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54
Level set methods: An overview and some recent results
 J. Comput. Phys
, 2001
"... The level set method was devised by Osher and Sethian in [64] as a simple and versatile method for computing and analyzing the motion of an interface Γ in two or three dimensions. Γ bounds a (possibly multiply connected) region Ω. The goal is to compute and analyze the subsequent motion of Γ under a ..."
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Cited by 226 (11 self)
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The level set method was devised by Osher and Sethian in [64] as a simple and versatile method for computing and analyzing the motion of an interface Γ in two or three dimensions. Γ bounds a (possibly multiply connected) region Ω. The goal is to compute and analyze the subsequent motion of Γ under a velocity field �v. This velocity can depend on position, time, the geometry of the interface and the external physics. The interface is captured for later time as the zero level set of a smooth (at least Lipschitz continuous) function ϕ(�x,t), i.e., Γ(t)={�xϕ(�x,t)=0}. ϕ is positive inside Ω, negative outside Ω andiszeroonΓ(t). Topological merging and breaking are well defined and easily performed. In this review article we discuss recent variants and extensions, including the motion of curves in three dimensions, the Dynamic Surface Extension method, fast methods for steady state problems, diffusion generated motion and the variational level set approach. We also give a user’s guide to the level set dictionary and technology, couple the method to a wide variety of problems involving external physics, such as compressible and incompressible (possibly reacting) flow, Stefan problems, kinetic crystal growth, epitaxial growth of thin films,
A FAST SWEEPING METHOD FOR EIKONAL EQUATIONS
, 2004
"... In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses GaussSeidel iterations with alternating sweeping ordering to solve the discr ..."
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Cited by 181 (7 self)
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In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses GaussSeidel iterations with alternating sweeping ordering to solve the discretized system. The crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding Eikonal equation in a certain direction simultaneously. The method has an optimal complexity of O(N) for N grid points and is extremely simple to implement in any number of dimensions. Monotonicity and stability properties of the fast sweeping algorithm are proven. Convergence and error estimates of the algorithm for computing the distance function is studied in detail. It is shown that 2n GaussSeidel iterations is enough for the distance function in n dimensions. An estimation of the number of iterations for general Eikonal equations is also studied. Numerical examples are used to verify the analysis.
Fast surface reconstruction using the level set method
 In VLSM ’01: Proceedings of the IEEE Workshop on Variational and Level Set Methods
, 2001
"... In this paper we describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from scattered data ..."
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Cited by 151 (12 self)
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In this paper we describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from scattered data set. The data set might consist of points, curves and/or surface patches. A weighted minimal surfacelike model is constructed and its variational level set formulation is implemented with optimal efficiency. The reconstructed surface is smoother than piecewise linear and has a natural scaling in the regularization that allows varying flexibility according to the local sampling density. As is usual with the level set method we can handle complicated topology and deformations, as well as noisy or highly nonuniform data sets easily. The method is based on a simple rectangular grid, although adaptive and triangular grids are also possible. Some consequences, such as hole filling capability, are demonstrated, as well as the viability and convergence of our new fast tagging algorithm.
Fast Sweeping Algorithms for a Class of HamiltonJacobi Equations
 SIAM Journal on Numerical Analysis
, 2003
"... We derive a Godunovtype numerical flux for the class of strictly convex, homogeneous Hamiltonians that includes H(p, q) = � ap 2 + bq 2 − 2cpq, c 2 < ab. We combine our Godunov numerical fluxes with simple GaussSeidel type iterations for solving the corresponding HamiltonJacobi Equations. Th ..."
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Cited by 136 (20 self)
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We derive a Godunovtype numerical flux for the class of strictly convex, homogeneous Hamiltonians that includes H(p, q) = � ap 2 + bq 2 − 2cpq, c 2 < ab. We combine our Godunov numerical fluxes with simple GaussSeidel type iterations for solving the corresponding HamiltonJacobi Equations. The resulting algorithm is fast since it does not require a sorting strategy as found, e.g., in the fast marching method. In addition, it provides a way to compute solutions to a class of HJ equations for which the conventional fast marching method is not applicable. Our experiments indicate convergence after a few iterations, even in rather difficult cases. 1
Level Set Methods
 in Imaging, Vision and Graphics
, 2000
"... The level set method was devised by Osher and Sethian in [56] as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions. bounds a (possibly multiply connected) region The goal is to compute and analyze the subsequent motion of under a velocity ..."
