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136
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
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.
Transport and diffusion of material quantities on propagating interfaces via level set methods
 Journal of Computational Physics
, 2003
"... We develop theory and numerical algorithms to apply level set methods to problems involving the transport and diffusion of material quantities in a level set framework. Level set methods are computational techniques for tracking moving interfaces; they work by embedding the propagating interface as ..."
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Cited by 38 (4 self)
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We develop theory and numerical algorithms to apply level set methods to problems involving the transport and diffusion of material quantities in a level set framework. Level set methods are computational techniques for tracking moving interfaces; they work by embedding the propagating interface as the zero level set of a higher dimensional function, and then approximate the solution of the resulting initial value partial differential equation using upwind finite difference schemes. The traditional level set method works in the trace space of the evolving interface, and hence disregards any parameterization in the interface description. Consequently, material quantities on the interface which themselves are transported under the interface motion are not easily handled in this framework. We develop model equations and algorithmic techniques to extend the level set method to include these problems. We demonstrate the accuracy of our approach through a series of test examples and convergence studies. 1
A Toolbox of HamiltonJacobi Solvers for Analysis of Nondeterministic Continuous and Hybrid Systems
 In HSCC 2005, LNCS 3414
, 2005
"... Submitted to HSCC 2005. Please do not redistribute Abstract. HamiltonJacobi partial differential equations have many applications in the analysis of nondeterministic continuous and hybrid systems. Unfortunately, analytic solutions are seldom available and numerical approximation requires a great de ..."
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Cited by 36 (3 self)
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Submitted to HSCC 2005. Please do not redistribute Abstract. HamiltonJacobi partial differential equations have many applications in the analysis of nondeterministic continuous and hybrid systems. Unfortunately, analytic solutions are seldom available and numerical approximation requires a great deal of programming infrastructure. In this paper we describe the first publicly available toolbox for approximating the solution of such equations, and discuss three examples of how these equations can be used in systems analysis: cost to go, stochastic differential games, and stochastic hybrid systems. For each example we briefly summarize the relevant theory, describe the toolbox implementation, and provide results. 1
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.
Finsler Active Contours
, 2007
"... In this paper, we propose an image segmentation technique based on augmenting the conformal (or geodesic) active contour framework with directional information. In the isotropic case, the Euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the wei ..."
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Cited by 27 (6 self)
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In this paper, we propose an image segmentation technique based on augmenting the conformal (or geodesic) active contour framework with directional information. In the isotropic case, the Euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest (typically edges) is small. The conformal factor which is chosen depends only upon position and is in this sense isotropic. While directional information has been studied previously for other segmentation frameworks, here we show that if one desires to add directionality in the conformal active contour framework, then one gets a welldefined minimization problem in the case that the factor defines a Finsler metric. Optimal curves may be obtained using the calculus of variations or dynamic programming based schemes. Finally we demonstrate the technique by extracting roads from aerial imagery, blood vessels from medical angiograms, and neural tracts from diffusionweighted magnetic resonance imagery.
A FAST ITERATIVE METHOD FOR EIKONAL EQUATIONS
, 2008
"... In this paper we propose a novel computational technique to solve the Eikonal equation efficiently on parallel architectures. The proposed method manages the list of active nodes and iteratively updates the solutions on those nodes until they converge. Nodes are added to or removed from the list ba ..."
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Cited by 25 (6 self)
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In this paper we propose a novel computational technique to solve the Eikonal equation efficiently on parallel architectures. The proposed method manages the list of active nodes and iteratively updates the solutions on those nodes until they converge. Nodes are added to or removed from the list based on a convergence measure, but the management of this list does not entail an extra burden of expensive ordered data structures or special updating sequences. The proposed method has suboptimal worstcase performance but, in practice, on real and synthetic datasets, runs faster than guaranteedoptimal alternatives. Furthermore, the proposed method uses only local, synchronous updates and therefore has better cache coherency, is simple to implement, and scales efficiently on parallel architectures. This paper describes the method, proves its consistency, gives a performance analysis that compares the proposed method against the stateoftheart Eikonal solvers, and describes the implementation on a single instruction multiple datastream (SIMD) parallel architecture.
N.: An anisotropic multifront fast marching method for realtime simulation of cardiac electrophysiology
 In: FIMH’07. Volume 4466 of LNCS
, 2007
"... Abstract. Cardiac arrhythmias can develop complex electrophysiological patterns which complexify the planning and control of therapies, especially in the context of radiofrequency ablation. The development of electrophysiology models aims at testing different therapy strategies. However, current mo ..."
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Cited by 25 (21 self)
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Abstract. Cardiac arrhythmias can develop complex electrophysiological patterns which complexify the planning and control of therapies, especially in the context of radiofrequency ablation. The development of electrophysiology models aims at testing different therapy strategies. However, current models are computationally expensive and often too complex to be adjusted with limited clinical data. In this paper, we propose a realtime method to simulate cardiac electrophysiology on triangular meshes. This model is based on a multifront integration of the Fast Marching Method. This efficient approach opens new possibilities, including the ability to directly integrate modelling in the interventional room. 1
Control Theory and Fast Marching Techniques for Brain Connectivity Mapping
 Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern RecognitionVolume
, 2006
"... We propose a novel, fast and robust technique for the computation of anatomical connectivity in the brain. Our approach exploits the information provided by Diffusion Tensor Magnetic Resonance Imaging (or DTI) and models the white matter by using Riemannian geometry and control theory. We show that ..."
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Cited by 24 (3 self)
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We propose a novel, fast and robust technique for the computation of anatomical connectivity in the brain. Our approach exploits the information provided by Diffusion Tensor Magnetic Resonance Imaging (or DTI) and models the white matter by using Riemannian geometry and control theory. We show that it is possible, from a region of interest, to compute the geodesic distance to any other point and the associated optimal vector field. The latter can be used to trace shortest paths coinciding with neural fiber bundles. We also demonstrate that no explicit computation of those 3D curves is necessary to assess the degree of connectivity of the region of interest with the rest of the brain. We finally introduce a general local connectivity measure whose statistics along the optimal paths may be used to evaluate the degree of connectivity of any pair of voxels. All those quantities can be computed simultaneously in a Fast Marching framework, directly yielding the connectivity maps. Apart from being extremely fast, this method has other advantages such as the strict respect of the convoluted geometry of white matter, the fact that it is parameterfree, and its robustness to noise. We illustrate our technique by showing results on real and synthetic datasets. Our GCM (Geodesic Connectivity Mapping) algorithm is implemented in C++ and will be soon available on the web. 1.