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## A Fast Marching Level Set Method for Monotonically Advancing Fronts (1995)

Venue: | PROC. NAT. ACAD. SCI |

Citations: | 626 - 24 self |

### Citations

1371 |
Numerical Recipes in C
- Press
- 2002
(Show Context)
Citation Context ...ient version of the above technique lies in a fast way of locating the grid point in the narrow band with the smallest value for T . We use a variation on a heapsort algorithm with back pointers, see =-=[21, 24]-=-. In more detail, imagine that the list of narrow band points is initially sorted in a heapsort so that the smallest member can be easily located. We store the values of these points in the heapsort, ... |

1181 | Front propagating with curvature depending speed
- Osher, Sethian
- 1988
(Show Context)
Citation Context ...numerical technique, this suggests using the technology from hyperbolic conservation laws to solve the equations of motion, as described in [32]. This leads to the level set formulation introduced in =-=[20]-=-, which we now describe. 2.1 Level Set Methods Given an initial position for an interface , where is a closed curve in R 2 , and a speed function F which gives the speed of in its normal direction, th... |

427 | Motion of level sets by mean curvature
- Evans, Spruck
- 1991
(Show Context)
Citation Context ...ions in [26], and grid generation in [30]. The fundamental Eulerian perspective presented by this approach has since been adopted in many theoretical analyses of mean curvaturesow, in particular, see =-=[12, 7]-=-. 2.2 Numerical Approximation As mentioned above, a careful approximation to the gradient in the level set equation (Eqn. 1) is required in order to produce the correct weak solution. One of the simpl... |

422 | A fast level set method for propagating interfaces
- Adalsteinsson, Sethian
- 1995
(Show Context)
Citation Context ...ation [22, 36],suid interfaces [6, 18], shape reconstruction [17, 16], as well as etching, deposition and lithography calculations in [2, 3]. Extensions of the basic technique include fast methods in =-=[1]-=-, level set techniques for multiplesuid interfaces and triple point junctions in [26], and grid generation in [30]. The fundamental Eulerian perspective presented by this approach has since been adopt... |

362 | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations
- Chen, Giga, et al.
- 1991
(Show Context)
Citation Context ...ions in [26], and grid generation in [30]. The fundamental Eulerian perspective presented by this approach has since been adopted in many theoretical analyses of mean curvaturesow, in particular, see =-=[12, 7]-=-. 2.2 Numerical Approximation As mentioned above, a careful approximation to the gradient in the level set equation (Eqn. 1) is required in order to produce the correct weak solution. One of the simpl... |

332 |
Generalized Solutions of Hamilton-Jacobi Equations
- Lions
- 1982
(Show Context)
Citation Context ...casting of the problem into a stationary one is common in a variety of applications. The notion of viscosity solutions is intimately connected to this equation; a central idea, discussed in detail in =-=[15-=-], is that the use of monotone, consistent schemes will lead to schemes which select the correct viscous limit of the partial dierential equation, as was done in the level set scheme in [20]. We refer... |

300 |
Souganidis. Convergence of approximation schemes for fully non-linear equations
- Barles, E
- 1991
(Show Context)
Citation Context ...orrect viscous limit of the partial dierential equation, as was done in the level set scheme in [20]. We refer the interested reader to large literature on this subject, including relevant theory in [=-=4, 10, 15, 35]-=-, and numerical algorithms in [5, 13, 14, 23, 19]. Roughly speaking, two possible ways to view these solution techniques are either iteration towards the solution or direct construction of the station... |

248 |
Some properties of viscosity solutions of Hamilton-Jacobi equations
- Crandall, Evans, et al.
- 1984
(Show Context)
Citation Context ...orrect viscous limit of the partial dierential equation, as was done in the level set scheme in [20]. We refer the interested reader to large literature on this subject, including relevant theory in [=-=4, 10, 15, 35]-=-, and numerical algorithms in [5, 13, 14, 23, 19]. Roughly speaking, two possible ways to view these solution techniques are either iteration towards the solution or direct construction of the station... |

