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A Fast Iterative Method for Solving the Eikonal Equation on Triangulated Surfaces
2011
SIAM Journal on Scientific Computing
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In this paper we extend the fast iterative method to solve Eikonal equations efficiently on triangulated domains on the CPU and on parallel architectures, including graphics processors. ...
Comput., 30 (2008), pp. 2512-2534], the authors proposed the fast iterative method (FIM) to efficiently solve the Eikonal equation on regular grids. ...
The dragon model is provided by the Stanford University Computer Graphics Laboratory. ...
doi:10.1137/100788951
pmid:22641200
pmcid:PMC3360588
fatcat:xco5som6irdvngc45n3p37lkoa
Fast Sweeping Methods for Eikonal Equations on Triangular Meshes
2007
SIAM Journal on Numerical Analysis
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The original fast sweeping method, which is an efficient iterative method for stationary Hamilton-Jacobi equations, relies on natural ordering provided by a rectangular mesh. ...
The computational complexity of the new algorithm is nearly optimal in the sense that the total computational cost consists of O(M ) flops for iteration steps and O(M logM ) flops for sorting at the predetermined ...
In this work, we design a class of fast sweeping methods on triangulated domains for the eikonal equation of the following form: |∇T (x)| = f (x), x ∈ Ω \ Γ, T (x) = g(x), x ∈ Γ ⊂ Ω, (1.1) where f (x ...
doi:10.1137/050627083
fatcat:s262h64banf4ha2xek5z57enpe
A Fast-Marching Approach to Cardiac Electrophysiology Simulation for XMR Interventional Imaging
[chapter]
2005
Lecture Notes in Computer Science
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In this paper, we have proposed a fast algorithm to solve Eikonal-based models on triangular meshes. ...
We propose a first approach to estimate these parameters, and have tested it on synthetic and real data derived using XMR imaging. ...
The static solving of Eikonal equations 2 and 3, first proposed by [5] with a Newton's method, requires to solve only one non-linear system of equations. ...
doi:10.1007/11566489_75
fatcat:rowxgsnb5zdtpeadl2fojw45p4
Fast methods for the Eikonal and related Hamilton- Jacobi equations on unstructured meshes
2000
Proceedings of the National Academy of Sciences of the United States of America
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The Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. ...
The scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a Dijkstra-like programming approach. ...
The Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. ...
doi:10.1073/pnas.090060097
pmid:10811874
pmcid:PMC18495
fatcat:2dhmyvponfcyjkjavk26ayrlra
An efficient solution to the eikonal equation on parametric manifolds
2004
Interfaces and free boundaries (Print)
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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 non-orthogonal coordinate system on the manifold by creating an appropriate numerical stencil. ...
Blayvas for helpful suggestions, M. Bronstein and A. Bronstein for the facial manifold metric data, and the anonymous reviewers for their good comments. ...
doi:10.4171/ifb/102
fatcat:lgtvri37unhclo6u6t7ae422xm
A Short- Time Beltrami Kernel for Smoothing Images and Manifolds
2007
IEEE Transactions on Image Processing
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In addition, the kernel works with equal ease on regular 2-D images and on images painted on parametric or triangulated manifolds. ...
The flow is implemented by "convolving" the image with a space dependent kernel in a similar fashion to the solution of the heat equation by a convolution with a Gaussian kernel. ...
ACKNOWLEDGMENT The authors would like to thank T. Treibitz and S. Rozenfeld for their assistance in the implementation of triangulated manifold smoothing. ...
doi:10.1109/tip.2007.894253
pmid:17547140
fatcat:bjfxfzuokra55fpizzfsensr4i
A Fast Sweeping Method for Static Convex Hamilton–Jacobi Equations
2007
Journal of Scientific Computing
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We develop a fast sweeping method for static Hamilton-Jacobi equations with convex Hamiltonians. Local solvers and fast sweeping strategies apply to structured and unstructured meshes. ...
With causality correctly enforced during sweepings numerical evidence indicates that the fast sweeping method converges in a finite number of iterations independent of mesh size. ...
ACKNOWLEDGMENTS The research of H.-K. Zhao is partially supported by ONR Grant #N00014-02-1-0090, DARPA Grant #N00014-02-1-0603 and Sloan Foundation Fellowship. The research of J. ...
doi:10.1007/s10915-006-9124-6
fatcat:c6eoejtoqrgmxngtn7anz6cu3q
Simple finite element-based computation of distance functions in unstructured grids
2007
International Journal for Numerical Methods in Engineering
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Therefore, we propose an extension of the fast marching method for computing a distance field in a finite element context employing the element interpolation to hold the Eikonal property ( ∇ = 1). ...
In this work the authors considered a stabilized space-time Galerkin least-squares finite element method with shock capturing to solve the Hamilton-Jacobi and Eikonal equations. ...
