## Computing Geodesic Paths on Manifolds (1998)

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Venue: | Proc. Natl. Acad. Sci. USA |

Citations: | 223 - 26 self |

### BibTeX

@INPROCEEDINGS{Kimmel98computinggeodesic,

author = {R. Kimmel and J. A. Sethian},

title = {Computing Geodesic Paths on Manifolds},

booktitle = {Proc. Natl. Acad. Sci. USA},

year = {1998},

pages = {8431--8435}

}

### Years of Citing Articles

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### Abstract

The Fast Marching Method [8] is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. In this paper we extend the Fast Marching Method to triangulated domains with the same computational complexity. As an application, we provide an optimal time algorithm for computing the geodesic distances and thereby extracting shortest paths on triangulated manifolds. 1 Introduction Sethian`s Fast Marching Method [8], is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M ) steps, where M is the total number of grid points in the domain. The technique hinges on producing numerically consistent approximations to the operators in the Eikonal equation that select the correct viscosity solution; this is done through the use of upwind nite dierence operators. The structure of this upwinding is then used to systematically construct the solution to the Eik...

### Citations

664 | Shape modeling with front propagation: a level set approach
- Malladi, Sethian, et al.
(Show Context)
Citation Context ...made fast by conning the \building zone" to a narrow band around the front, motivated by the narrow band introduced in Chopp [3], used in recovering shapes from images in Malladi, Sethian and Vem=-=uri [5]-=-, and analyzed extensively by Adalsteinsson and Sethian in [1]. The idea is to sweep the front ahead in an upwind fashion by considering a set of points in a narrow band around the existing front, and... |

466 | fast marching level set method for monotonically advancing fronts
- SETHIAN, “A
- 1996
(Show Context)
Citation Context ...awrence Berkeley Laboratory University of California, Berkeley, California 94720 Accepted for publication, Proc. National. Academy of Sciences, To appear: July, 1997 Abstract The Fast Marching Method =-=[8]-=- is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. In this paper we extend the Fast Marching M... |

352 | A Fast Level Set Method for Propagating Interfaces
- Adalsteinsson, Sethian
- 1995
(Show Context)
Citation Context ...und the front, motivated by the narrow band introduced in Chopp [3], used in recovering shapes from images in Malladi, Sethian and Vemuri [5], and analyzed extensively by Adalsteinsson and Sethian in =-=[1]-=-. The idea is to sweep the front ahead in an upwind fashion by considering a set of points in a narrow band around the existing front, and to march this narrow band forward, freezing the values of exi... |

210 |
A viscosity solutions approach to shape-from-shading
- Rouy, Tourin
- 1992
(Show Context)
Citation Context ...idea carefully considering the nature of upwind, entropy{satisfying approximations to the Eikonal equation. In more detail, consider one particular upwind approximation to the gradient, given by (see =-=[-=-6]) max(D x ij T; D +x ij T; 0) 2 + max(D y ij T; D +y ij T; 0) 2 1=2 = F ij ; (2) The central idea behind Fast Marching Methods is to systematically advance the front in an upwind fashion to produce... |

207 |
Level Set Methods: Evolving Interfaces
- Sethian
- 1996
(Show Context)
Citation Context ... point the one that produces the smallest value of T . Judicious programming and careful attention to \Far" values makes the above search extremely fast. This is the Fast Marching Method describe=-=d in [8, 9]-=-. 3 Fast Marching on a Particular Triangulated Planar Domain Our goal now is to extend this method to triangular domains. In order to do so, we shall build a monotone update procedure on the triangula... |

158 |
A Note on Two
- Dijkstra
- 1959
(Show Context)
Citation Context ...rdering of points during the update. The optimal ordering is executed using a heap operator to extract the next point in the update sweep. As such, the technique is a reminiscent of Dijkstra's method =-=[4]; h-=-owever, the resulting ron@math.lbl.gov y Please send all correspondence to J.A. Sethian at sethian@math.berkeley.edu z Supported in part by the Applied Mathematics Subprogram of the O��ce of Ener... |

115 |
Computing minimal surfaces via level set curvature flow
- Chopp
- 1993
(Show Context)
Citation Context ...building the solution outward from the smallest T value. The algorithm is made fast by conning the \building zone" to a narrow band around the front, motivated by the narrow band introduced in Ch=-=opp [3]-=-, used in recovering shapes from images in Malladi, Sethian and Vemuri [5], and analyzed extensively by Adalsteinsson and Sethian in [1]. The idea is to sweep the front ahead in an upwind fashion by c... |

115 |
Curvature and the Evolution of Fronts
- Sethian
- 1985
(Show Context)
Citation Context ...that the above Eikonal equation becomes non{dierentiable, and an appropriate weak solution must be built. The appropriate weak solution comes from satisfying the entropy condition; see, for example, [=-=7]-=-. The crucial point in this (or any such appropriate) numerical scheme is the correct direction of the upwinding and treatment of sonic points. For details and an extensive review, see [9]. The Fast M... |

62 | Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains
- Barth, Sethian
- 1998
(Show Context)
Citation Context ... DARPA under grant DMS{8919074. approximation is consistent in that it produces the correct shortest path on an orthogonal grid. For details about Fast Marching Methods, see [8, 9]. Barth and Sethian =-=[2]-=- have recently constructed operators for viscosity solutions to both the Eikonal equation and Hamilton-Jacobi equations on arbitrary triangulated domains. These operators exploit the upwind nature to ... |