## Fast Marching Methods (1998)

Venue: | SIAM Review |

Citations: | 147 - 4 self |

### BibTeX

@ARTICLE{Sethian98fastmarching,

author = {J. A. Sethian},

title = {Fast Marching Methods},

journal = {SIAM Review},

year = {1998},

volume = {41},

pages = {199--235}

}

### Years of Citing Articles

### OpenURL

### Abstract

Fast Marching Methods are numerical schemes for computing solutions to the non-linear Eikonal equation and related static Hamilton-Jacobi equations. Based on entropy-satisfying upwind schemes and fast sorting techniques, they yield consistent, accurate, and highly efficient algorithms. They are optimal in the sense that the computational complexity of the algorithms is O(N log N ), where N is the total number of points in the domain. The schemes are of use in a variety of applications, including problems in shape offsetting, computing distances from complex curves and surfaces, shape-from-shading, photolithographic development, computing rst arrivals in seismic travel times, construction of shortest geodesics on surfaces, optimal path planning around obstacles, and visibility and reection calculations. In this paper, we review the development of these techniques, including the theoretical and numerical underpinnings, provide details of the computational schemes including higher order versions,...

### Citations

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Introduction to Algorithms
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1989 |
Robot Motion Planning
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- 1990
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Citation Context ...h R 2 and employing periodic boundary conditions for . Thus, we solve the Eikonal equation h u 2 x + u 2 y + u 2 i 1=2 = 1: (27) In the presence of obstacles, we take the following approach; see [14=-=-=-]. Rather than maneuver an oddly shaped robot, we instead consider the robot as a point, and, for every discretized angle i , alter the shape of the obstacles corresponding to that angle. To do this ... |

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- 1988
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Citation Context ...0]. 2.3 Schemes for viscosity solutions Continuing with the example of an evolving interface, we now focus on the gradient term (1 +sx 2 ). Consider now thesnite dierence approximation introduced in [=-=1-=-8], namelysx 2 (max(D +x is; 0) 2 + min(D x is; 0) 2 ) (10) 9 where again we have used standardsnite dierence notation that D x is=sisi 1 h D +x is=si+1si h : (11) Here,si is the value ofson a grid a... |

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Citation Context ...dierentiability by constructing accurate and ecient approximation schemes that admit physically correct non-smooth solutions. The main techniques we will use are Fast Marching Methods, introduced in [=-=27]. Th-=-ese consistent and highly ecient techniques are based on two key components. First, by exploiting upwind \viscosity schemes", they automatically select solutions which include non-dierentiability... |

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Citation Context ...robot with rotational angle navigating in two-dimensional space around obstacles. For further details about applications of Fast Marching Methods to path planning and robotic navigation, see [12] and =-=[31]-=-. 20 40 60 80 20 40 60 80 20 40 60 80 20 40 60 80 20 40 60 80 20 40 60 80 20 40 60 80 20 40 60 80 Figure 29: Two-dimensional navigation around obstacles with rotation 46 6.6 Visibility calculations an... |

425 |
Numerical Methods for Conservation laws
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- 1990
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Citation Context ...andall and Lions [8] proved that consistent monotone schemes must converge to the correct viscosity solution; this result parallels similar results for hyperbolic conservation laws (see, for example, =-=[4, 15, 3-=-4]). Thus, we need only check that this scheme satises the necessary requirements for convergence to viscosity solutions. The operator is easily seen to be consistent, since it usessrst ordersnite die... |

328 | A Fast Level Set Method for Propagating Interfaces
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Citation Context ...he instructive example of an curve evolving in time whose position can always be described as the graph of a function. Consider an initial front given by the graph of g(x), with g and g 0 periodic on =-=[0; 1]-=-, and suppose that the front (i) propagates with speed S in its normal direction and (ii) remains a function for all time. Letsbe the height of the propagating function at time t, thuss(x; 0) = g(x). ... |