They have shown, that in the finite element simulation of twophase incompressible flow with surface tension, FMM outperforms methods based on pseudo-time stepping of the Eikonal equation. ...
doi:10.1002/nme.2079
fatcat:ohhqtwghbfewzilj5k7tvpyqiy
Ordered upwind methods for static Hamilton-Jacobi equations
2001
Proceedings of the National Academy of Sciences of the United States of America
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Our fast methods avoid iteration through a careful use of information about the characteristic directions of the underlying partial differential equation. ...
We introduce a family of fast ordered upwind methods for approximating solutions to a wide class of static Hamilton-Jacobi equations with Dirichlet boundary conditions. ...
Since the latter equation is Eikonal, the Fast Marching Method can be used to solve it efficiently. ...
doi:10.1073/pnas.201222998
pmid:11572970
pmcid:PMC58684
fatcat:zext7io3nrakfexhwsbc2bzjfe
Geodesic Remeshing Using Front Propagation
2006
International Journal of Computer Vision
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A geodesic Delaunay triangulation of the set of points is then created, using a Voronoi diagram construction based on Fast Marching. ...
Abstract In this paper, we present a method for remeshing triangulated manifolds by using geodesic path calculations and distance maps. ...
A similar algorithm was also proposed in [21] . Fast Marching enables solving the nonlinear Eikonal equation: ¡ ¤ ! " $ # & % ¡ ! ...
doi:10.1007/s11263-006-6859-3
fatcat:44enbr3mwbdnpbfbh7d4bggfea
Equivalent extensions of Hamilton-Jacobi-Bellman equations on hypersurfaces
[article]
2020
arXiv
pre-print
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This framework also provides a systematic way for solving HJB equations on the unstructured point clouds that are sampled from the surface. ...
With this framework, one can easily use existing (high order) numerical methods developed on Cartesian grids to solve HJB equations on surfaces, with a computational cost that scales with the dimension ...
Tsai also thanks National Center for Theoretical Study, Taipei for hosting his visits, in which some of the ideas presented in this paper originated. ...
arXiv:1903.11173v2
fatcat:dfehsqyi2rejvao3443ht5uh4e
A Two Stage CVT / Eikonal Convection Mesh Deformation Approach for Large Nodal Deformations
[article]
2014
arXiv
pre-print
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The ray direction is computed by solving an Eikonal equation. ...
The methodology is available as a FEniCS shape optimization add-on at http://www.mathematik.uni-wuerzburg.de/~schmidt/femorph. ...
The above equation is solved using 2nd order SUPG-Finite elements. The non-linearity in the Eikonal equation is solved via Newton's method. ...
arXiv:1411.7663v1
fatcat:z3mfxhlp5fb7lmdpyd5eok42ne
An adjoint state method for three-dimensional transmission traveltime tomography using first-arrivals
2006
Communications in Mathematical Sciences
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We start from the eikonal equation, define a mismatching functional and derive the gradient of the nonlinear functional by an adjoint state method. ...
The resulting forward and adjoint problems can be efficiently solved by using the fast sweeping method; a limited memory BFGS method is used to drive the mismatching functional to zero with quadratic convergence ...
This work was finished when the second author was visiting the Center for Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, and he is grateful to the Director ...
doi:10.4310/cms.2006.v4.n1.a10
fatcat:qbq7gownczafbp23g2je4v3ozm
Legendre-transform-based fast sweeping methods for static Hamilton–Jacobi equations on triangulated meshes
2008
Journal of Computational Physics
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The algorithm is a general extension of the previous proposed algorithm by Kao et al. [C.Y. Kao, S.J. Osher, Y.-H. Tsai, Fast sweeping method for static Hamilton-Jacobi equations, SIAM J. Numer. ...
We propose a new sweeping algorithm which utilizes the Legendre transform of the Hamiltonian on triangulated meshes. ...
In [26] the fast sweeping method was systematically analyzed for eikonal equations. ...
doi:10.1016/j.jcp.2008.08.016
fatcat:4wsr3edsengqzdgdyitvsdje6y
Parallel algorithms for approximation of distance maps on parametric surfaces
2008
ACM Transactions on Graphics
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We present an efficient O(n) numerical algorithm for first-order approximation of geodesic distances on parametric surfaces, where n is the number of points on the surface. ...
The structure of our algorithm allows efficient implementation on parallel architectures. Two implementations on a SIMD processor and on a GPU are discussed. ...
do Update (v) A distance map on the surface is computed by solving the eikonal equation, expressed in our notation as ∇ G t 2 = ∇ T u t G(u) −1 ∇ u t = 1 (4) on a discrete grid obtained by sampling the ...
doi:10.1145/1409625.1409626
fatcat:3cl3hd256vas5lvtkm4jgarwve
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