## An efficient solution to the eikonal equation on parametric manifolds (2004)

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Venue: | INTERFACES AND FREE BOUNDARIES 6 (2004), 315–327 |

Citations: | 28 - 16 self |

### BibTeX

@MISC{Spira04anefficient,

author = {Alon Spira and Ron Kimmel},

title = { An efficient solution to the eikonal equation on parametric manifolds },

year = {2004}

}

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### Abstract

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. 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.

### Citations

976 | Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations
- Osher, Sethian
- 1988
(Show Context)
Citation Context ...025 2 . Assuming that v is of the form v = u n +Ch r +O(h r+1 ), where h = 1 , the order of accuracy of the numerical scheme according to the Lk n−1 norm at grid size n2 can be estimated according to =-=[11]-=- r n � n ek k = log2 e2n � . (12) k size: 17 2 33 2 65 2 129 2 257 2 513 2 e n 2: 6.2 · 10 −3 2.3 · 10 −3 8.4 · 10 −4 3.0 · 10 −4 4.1 · 10 −5 5.7 · 10 −6 r n 2 : 1.43 1.45 1.50 2.86 2.85 e n ∞: 0.4425... |

466 | fast marching level set method for monotonically advancing fronts
- SETHIAN, “A
- 1996
(Show Context)
Citation Context ...eights are given by the scalar positive function F (x, y). 1sEfficient solutions to the Eikonal equation on the plane parameterized by a regular (orthogonal) numerical grid were introduced by Sethian =-=[12]-=- and by Tsitsiklis [20]. Sethian’s fast marching method was extended by Kimmel and Sethian [7] to the solution of the Eikonal equation on triangulated manifolds �∇Mφ� = F, (2) with M the manifold and ... |

316 | Efficient algorithms for globally optimal trajectories
- Tsitsiklis
- 1995
(Show Context)
Citation Context ... scalar positive function F (x, y). 1sEfficient solutions to the Eikonal equation on the plane parameterized by a regular (orthogonal) numerical grid were introduced by Sethian [12] and by Tsitsiklis =-=[20]-=-. Sethian’s fast marching method was extended by Kimmel and Sethian [7] to the solution of the Eikonal equation on triangulated manifolds �∇Mφ� = F, (2) with M the manifold and ∇Mφ the gradient on the... |

223 | Computing geodesic paths on manifolds
- Kimmel, Sethian
- 1998
(Show Context)
Citation Context ... equation on the plane parameterized by a regular (orthogonal) numerical grid were introduced by Sethian [12] and by Tsitsiklis [20]. Sethian’s fast marching method was extended by Kimmel and Sethian =-=[7]-=- to the solution of the Eikonal equation on triangulated manifolds �∇Mφ� = F, (2) with M the manifold and ∇Mφ the gradient on the manifold. This extension enables a fast calculation of geodesic paths ... |

188 | Framework for low level vision
- Sochen, Malladi
- 1998
(Show Context)
Citation Context ...ny applications. In this section we demonstrate its use in the areas of image processing and computer vision. The first application consists of the acceleration of the image enhancing Beltrami filter =-=[6, 16]-=- by using a short time kernel [18]. Calculating the kernel requires the solution to the Eikonal equation on the image manifold. The second application is the implementation 13sof face recognition [1] ... |

124 |
On Bending Invariant Signatures for Surfaces
- Elad, Kimmel
(Show Context)
Citation Context ...ernel [18]. Calculating the kernel requires the solution to the Eikonal equation on the image manifold. The second application is the implementation 13sof face recognition [1] by geometric invariants =-=[4, 5]-=- without reconstruction of the facial surface [2]. In this case a signature of the face is computed from geodesic distances between points on the facial manifold. The geodesic distances are calculated... |

98 |
Ordered upwind methods for static Hamilton-Jacobi equations: Theory and algorithms
- 15Sethian, Vladimirsky
(Show Context)
Citation Context ...manifold and ∇Mφ the gradient on the manifold. This extension enables a fast calculation of geodesic paths [7], Voronoi diagrams, and offsets [8, 9] on triangulated manifolds. Sethian and Vladimirsky =-=[15]-=- presented Ordered Upwind Methods (OUM) for static Hamilton-Jacobi equations. These methods enable the solution of equations where the directions of the characteristics are different from those of the... |

96 | Fast sweeping algorithms for a class of Hamilton-Jacobi equations
- Tsai, Cheng, et al.
(Show Context)
Citation Context ...s of the characteristics are different from those of the gradients of φ. As an example, they demonstrate the solution of the Eikonal equation for manifolds which are function graphs. Also Tsai et al. =-=[19]-=- solved the equation on function graphs, but they used an iterative sweeping method. A similar sweeping approach was previously used by Danielson [3] to compute Euclidean distance maps on flat domains... |

92 | Expression-invariant 3D face recognition
- Bronstein, Bronstein, et al.
- 2003
(Show Context)
Citation Context ... 16] by using a short time kernel [18]. Calculating the kernel requires the solution to the Eikonal equation on the image manifold. The second application is the implementation 13sof face recognition =-=[1]-=- by geometric invariants [4, 5] without reconstruction of the facial surface [2]. In this case a signature of the face is computed from geodesic distances between points on the facial manifold. The ge... |