197 |
A viscosity solutions approach to shape-from-shading
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Citation Context ...nd backwards operators D y , D +y , D z , and D +z in the other coordinate directions are similar to the one dened earlier for the x direction. A slightly dierent upwind scheme due to Godunov, (see [2=-=0]-=-), which will turn out to be more convenient, is given by 2 6 4 max(D x ijk u; D +x ijk u; 0) 2 + max(D y ijk u; D +y ijk u; 0) 2 + max(D z ijk u; D +z ijk u; 0) 2 3 7 5 1=2 = F ijk ; (14) where we us... |

190 |
Set Methods: Evolving Interfaces
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- 1996
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Citation Context ...one to eciently construct the solution outward from the boundary data. 3 The majority of this review is taken from the original work on Fast Marching Methods [27], and two recent books on the subject =-=[30, 31]-=-. We refer the interested reader to these resources for many computational schemes which can exploit Fast Marching Methods, as well as many more applications and examples. 2 The Eikonal and Static Ham... |

161 |
Some properties of viscosity solutions of the Hamilton-Jacobi equations
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- 1984
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Citation Context ...n produced by taking the limit of the smooth solutions u as vanishes is indeed this viscosity solution. We shall prove none of these statements here. Precise statements and proofs may be found in [6=-=, 8, -=-7]. The salient point is that for both timedependent and static (e.g., the Eikonal equation) Hamilton-Jacobi equations, the viscosity solution can be dened in a way that does not require dierentiation... |

143 |
A Note on Two
- Dijkstra
- 1959
(Show Context)
Citation Context ...) A A Heap property restored Figure 9: Heap structure and UpHeap+ operation 19 4 Related Algorithms 4.1 Network path algorithms The Fast Marching Method is reminiscent of Dijkstra's algorithm [10]=-=-=- (see also [5] and [22]), which is a method forsnding the shortest path on a network with prescribed weights between each link. As illustration, imagine one is given a rectangular network with equal u... |

111 |
Computing minimal surfaces via level set curvature flow
- Chopp
- 1993
(Show Context)
Citation Context ...ion in a downwind direction. The algorithm is made fast by conning the \building zone" to a narrow band around the front; this approach is motivated by the narrow band technology introduced by Ch=-=opp [3]-=-, used in recovering shapes in images by Malladi, Sethian and Vemuri [16], and analyzed extensively by Adalsteinsson and Sethian in [30]. The idea is to sweep the front ahead in a downwind fashion by ... |

103 |
Curvature and the evolution of fronts
- Sethian
- 1985
(Show Context)
Citation Context ...nding to the shortest distance or \rst arrival", is the one obtained through the Huygens construction. Another way to obtain this solution is through the notion of an entropy condition. As dened =-=in [23, 24-=-], we imagine the boundary 5 curve as a source for a propagatingsame, and the expandingsame satises the requirement that once a point in the domain is ignited by the expanding front, it stays burnt. T... |

56 | Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains - Barth, Sethian - 1998 |

51 |
Evolutionary fronts for topologyindependent shape modeling and recovery
- Malladi, Sethian, et al.
- 1994
(Show Context)
Citation Context ...\building zone" to a narrow band around the front; this approach is motivated by the narrow band technology introduced by Chopp [3], used in recovering shapes in images by Malladi, Sethian and Ve=-=muri [16]-=-, and analyzed extensively by Adalsteinsson and Sethian in [30]. The idea is to sweep the front ahead in a downwind fashion by considering a set of points in narrow band around the existing front, and... |

47 |
An Analysis of Flame Propagation
- Sethian
- 1982
(Show Context)
Citation Context ...nding to the shortest distance or \rst arrival", is the one obtained through the Huygens construction. Another way to obtain this solution is through the notion of an entropy condition. As dened =-=in [23, 24-=-], we imagine the boundary 5 curve as a source for a propagatingsame, and the expandingsame satises the requirement that once a point in the domain is ignited by the expanding front, it stays burnt. T... |

42 | Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws - Sethian |

33 | Finite-difference calculation of traveltimes in three dimensions - Vidale - 1990 |

32 |
Fast marching level set methods for three dimensional photolithography development
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- 1996
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Citation Context ...ivity of the surface. In Figure 24(b) a view of the developed prole is shown from underneath; the etching of the holes and the presence of standing waves can easily be seen. For further results, see [=-=29]-=-. 38 (a) Masking pattern (b) Lithographic development: View from below Figure 24: Lithographic development using Fast Marching Method 39 6.3 Seismic traveltimes Next, we apply Fast Marching Methods to... |

