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Solving PDEs on Manifolds with Global Conformal Parametriazation

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Abstract

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.

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Authors and Affiliations

  1. Mathematics Department, UCLA,  

    Lok Ming Lui, Yalin Wang & Tony F. Chan

Authors
  1. Lok Ming Lui

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  2. Yalin Wang

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  3. Tony F. Chan

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Editor information

Editors and Affiliations

  1. MAS - Ecole Centrale Paris, Grande Voie des Vignes, 92295, Chatenay-Malabry, France

    Nikos Paragios

  2. I.N.R.I.A, 2004 route des lucioles,, 06902, Sophia-Antipolis, France

    Olivier Faugeras

  3. Department of Mathematics, UCLA,  

    Tony Chan

  4. Image and Pattern Analysis Group, Heidelberg Collaboratory for Image Processing, University of Heidelberg, Germany

    Christoph Schnörr

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© 2005 Springer-Verlag Berlin Heidelberg

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Lui, L.M., Wang, Y., Chan, T.F. (2005). Solving PDEs on Manifolds with Global Conformal Parametriazation. In: Paragios, N., Faugeras, O., Chan, T., Schnörr, C. (eds) Variational, Geometric, and Level Set Methods in Computer Vision. VLSM 2005. Lecture Notes in Computer Science, vol 3752. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11567646_26

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