Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces

Facundo Mémoli, Guillermo Sapiro

Research output: Contribution to journalArticlepeer-review

92 Scopus citations

Abstract

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 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, 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 efficient 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 for the computations to be performed 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.

Original languageEnglish (US)
Pages (from-to)730-764
Number of pages35
JournalJournal of Computational Physics
Volume173
Issue number2
DOIs
StatePublished - Nov 1 2001
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Distance functions
  • Fast computations
  • Geodesics
  • Hamilton-Jacobi equations
  • Implicit hyper-surfaces

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