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90
Shape modeling with front propagation: A level set approach
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1995
"... Abstract Shape modeling is an important constituent of computer vision as well as computer graphics research. Shape models aid the tasks of object representation and recognition. This paper presents a new approach to shape modeling which retains some of the attractive features of existing methods ..."
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Cited by 664 (18 self)
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Abstract Shape modeling is an important constituent of computer vision as well as computer graphics research. Shape models aid the tasks of object representation and recognition. This paper presents a new approach to shape modeling which retains some of the attractive features of existing methods and overcomes some of their limitations. Our techniques can be applied to model arbitrarily complex shapes, which include shapes with significant protrusions, and to situations where no a priori assumption about the object’s topology is made. A single instance of our model, when presented with an image having more than one object of interest, has the ability to split freely to represent each object. This method is based on the ideas developed by Osher and Sethian to model propagating solidhiquid interfaces with curvaturedependent speeds. The interface (front) is a closed, nonintersecting, hypersurface flowing along its gradient field with constant speed or a speed that depends on the curvature. It is moved by solving a “HamiltonJacob? ’ type equation written for a function in which the interface is a particular level set. A speed term synthesizpd from the image is used to stop the interface in the vicinity of object boundaries. The resulting equation of motion is solved by employing entropysatisfying upwind finite difference schemes. We present a variety of ways of computing evolving front, including narrow bands, reinitializations, and different stopping criteria. The efficacy of the scheme is demonstrated with numerical experiments on some synthesized images and some low contrast medical images. Index Terms Shape modeling, shape recovery, interface motion, level sets, hyperbolic conservation laws, HamiltonJacobi
A Fast Marching Level Set Method for Monotonically Advancing Fronts
 Proc. Nat. Acad. Sci
, 1995
"... We present a fast marching level set method for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation. Level set methods are numerical techniques for computing the position of propagating fronts. They rely on an initial value partial dierential equa ..."
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Cited by 466 (21 self)
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We present a fast marching level set method for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation. Level set methods are numerical techniques for computing the position of propagating fronts. They rely on an initial value partial dierential equation for a propagating level set function, and use techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. In this paper, we describe a particular case of such methods for interfaces whose speed depends only on local position. The technique works by coupling work on entropy conditions for interface motion, the theory of viscosity solutions for HamiltonJacobi equations and fast adaptive narrow band level set methods. The technique is applicable to a variety of problems, including shapefromshading problems, lithog...
A NonOscillatory Eulerian Approach to Interfaces in Multimaterial Flows (The Ghost Fluid Method)
, 2000
"... While Eulerian schemes work well for most gas flows, they have been shown to admit nonphysical oscillations near some material interfaces. In contrast,... ..."
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Cited by 223 (43 self)
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While Eulerian schemes work well for most gas flows, they have been shown to admit nonphysical oscillations near some material interfaces. In contrast,...
A PDEBased Fast Local Level Set Method
 Journal of Computational Physics
, 1999
"... this paper we localize the level set method. Our localization works in as much generality as does the original method and all of its recent variants [27, 28], but requires an order of magnitude less computing effort. Earlier work on localization was done by Adalsteinsson and Sethian [1]. Our approac ..."
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Cited by 209 (24 self)
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this paper we localize the level set method. Our localization works in as much generality as does the original method and all of its recent variants [27, 28], but requires an order of magnitude less computing effort. Earlier work on localization was done by Adalsteinsson and Sethian [1]. Our approach differs from theirs in that we use only the values of the level set function (or functions, for multiphase flow) and not the explicit location of points in the domain. Our implementation is easy and straightforward and has been used in [9, 14, 27, 28]. Our approach is partial differential equation (PDE) based, in the sense that our localization, extension, and reinitialization are all based on solving different PDEs. This leads to a simple, accurate, and flexible method. Equations (10) and (11) of Section 2 enable us to update the level set function (or functions in the multiphase case) without any stability problems at the boundary of the tube used to localize the evolution. Such equations are new and do not appear in [1]. In fact, the technique used in [1] has boundary stability problems because Eq. (2) or (3) (the evolution equation of the level set function) is solved right up to this boundary. In contrast, in our method, the speed of evolution degenerates smoothly to 0 at this boundary. This is achieved by modifying the evolution of the level set function near the tube boundary but away from the interface. This modification effectively eliminates the boundary stability issues in [1] and has no impact on the correct evolution of the interface. The reinitialization step will reset the level set function to be a signed distance function to the front. There are no boundary issues in our distance reinitialization or extension of velocity field off the interface. Both of the...