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Cited by 74 (7 self)
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The level set method was devised by Osher and Sethian in [56] as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions. bounds a (possibly multiply connected) region The goal is to compute and analyze the subsequent motion of under a velocity field ~v. This velocity can depend on position, time, the geometry of the interface and the external physics. The interface is captured for later time as the zero level set of a smooth (at least Lipschitz continuous) function '(~x; t), i.e., (t) = f~xj'(~x; t) = 0g. ' is positive inside negative outside and is zero on (t). Topological merging and breaking are well defined and easily performed. In this review article we discuss recent variants and extensions, including the motion of curves in three dimensions, the Dynamic Surface Extension method, fast methods for steady state problems, diffusion generated motion and the variational level set approach. We also give a user's gui...
O(N) Implementation of the Fast Marching Algorithm
 Journal of Computational Physics
, 2005
"... In this note we present an implementation of the fast marching algorithm for solving Eikonal equations that reduces the original runtime from O(N log N) to linear. This lower runtime cost is obtained while keeping an error bound of the same order of magnitude as the original algorithm. This improv ..."
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Cited by 69 (11 self)
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In this note we present an implementation of the fast marching algorithm for solving Eikonal equations that reduces the original runtime from O(N log N) to linear. This lower runtime cost is obtained while keeping an error bound of the same order of magnitude as the original algorithm. This improvement is achieved introducing the straight forward untidy priority queue, obtained via a quantization of the priorities in the marching computation. We present the underlying framework, estimations on the error, and examples showing the usefulness of the proposed approach. Key words: Fast marching, HamiltonJacobi and Eikonal equations, distance functions, bucket sort, untidy priority queue.
Fast computation of weighted distance functions and geodesics on implicit hypersurfaces
 J. Comput. Phys
"... An algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hypersurfaces 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 h ..."
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Cited by 63 (8 self)
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An algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hypersurfaces 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 hypersurface 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–Jacobibased, 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 hypersurfaces 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 hypersurface. c ○ 2001 Academic Press Key Words: implicit hypersurfaces; distance functions; geodesics; Hamilton– Jacobi equations; fast computations.
LaxFriedrichs Sweeping Scheme for Static HamiltonJacobi Equations
 Journal of Computational Physics
, 2003
"... We propose a simple, fast sweeping method based on the LaxFriedrichs monotone numerical Hamiltonian to approximate the viscosity solution of arbitrary static HamiltonJacobi equations in any number of spatial dimensions. ..."
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Cited by 60 (6 self)
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We propose a simple, fast sweeping method based on the LaxFriedrichs monotone numerical Hamiltonian to approximate the viscosity solution of arbitrary static HamiltonJacobi equations in any number of spatial dimensions.
Fast sweeping methods for static hamiltonjacobi equations
 Society for Industrial and Applied Mathematics
, 2005
"... Abstract. We propose a new sweeping algorithm which discretizes the Legendre transform of the numerical Hamiltonian using an explicit formula. This formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically. The minimiz ..."
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Cited by 55 (5 self)
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Abstract. We propose a new sweeping algorithm which discretizes the Legendre transform of the numerical Hamiltonian using an explicit formula. This formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically. The minimization that is related to the Legendre transform in our sweeping scheme can either be solved analytically or numerically. We illustrate the efficiency and accuracy approach with several numerical examples in two and three dimensions.
Rapid and Accurate Computation of the Distance Function Using Grids
 J. Comput. Phys
, 2002
"... We present two fast and simple algorithms for approximating the distance function for given isolated points on uniform grids. The algorithms are then generalized to compute the distance to piecewise linear objects. By incorporating the geometry of Huygens ’ principle in the reverse order with the cl ..."
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Cited by 53 (3 self)
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We present two fast and simple algorithms for approximating the distance function for given isolated points on uniform grids. The algorithms are then generalized to compute the distance to piecewise linear objects. By incorporating the geometry of Huygens ’ principle in the reverse order with the classical viscosity solution theory for the eikonal equation ∇u=1, the algorithms become almost purely algebraic and yield very accurate approximations. The generalized closest point formulation used in the second algorithm provides a framework for further extension to compute the distance accurately to smooth geometric objects on different grid geometries, without the construction of the Voronoi diagrams. This method provides a fast and simple translator of data commonly given in computational geometry to the volumetric representation used in level set methods. c ○ 2002 Elsevier Science (USA) 1.