27 |
Finite-difference calculation of travel times
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- 1988
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Citation Context ...ources positioned every 200 meters, requires 1 terabyte of traveltime volumes. Thus, speed is an important issue. Finite dierence approximations to traveltime computations include the work of Vidale [=-=38-=-] and van Trier and Symes [37]. Because of their speed, robustness, and the fact that they are unconditionally stable, Fast Marching Methods oer attractive methods of computing travel times. 6.3.2 Mig... |

25 |
Fast marching methods on triangulated domains
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- 1998
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Citation Context ...riangulated meshing so that the non-etchable material boundaries are accurately represented. Second, surfaces are often described by triangulated patches, and hence this discretization is natural. In =-=[13]-=-, a Fast Marching Method on triangulated unstructured meshes was introduced, using the unstructured mesh methodology for level set methods developed by Barth and Sethian [2], and applied to the proble... |

25 |
Numerical Methods in Fluid Dynamics
- Sod
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Citation Context ...andall and Lions [8] proved that consistent monotone schemes must converge to the correct viscosity solution; this result parallels similar results for hyperbolic conservation laws (see, for example, =-=[4, 15, 3-=-4]). Thus, we need only check that this scheme satises the necessary requirements for convergence to viscosity solutions. The operator is easily seen to be consistent, since it usessrst ordersnite die... |

23 | An O(N log N) algorithm for shape modeling
- Malladi, Sethian
- 1996
(Show Context)
Citation Context ...ach direction, the Fast Marching Method reduces the total operation count from N 4 to N 3 log N ; essentially, each grid point is visited once to compute its arrival time value. For more details, see =-=[27, 28, 30, -=-17]. 18 u = 0:6(i = 2; j = 8) """" ` u = 1:3(3; 5) u = 2:0(4; 5) A A @ @@ u = 3:0(4; 5) A A u = 2:3(6; 8) u = 3:1(2; 7) A A @ @@ u = 2:9(3; 2) A A Step 1: Chan... |

22 | Three-dimensional traveltime computation using the fast marching method: submitted to Geophysics
- Sethian, Popovici
- 1997
(Show Context)
Citation Context ...rom below Figure 24: Lithographic development using Fast Marching Method 39 6.3 Seismic traveltimes Next, we apply Fast Marching Methods to problems involving the imaging of geophysical data sets. In =-=[32]-=-, Sethian and Popovici used the Fast Marching Method to rapidly constructsrst arrival times in seismic analysis, and then coupled this work to prestack migration. Here, we summarize that work. For fur... |

14 |
Numerical methods for propagating fronts, in Variational Methods for Free Surface Interfaces, edited by P. Concus and R. Finn (Springer-Verlag
- Sethian
- 1987
(Show Context)
Citation Context ...n, and can be proven to be the unique viscous limit of the smoothed Hamilton{Jacobi equation. Our goal is to develop numerical approximations which correctly select this viscous limit. As proposed in =-=[25]-=-, since the entropy condition is similar to the one for hyperbolic conservation laws, it suggests using the numerical methodologies associated with hyperbolic equation. 2.2 Upwind schemes and numerica... |

12 |
Users Guide to Viscosity
- Crandall, Iskii, et al.
- 1992
(Show Context)
Citation Context ...n produced by taking the limit of the smooth solutions u as vanishes is indeed this viscosity solution. We shall prove none of these statements here. Precise statements and proofs may be found in [6=-=, 8, -=-7]. The salient point is that for both timedependent and static (e.g., the Eikonal equation) Hamilton-Jacobi equations, the viscosity solution can be dened in a way that does not require dierentiation... |

12 |
Prestack migration by split-step DSR
- Popovici
- 1996
(Show Context)
Citation Context ...een's functions of the single scattering wave-propagation experiment (see, for example, [21]). 6.3.1 Background equations In some more detail, the essence of 3-D prestack migration (see, for example, =-=[19]-=-), is expressed by the following integral equation: Image(x) = Z Z xs Z xr G(x s ; x; !)G(x; x r ; !)Data(x s ; x r ; !)dx r dx s d!; where x is the image output location, and x s and x r are the data... |

