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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.
Continuum Crowds
 ACM Trans. Graph
, 2006
"... classroom use is granted without fee provided that copies are not made or distributed for commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitt ..."
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Cited by 145 (0 self)
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classroom use is granted without fee provided that copies are not made or distributed for commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee.
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.
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.
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.
Integral Invariants for Robust Geometry Processing
 IN: ICCV ’95: PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON COMPUTER VISION. IEEE COMPUTER SOCIETY
, 2005
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A Binary Level Set Model and some Applications to MumfordShah Image Segmentation
"... In this work we propose a variant of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead let the interfaces be represented by discontinuities of piecewise constant level set functions. Each level set function can ..."
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Cited by 34 (5 self)
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In this work we propose a variant of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead let the interfaces be represented by discontinuities of piecewise constant level set functions. Each level set function can at convergence only take two values, i.e. it can only be 1 or1. Some of the properties of the standard level set function are preserved in the proposed method, while others are not. Using this new level set method for interface problems, we need to minimize a smooth convex functional under a quadratic constraint. The level set functions are discontinuous at convergence, but the minimization functional is smooth and locally convex. We show numerical results using the method for segmentation of digital images.
An efficient solution to the eikonal equation on parametric manifolds
 INTERFACES AND FREE BOUNDARIES 6 (2004), 315–327
, 2004
"... We present an efficient solution to the eikonal equation on parametric manifolds, based on the fast marching approach. This method overcomes the problem of a nonorthogonal coordinate system on the manifold by creating an appropriate numerical stencil. The method is tested numerically and demonstrat ..."
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Cited by 31 (15 self)
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We present an efficient solution to the eikonal equation on parametric manifolds, based on the fast marching approach. This method overcomes the problem of a nonorthogonal coordinate system on the manifold by creating an appropriate numerical stencil. The method is tested numerically and demonstrated by calculating distances on various parametric manifolds. It is further used for two applications: image enhancement and face recognition.