Solving PDEs on Manifolds with Global Conformal Parametriazation

  • Lok Ming Lui
  • Yalin Wang
  • Tony F. Chan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3752)


In this paper, we propose a method to solve PDEs on surfaces with arbitrary topologies by using the global conformal parametrization. The main idea of this method is to map the surface conformally to 2D rectangular areas and then transform the PDE on the 3D surface into a modified PDE on the 2D parameter domain. Consequently, we can solve the PDE on the parameter domain by using some well-known numerical schemes on ℝ2. To do this, we have to define a new set of differential operators on the manifold such that they are coordinates invariant. Since the Jacobian of the conformal mapping is simply a multiplication of the conformal factor, the modified PDE on the parameter domain will be very simple and easy to solve. In our experiments, we demonstrated our idea by solving the Navier-Stoke’s equation on the surface. We also applied our method to some image processing problems such as segmentation, image denoising and image inpainting on the surfaces.


Riemann Surface Conformal Factor Parameter Domain Image Inpainting Conformal Parametrization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Turk, G.: Generating textures on arbitrary surfaces using reaction-diffusion. Computer Graphic 25, 289–298 (1991)CrossRefGoogle Scholar
  2. 2.
    Dorsey, J., Hanrahan, P.: Digital materials and virtual weathering. Scientific American 282, 46–53 (2000)CrossRefGoogle Scholar
  3. 3.
    Stam, J.: Flows on surfaces of arbitrary topology. In: Proceedings of ACM SIGGRAPH 2003, vol. 22, pp. 724–731 (2003)Google Scholar
  4. 4.
    Clarenza, U., Rumpfa, M., Teleaa, A.: Surface processing methods for point sets using finite elements. Computers and Graphics 28, 851–868 (2004)CrossRefGoogle Scholar
  5. 5.
    Bertalmio, M., Cheng, L.T., Osher, S., Sapiro, G.: Variational problems and partial differential equations on implicit surfaces. Journal of Computational Physics 174, 759–780 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Memoli, F., Sapiro, G., Thompson, P.: Implicit brain imaging. Neuroimage 23, 179–188 (2004)CrossRefGoogle Scholar
  7. 7.
    Schoen, R., Yau, S.T.: Lectures on Harmonic Maps. International Press (1997)Google Scholar
  8. 8.
    Gu, X., Yau, S.T.: Global conformal surface parameterization. In: ACM Symposium on Geometry Processing (2003)Google Scholar
  9. 9.
    Syngen, J., Schild, A.: Tensor Calculus. Dover Publication (1949)Google Scholar
  10. 10.
    Stam, J.: Stable fluids. In: Proceedings of ACM SIGGRAPH, pp. 121–128 (1999)Google Scholar
  11. 11.
    Aris, R.: Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover Publication (1989)Google Scholar
  12. 12.
    Vese, L.A., Chan, T.F.: Multiphase level set framework for image segmentation using the mumford and shah model. International Journal of Computer Vision 50, 271–293 (2002)zbMATHCrossRefGoogle Scholar
  13. 13.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena 60, 259–268 (1992)zbMATHCrossRefGoogle Scholar
  14. 14.
    Chan, T.F., Shen, J.: Non-texture inpainting by curvature-driven diffusions (cdd). J. Visual Comm. Image Rep. 12, 436–449 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Lok Ming Lui
    • 1
  • Yalin Wang
    • 1
  • Tony F. Chan
    • 1
  1. 1.Mathematics DepartmentUCLA 

Personalised recommendations