An algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hyper-surfaces 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 hyper-surface in the embedding space, thereby performing all the computations in a Cartesian grid with classical and computationally optimal 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, Hamilton-Jacobi based, 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 hyper-surfaces a computationally optimal technique. The approach can be extended to solve a more general class of Hamilton-Jacobi 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 to perform the computations on a Cartesian grid with computationally optimal algorithms, in spite of the fact that the distance and Hamilton-Jacobi equations are intrinsic to the implicit hyper-surface. For other surface representations like triangulated or unorganized points ones, the algorithm here introduced can be used after simple pre-processing of the data.
Institute for Mathematics and Its Applications>IMA Preprints Series
Memoli, Facundo; Sapiro, Guillermo.
Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces.
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