245 |
A viscosity solutions approach to shape from shading
- ROUY, TOURIN
- 1992
(Show Context)
Citation Context ...al equation, as was done in the level set scheme in [20]. We refer the interested reader to large literature on this subject, including relevant theory in [4, 10, 15, 35], and numerical algorithms in =-=[5, 13, 14, 23, 19]-=-. Roughly speaking, two possible ways to view these solution techniques are either iteration towards the solution or direct construction of the stationary solution surface T (x; y). We now discuss a t... |

179 | A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows,
- Chang, Hou, et al.
- 1996
(Show Context)
Citation Context ...oving interfaces. Some of these applications include the generation of minimal surfaces [8], singularities and geodesics in moving curves and surfaces in [9],same propagation [22, 36],suid interfaces =-=[6, 18]-=-, shape reconstruction [17, 16], as well as etching, deposition and lithography calculations in [2, 3]. Extensions of the basic technique include fast methods in [1], level set techniques for multiple... |

132 |
Computing minimal surfaces via level set curvature flow.
- Chopp
- 1993
(Show Context)
Citation Context ...ts introduction in [20], the above level set approach has been used in a wide collection of problems involving moving interfaces. Some of these applications include the generation of minimal surfaces =-=[8]-=-, singularities and geodesics in moving curves and surfaces in [9],same propagation [22, 36],suid interfaces [6, 18], shape reconstruction [17, 16], as well as etching, deposition and lithography calc... |

132 |
Curvature and the Evolution of Fronts,
- Sethian
- 1985
(Show Context)
Citation Context ...on F . The goal is to track the motion of this interface as it evolves. We are only concerned with the motion of the interface in its normal direction, and shall ignore tangential motion. As shown in =-=[27, 28, 31]-=-, a propagating interface can develop corners and discontinuities as it evolves, which require the introduction of a weak solution in order to proceed. The correct weak solution comes from enforcing a... |

102 | Computing interface motion in compressible gas dynamics.
- Mulder, Osher, et al.
- 1992
(Show Context)
Citation Context ...oving interfaces. Some of these applications include the generation of minimal surfaces [8], singularities and geodesics in moving curves and surfaces in [9],same propagation [22, 36],suid interfaces =-=[6, 18]-=-, shape reconstruction [17, 16], as well as etching, deposition and lithography calculations in [2, 3]. Extensions of the basic technique include fast methods in [1], level set techniques for multiple... |

87 |
Approximation schemes for viscosity solutions of Hamilton-Jacobi equations
- Souganidis
- 1983
(Show Context)
Citation Context ...orrect viscous limit of the partial dierential equation, as was done in the level set scheme in [20]. We refer the interested reader to large literature on this subject, including relevant theory in [=-=4, 10, 15, 35]-=-, and numerical algorithms in [5, 13, 14, 23, 19]. Roughly speaking, two possible ways to view these solution techniques are either iteration towards the solution or direct construction of the station... |

55 |
B.C.: Evolutionary fronts for topology-independent shape modeling and recovery
- Malladi, Sethian, et al.
- 1994
(Show Context)
Citation Context ...e applications include the generation of minimal surfaces [8], singularities and geodesics in moving curves and surfaces in [9],same propagation [22, 36],suid interfaces [6, 18], shape reconstruction =-=[17, 16]-=-, as well as etching, deposition and lithography calculations in [2, 3]. Extensions of the basic technique include fast methods in [1], level set techniques for multiplesuid interfaces and triple poin... |

55 |
An Analysis of Flame Propagation,
- Sethian
- 1982
(Show Context)
Citation Context ...on F . The goal is to track the motion of this interface as it evolves. We are only concerned with the motion of the interface in its normal direction, and shall ignore tangential motion. As shown in =-=[27, 28, 31]-=-, a propagating interface can develop corners and discontinuities as it evolves, which require the introduction of a weak solution in order to proceed. The correct weak solution comes from enforcing a... |

