Solving PDEs on Manifolds with Global Conformal Parameterization
Lok Ming Lui, Yalin Wang, Tony F. Chan
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.
Figures (click on each for a larger version):
Related Publications
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L.M. Lui, Y. Wang, Tony F. Chan,
"
Solving PDEs on Manifold using Global Conformal Parameterization",
Variational, Geometric, and Level Set Methods in Computer Vision:
Third International Workshop, VLSM 2005, Beijing, China, Oct. 16, 2005,
pp. 307-319