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 159 (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 Boundary Condition Capturing Method for Poisson's Equation on Irregular Domains
, 1999
"... Interfaces have a variety of boundary conditions (or jump conditions) that need to be enforced. In [3], the Ghost Fluid Method (GFM) was developed to capture the boundary conditions at a contact discontinuity in the inviscid Euler equations. This method was extended to treat more general discontinui ..."
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Cited by 76 (19 self)
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Interfaces have a variety of boundary conditions (or jump conditions) that need to be enforced. In [3], the Ghost Fluid Method (GFM) was developed to capture the boundary conditions at a contact discontinuity in the inviscid Euler equations. This method was extended to treat more general discontinuities such as shocks, detonations, and deflagrations in [2] and compressible viscous flows in [4]. In this paper, a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient Poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous. This new method is robust and easy to implement even in three spatial dimensions. Furthermore, the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient Poisson equation in the absence of interfaces allowing for straightforward application of standard "black box" solvers.
Three Dimensional Front Tracking
 SIAM J. Sci. Comp
, 1995
"... . We describe a three dimensional front tracking algorithm, discuss its numerical implementation, and present studies to validate the correctness of this approach. Based on the results of the two dimensional computations, we expect three dimensional front tracking to improve significantly computatio ..."
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Cited by 68 (18 self)
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. We describe a three dimensional front tracking algorithm, discuss its numerical implementation, and present studies to validate the correctness of this approach. Based on the results of the two dimensional computations, we expect three dimensional front tracking to improve significantly computational efficiencies for problems dominated by discontinuities. In some cases, for which the interface computations display considerable numerical sensitivity, we expect a greatly enhanced capability. 1. Introduction Front tracking is a numerical method in which surfaces of discontinuity are given explicit computational degrees of freedom; these degrees of freedom are supplemented by degrees of freedom representing continuous solution values at regular grid points. This method is ideal for solutions in which discontinuities are an important feature, and especially where their accurate computation is difficult by other methods. Computational continuum mechanics abounds in such problems, which in...
An efficient, interfacepreserving level set redistancing algorithm and its application to interfacial incompressible fluid flow
 SIAM Journal on Scientific Computing
, 1999
"... This paper is dedicated to the memory of Dr. Emad Fatemi, a very kind person and a truly original scientist. Abstract. In Sussman, Smereka, and Osher [J. Comp. Phys., 94 (1994), pp. 146–159], a numerical scheme was presented for computing incompressible air–water flows using the level set method. Cr ..."
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Cited by 61 (0 self)
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This paper is dedicated to the memory of Dr. Emad Fatemi, a very kind person and a truly original scientist. Abstract. In Sussman, Smereka, and Osher [J. Comp. Phys., 94 (1994), pp. 146–159], a numerical scheme was presented for computing incompressible air–water flows using the level set method. Crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set. In this paper we implement a “constraint ” along with higher order difference schemes in order to make the iteration method more accurate and efficient. Accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set. We apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost. We validate our results with experiment and theory. We show that our “distance level set scheme ” with the added constraint competes well with available interface tracking schemes for basic advection of an interface. We perform basic accuracy checks and more stringent tests involving complicated interfacial structures. As with all level set schemes, our method is easy to implement.
Spatially adaptive techniques for level set methods and incompressible flow
 Comput. Fluids
"... Since the seminal work of [92] on coupling the level set method of [69] to the equations for twophase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes ..."
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Cited by 60 (12 self)
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Since the seminal work of [92] on coupling the level set method of [69] to the equations for twophase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interestingly, this work also demonstrated the largest weakness of the level set method, i.e. mass or information loss characteristic of most Eulerian capturing techniques. In fact, [92] introduced a partial differential equation for battling this weakness, without which their work would not have been possible. In this paper, we discuss both historical and most recent works focused on improving the computational accuracy of the level set method focusing in part on applications related to incompressible flow due to both its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as HamiltonJacobi WENO [46], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [27] and the coupled level set volume of fluid method [91], and adaptive gridding techniques such as the octree approach to free surface flows proposed in [56].
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 60 (6 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...