53 |
Numerical algorithm for propagating interfaces: HamiltonJacobi equations and conservation
- Sethian
- 1990
(Show Context)
Citation Context ...on F . The goal is to track the motion of this interface as it evolves. We are only concerned with the motion of the interface in its normal direction, and shall ignore tangential motion. As shown in =-=[27, 28, 31]-=-, a propagating interface can develop corners and discontinuities as it evolves, which require the introduction of a weak solution in order to proceed. The correct weak solution comes from enforcing a... |

40 |
An Approximation Scheme for the Minimum Time Function
- Bardi, Falcone
- 1990
(Show Context)
Citation Context ...al equation, as was done in the level set scheme in [20]. We refer the interested reader to large literature on this subject, including relevant theory in [4, 10, 15, 35], and numerical algorithms in =-=[5, 13, 14, 23, 19]-=-. Roughly speaking, two possible ways to view these solution techniques are either iteration towards the solution or direct construction of the stationary solution surface T (x; y). We now discuss a t... |

35 | Flow under curvature: Singularity formation, minimal surfaces, and geodesics
- Chopp, Sethian
- 1993
(Show Context)
Citation Context ...d in a wide collection of problems involving moving interfaces. Some of these applications include the generation of minimal surfaces [8], singularities and geodesics in moving curves and surfaces in =-=[9]-=-,same propagation [22, 36],suid interfaces [6, 18], shape reconstruction [17, 16], as well as etching, deposition and lithography calculations in [2, 3]. Extensions of the basic technique include fast... |

28 |
A Review of the Theory, Algorithms, and Applications of Level Set Methods for Propagating Interfaces, Acta Numerica,
- Sethian
- 1996
(Show Context)
Citation Context ...om control theory, as well as surface advancement in lithographic development and isotropic etching and deposition in the manufacturing of microelectronic structures. This scheme wassrst described in =-=[25]-=-; in this paper, we present the details of this scheme and show results and timings. 2 Background Consider a boundary, either a curve in two dimensions or a surface in three dimensions, separating one... |

26 |
Image processing via level set curvature flow.
- Malladi, Sethian
- 1995
(Show Context)
Citation Context ...e applications include the generation of minimal surfaces [8], singularities and geodesics in moving curves and surfaces in [9],same propagation [22, 36],suid interfaces [6, 18], shape reconstruction =-=[17, 16]-=-, as well as etching, deposition and lithography calculations in [2, 3]. Extensions of the basic technique include fast methods in [1], level set techniques for multiplesuid interfaces and triple poin... |

23 | Curvature flow and entropy conditions applied to grid generation.”
- Sethian
- 1994
(Show Context)
Citation Context ...roduced [20, 31], namely t + Fjrj = 0; (1) (x; t = 0) = given: (2) This is an initial value partial dierential equation in one higher dimension than the original problem. In Figure 2 (taken from [30]), we show the outward propagation of an initial curve and the accompanying motion of the level set function . x y x y x y x y g g (0) (t) g g (0) = Level Set (t) = Level Set f = 0 f = 0 f = f = C z ... |

22 |
Projection methods coupled to level set interface techniques,”
- Zhu, Sethian
- 1992
(Show Context)
Citation Context ...n of problems involving moving interfaces. Some of these applications include the generation of minimal surfaces [8], singularities and geodesics in moving curves and surfaces in [9],same propagation =-=[22, 36]-=-,suid interfaces [6, 18], shape reconstruction [17, 16], as well as etching, deposition and lithography calculations in [2, 3]. Extensions of the basic technique include fast methods in [1], level set... |

