Math @ Duke

Publications [#264950] of Guillermo Sapiro
Papers Published
 Mémoli, F; Sapiro, G, Fast computation of weighted distance functions and geodesics on implicit hypersurfaces,
Journal of Computational Physics, vol. 173 no. 2
(2001),
pp. 730764, ISSN 00219991 [doi]
(last updated on 2018/02/18)
Abstract: An algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hypersurfaces is introduced in this paper. The basic idea is to approximate the intrinsic weighted distance by the Euclidean weighted distance computed in a band surrounding the implicit hypersurface in the embedding space, thereby performing all the computations in a Cartesian grid with classical and efficient numerics. Based on work on geodesics on Riemannian manifolds with boundaries, we bound the error between the two distance functions. We show that this error is of the same order as the theoretical numerical error in computationally optimal, HamiltonJacobibased, algorithms for computing distance functions in Cartesian grids. Therefore, we can use these algorithms, modified to deal with spaces with boundaries, and obtain also for the case of intrinsic distance functions on implicit hypersurfaces a computationally efficient technique. The approach can be extended to solve a more general class of HamiltonJacobi equations defined on the implicit surface, following the same idea of approximating their solutions by the solutions in the embedding Euclidean space. The framework here introduced thereby allows for the computations to be performed on a Cartesian grid with computationally optimal algorithms, in spite of the fact that the distance and HamiltonJacobi equations are intrinsic to the implicit hypersurface. © 2001 Academic Press.


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