle Scholar
  11. 11.
    Haker, S., Angenent, S., Tannenbaum, A., Kikinis, R., Sapiro, G., Halle, M.: Conformal surface parameterization for texture mapping. IEEE Trans. Vis. Comput. Graph. 6(2), 181–189 (2000)CrossRefGoogle Scholar
  12. 12.
    Springborn, B., Schröder, P., Pinkall, U.: Conformal equivalence of triangle meshes. ACM Transactions on Graphics (TOG)—Proceedings of ACM SIGGRAPH 2008, vol. 27(3) (2008)Google Scholar
  13. 13.
    Gu X., Yau S.T.: Global conformal surface parameterization. Symposium on Geometry Processing, pp. 127–137 (2003)Google Scholar
  14. 14.
    Jin M., Wang Y., Yau S.T., Gu X.: Optimal global conformal surface parameterization. IEEE Visualization, Austin, TX, pp. 267–274 (2004)Google Scholar
  15. 15.
    Jin, M., Kim, J., Luo, F., Gu, X.: Discrete surface ricci flow. IEEE Trans. Vis. Comput. Graph. 14(5), 1030–1043 (2008)CrossRefGoogle Scholar
  16. 16.
    Yang, Y., Kim, J., Luo, F., Hu, S., Gu, X.: Optimal surface parameterization using inverse curvature map. IEEE Trans. Vis. Comput. Graph. 14(5), 1054–1066 (2008)CrossRefGoogle Scholar
  17. 17.
    Schoen, R., Yau, S.-T.: Lectures on Harmonic Maps. International Press, Cambridge (1997)Google Scholar
  18. 18.
    Wen Z., Yin W.: A feasible method for optimization with orthogonality constraints. Math. Program. (2013, in press)Google Scholar
  19. 19.
    Lui, L.M., Thiruvenkadam, S., Wang, Y., Thompson, P.M., Chan, T.F.: Optimized conformal surface registration with shape-based landmark matching. SIAM J. Imaging Sci. 3(1), 52–78 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Jost, J.: Riemannian Geometry and Geometric Analysis, 3rd edn. Springer, Berlin (2001)Google Scholar
  21. 21.
    Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ (2008)zbMATHGoogle Scholar
  22. 22.
    Helmke, U., Moore, J.B.: Optimization and Dynamical Systems. Communications and Control Engineering Series. Springer-Verlag London Ltd., London (1994). With a foreword by R. BrockettGoogle Scholar
  23. 23.
    Udrişte, Constantin: Convex Functions and Optimization Methods on Riemannian Manifolds, vol. 297 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht (1994)CrossRefGoogle Scholar
  24. 24.
    Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8(1), 141–148 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Zhang, H., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14(4), 1043–1056 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Hege, H.C., Polthier, K. (eds.) Visualization and Mathematics III, pp. 35–57. Springer, Berlin (2003)CrossRefGoogle Scholar
  27. 27.
    Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Discrete differential geometry operators in nd. In: Proceedings of VisMath’02, Berlin, Germany (2002)Google Scholar
  28. 28.
    Xu, G.: Convergent discrete Laplace–Beltrami operator over triangular surfaces. Proceedings of Geometric Modelling and Processing, pp. 195–204 (2004)Google Scholar
  29. 29.
    Chavel, I.: Eigenvalues in Riemannian Geometry. Academic Press, Inc., London (1984)Google Scholar
  30. 30.
    Ben-Chen, M., Gotsman, C.: Characterizing shape using conformal factors. Proceedings of Eurographics Workshop on Shape Retrieval, Crete, April (2008)Google Scholar
  31. 31.
    Reuter, M., Wolter, F.E., Peinecke, N.: Laplace–Beltrami spectra as Shape-DNA of surfaces and solids. Comput. Aided Des. 38, 342–366 (2006)CrossRefGoogle Scholar
  32. 32.
    Shi, Y., Lai, R., Krishna, S., Sicotte, N., Dinov, I., Toga, A.W.: Anisotropic Laplace–Beltrami eigenmaps: bridging reeb graphs and skeletons. In: Proceedings of MMBIA (2008)Google Scholar
  33. 33.
    Lai, R., Shi, Y., Scheibel, K., Fears, S., Woods, R., Toga, A.W., Chan, T.F.: Metric-induced optimal embedding for intrinsic 3D shape analysis. CVPR (2010)Google Scholar
  34. 34.
    Shi, Y., Lai, R., Gill, R., Pelletier, D., Mohr, D., Sicotte, N., Toga, A.W.: Conformal metric optimization on surface (CMOS) for deformation and mapping in Laplace–Beltrami embedding space. MICCAI (2011)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Rongjie Lai
    • 1
    Email author
  • Zaiwen Wen
    • 2
  • Wotao Yin
    • 3
  • Xianfeng Gu
    • 4
  • Lok Ming Lui
    • 5
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of Mathematics, MOE-LSC and Institute of Natural SciencesShanghai Jiaotong UniversityShanghaiChina
  3. 3.Department of Computational and Applied MathematicsRice UniversityHoustonUSA
  4. 4.Department of Computer ScienceState University of New York at Stony BrookStony BrookUSA
  5. 5.Department of MathematicsThe Chinese University of Hong KongShatin, New TerritoriesHong Kong

Personalised recommendations