21 |
Level Sets of Viscosity Solutions: Some Applications to Fronts and Rendez-Vous Problems
- Falcone, Giorgi, et al.
- 1994
(Show Context)
Citation Context ...al equation, as was done in the level set scheme in [20]. We refer the interested reader to large literature on this subject, including relevant theory in [4, 10, 15, 35], and numerical algorithms in =-=[5, 13, 14, 23, 19]-=-. Roughly speaking, two possible ways to view these solution techniques are either iteration towards the solution or direct construction of the stationary solution surface T (x; y). We now discuss a t... |

20 |
Numerical methods for propagating fronts, [in:] Variational Methods for Free Surface Interfaces,
- Sethian
- 1987
(Show Context)
Citation Context ...front and is irreversible [28]. For details, see [28]. As a numerical technique, this suggests using the technology from hyperbolic conservation laws to solve the equations of motion, as described in =-=[32]-=-. This leads to the level set formulation introduced in [20], which we now describe. 2.1 Level Set Methods Given an initial position for an interface , where is a closed curve in R 2 , and a speed fun... |

12 |
Shape from shading via level sets
- Kimmel, Bruckstein
- 1992
(Show Context)
Citation Context |

11 |
A uni level set approach to etching, deposition and lithography I: Algorithms and two-dimensional simulations
- Adalsteinsson, Sethian
- 1995
(Show Context)
Citation Context ...ies and geodesics in moving curves and surfaces in [9],same propagation [22, 36],suid interfaces [6, 18], shape reconstruction [17, 16], as well as etching, deposition and lithography calculations in =-=[2, 3]-=-. Extensions of the basic technique include fast methods in [1], level set techniques for multiplesuid interfaces and triple point junctions in [26], and grid generation in [30]. The fundamental Euler... |

10 | A Brief Overview of Vortex Methods, - Sethian - 1990 |

8 | Stable and entropy-satisfying approximations for transonic flow calculations - Engquist, Osher - 1980 |

5 |
Dynamical study of a premixed V
- Rhee, Talbot, et al.
- 1995
(Show Context)
Citation Context ...n of problems involving moving interfaces. Some of these applications include the generation of minimal surfaces [8], singularities and geodesics in moving curves and surfaces in [9],same propagation =-=[22, 36]-=-,suid interfaces [6, 18], shape reconstruction [17, 16], as well as etching, deposition and lithography calculations in [2, 3]. Extensions of the basic technique include fast methods in [1], level set... |

4 |
Rapid convergence of approximate solutions to shape-from-shading problem
- Osher, Rudin
- 1993
(Show Context)
Citation Context |

4 |
Algorithms for Tracking Interfaces
- Sethian
- 1995
(Show Context)
Citation Context ...ching, deposition and lithography calculations in [2, 3]. Extensions of the basic technique include fast methods in [1], level set techniques for multiplesuid interfaces and triple point junctions in =-=[26]-=-, and grid generation in [30]. The fundamental Eulerian perspective presented by this approach has since been adopted in many theoretical analyses of mean curvaturesow, in particular, see [12, 7]. 2.2... |

1 |
Extensions to a Fast Marching Level Set Method for Advancing Fronts
- Sethian
- 1995
(Show Context)
Citation Context ...llumination. At the same time, the above technique can be extended, with modication, to general convex speeds laws, and perhaps, non-convex speed functions. We shall report on these issues elsewhere [=-=33]-=-. Furth Acknowledgements: All calculations were performed at the University of California at Berkeley and the Lawrence Berkeley Laboratory. The author would like to thank D. Adalsteinsson and R. Malla... |

1 |
An E��cient Level Set Technique for Propagating Interfaces
- Sethian, Adalsteinsson, et al.
- 1995
(Show Context)
Citation Context ...array. This results in an O(logN) algorithm for the total amount of work, where N is the number of points in the narrow band. For implementation details and further application of this technique, see =-=[34]-=-. 4.2 Arbitrary Initial Fronts The above technique considered asat initial interface for which trial values at the narrow band points could be easily initialized. Suppose we are given an arbitrary clo... |