10 |
Partial Di erential Equations
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- 1998
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Citation Context ... H H H H H H H H Hj Level Set Perspective t + Fjrj = 0 Initial Value Problem [18] ? adaptivity ? NARROW BAND LEVEL SET METHODS [1] Stationary Perspective jrT j F = 1 Boundary Value Problem [11] ? adaptivity ? FAST MARCHING METHODS [27] Figure 14: Evolution of theory and algorithms for interface propagations. 25 5 Extensions of Fast Marching Methods 5.1 Higher accuracy Fast Marching Methods ... |

8 |
Fast marching methods for robotic navigation with constraints
- Kimmel, Sethian
- 1996
(Show Context)
Citation Context ... Shortest paths on a bead (a genus one 2D manifold). 44 6.5 Optimal path planning 6.5.1 Statement of problem The application of Fast Marching Methods to problems in path planning wassrst developed in =-=[12-=-]; here we summarize some of those results. Given a cost function F (x 1 ; x 2 ; ::; x n ), and a starting point A in R n , one goal in path planning is to construct the paths() : [0; 1) ! R n from A ... |

2 |
Computing and Imaging using Fast Marching
- Corporation
- 1998
(Show Context)
Citation Context ...ions, and close to the left side of the top of the salt, most probably because of the use ofsrst arrivals in the Fast Marching Method. For further results and discussion of other issues, see [32] and =-=[35]-=-. 6 The Green's function can be reconstructed from traveltime tables that describe traveltimes from all surface points (x; y) to all subsurface locations (x; y; z); thus the tables aresve-dimensional.... |

2 |
Upwind Finite-dierence
- Trier, Symes
- 1991
(Show Context)
Citation Context ...eters, requires 1 terabyte of traveltime volumes. Thus, speed is an important issue. Finite dierence approximations to traveltime computations include the work of Vidale [38] and van Trier and Symes [=-=37-=-]. Because of their speed, robustness, and the fact that they are unconditionally stable, Fast Marching Methods oer attractive methods of computing travel times. 6.3.2 Migration using the Fast Marchin... |

1 |
Modern Numerical Methods for Fluid Flow, Lecture Notes
- Colella, Puckett
- 1994
(Show Context)
Citation Context ...andall and Lions [8] proved that consistent monotone schemes must converge to the correct viscosity solution; this result parallels similar results for hyperbolic conservation laws (see, for example, =-=[4, 15, 3-=-4]). Thus, we need only check that this scheme satises the necessary requirements for convergence to viscosity solutions. The operator is easily seen to be consistent, since it usessrst ordersnite die... |

1 |
Viscosity
- Crandall, Lions
- 1983
(Show Context)
Citation Context ...ilton-Jacobi equations. Rather than dene the solution as the viscous limit, one instead analyzes the behavior of potential solutions when measured against possible test functions. Crandall and Lions [8] have developed the theory of viscosity solutions for time-dependent Hamilton-Jacobi equation. Brie y, following the denitions in [8], they dene a viscosity solution as follows. Denition: A functio... |

1 |
Robust and ecient upwind traveltime calculations in three dimensions
- Schneider
- 1995
(Show Context)
Citation Context ...ltidimensional surfaces; the shapes of the summation surfaces and the summation weights are computed from the Green's functions of the single scattering wave-propagation experiment (see, for example, =-=[21]-=-). 6.3.1 Background equations In some more detail, the essence of 3-D prestack migration (see, for example, [19]), is expressed by the following integral equation: Image(x) = Z Z xs Z xr G(x s ; x; !)... |

1 |
Extensions to Triangulated Fast Marching Methods
- Vladimirsky
- 1998
(Show Context)
Citation Context ...ined. We refer the interested reader there for further details. Further discussion of triangulated Fast Marching Methods, higher order versions, and extensions to the basic techniques may be found in =-=[33-=-]. 36 6 Applications In this section, we provide a collection of examples to demonstrate the applicability of Fast Marching Methods. 6.1 Shape-osetting Thesrst example, borrowed from [30], is the stra... |