Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces: 730
Journal of Computational Physics
Parallel algorithms for approximation of distance maps on parametric surfaces
ACM Transactions on Graphics (TOG)
Numerical Geometry of Non-Rigid Shapes
Numerical Geometry of Non-Rigid Shapes
Möbius voting for surface correspondence
ACM SIGGRAPH 2009 papers
Incompressible, Quasi-Isometric Deformations of 2-Dimensional Domains
SIAM Journal on Imaging Sciences
| Hi-index | 0.00 |
According to the Uniformization Theorem any surface can be conformally mapped into a flat domain, that is, a domain with zero Gaussian curvature. The conformal factor indicates the local scaling introduced by such a mapping. This process could be used to compute geometric quantities in a simplified flat domain. For example, the computation of geodesic distances on a curved surface can be mapped into solving an eikonal equation in a plane weighted by the conformal factor. Solving an eikonal equation on the weighted plane can then be done with regular sampling of the domain using, for example, the fast marching method. The connection between the conformal factor on the plane and the surface geometry can be justified analytically. Still, in order to construct consistent numerical solvers that exploit this relation one needs to prove that the conformal factor is bounded. In this paper we provide theoretical bounds over the conformal factor and introduce optimization formulations that control its behavior. It is demonstrated that without such a control the numerical results are unboundedly inaccurate. Putting all ingredients in the right order, we introduce a method for computing geodesic distances on a two dimensional manifold by using the fast marching algorithm on a weighed flat domain.