69 | Optimal algorithm for shape from shading and path planning
- Kimmel, Sethian
(Show Context)
Citation Context ...ation on triangulated manifolds �∇Mφ� = F, (2) with M the manifold and ∇Mφ the gradient on the manifold. This extension enables a fast calculation of geodesic paths [7], Voronoi diagrams, and offsets =-=[8, 9]-=- on triangulated manifolds. Sethian and Vladimirsky [15] presented Ordered Upwind Methods (OUM) for static Hamilton-Jacobi equations. These methods enable the solution of equations where the direction... |

51 | G.: Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces
- Memoli, Sapiro
- 2001
(Show Context)
Citation Context ..., but they used an iterative sweeping method. A similar sweeping approach was previously used by Danielson [3] to compute Euclidean distance maps on flat domains with regular grids. Mémoli and Sapiro =-=[10]-=- calculated distances on implicit manifolds by using orthogonal fast marching in a thin offset band surrounding the manifold. We present here an efficient solution to the Eikonal equation on parametri... |

42 |
Fast methods for the eikonal and related hamilton-jacobi equations on ustructured meshes
- Sethian, Vladimirsky
- 2000
(Show Context)
Citation Context ...he presented method is first order accurate as that of Kimmel and Sethian, but may be extended to higher orders by using Sethian and Vladimirsky’s higher order directional derivative � approximations =-=[14]-=-. The error of Mémoli and Sapiro’s method √h� is o . The derivatives of X with respect to u i are defined as Xi � ∂X ∂u i , and they constitute a non-orthogonal coordinate system on the parametric man... |

41 |
Level Set Methods and Fast Marching Methods. Cambridge Monograph on Applied and Computational Mathematics
- Sethian
- 1999
(Show Context)
Citation Context ... then φ(C) = min{φ(C), t + φ(A)}. cos θ Else, φ(C) = min{φ(C), bF + φ(A), aF + φ(B)}. 5 Marching on Manifolds After the pre-processing stage, the Eikonal equation is solved by the following algorithm =-=[13]-=-. Initialization: • The initial points are defined as Accepted and given their initial values. • All the other grid points are defined as Far and given the value infinity. Iterations: 1. Far ‘neighbor... |

38 |
Bending invariant representations for surfaces
- Elad, Kimmel
- 2001
(Show Context)
Citation Context ...ernel [18]. Calculating the kernel requires the solution to the Eikonal equation on the image manifold. The second application is the implementation 13sof face recognition [1] by geometric invariants =-=[4, 5]-=- without reconstruction of the facial surface [2]. In this case a signature of the face is computed from geodesic distances between points on the facial manifold. The geodesic distances are calculated... |

24 |
Euclidean distance mapping
- Danielson
- 1980
(Show Context)
Citation Context ...folds which are function graphs. Also Tsai et al. [19] solved the equation on function graphs, but they used an iterative sweeping method. A similar sweeping approach was previously used by Danielson =-=[3]-=- to compute Euclidean distance maps on flat domains with regular grids. Mémoli and Sapiro [10] calculated distances on implicit manifolds by using orthogonal fast marching in a thin offset band surrou... |

17 | Image processing via the Beltrami operator
- Kimmel, R, et al.
- 1998
(Show Context)
Citation Context ...ny applications. In this section we demonstrate its use in the areas of image processing and computer vision. The first application consists of the acceleration of the image enhancing Beltrami filter =-=[6, 16]-=- by using a short time kernel [18]. Calculating the kernel requires the solution to the Eikonal equation on the image manifold. The second application is the implementation 13sof face recognition [1] ... |

12 | Face recognition from facial surface metric,” presented at the Eur - Bronstein, Bronstein, et al. - 2004 |

9 | Fast voronoi diagrams and offsets on triangulated surfaces
- Kimmel, Sethian
- 1999
(Show Context)
Citation Context ...ation on triangulated manifolds �∇Mφ� = F, (2) with M the manifold and ∇Mφ the gradient on the manifold. This extension enables a fast calculation of geodesic paths [7], Voronoi diagrams, and offsets =-=[8, 9]-=- on triangulated manifolds. Sethian and Vladimirsky [15] presented Ordered Upwind Methods (OUM) for static Hamilton-Jacobi equations. These methods enable the solution of equations where the direction... |

5 | 3D face recognition without facial surface reconstruction
- Bronstein, Bronstein, et al.
(Show Context)
Citation Context ...tion to the Eikonal equation on the image manifold. The second application is the implementation 13sof face recognition [1] by geometric invariants [4, 5] without reconstruction of the facial surface =-=[2]-=-. In this case a signature of the face is computed from geodesic distances between points on the facial manifold. The geodesic distances are calculated from the surface metric using our method. 7.1 A ... |

5 | Efficient Beltrami flow using a short time kernel
- Spira, Kimmel, et al.
- 2003
(Show Context)
Citation Context ...emonstrate its use in the areas of image processing and computer vision. The first application consists of the acceleration of the image enhancing Beltrami filter [6, 16] by using a short time kernel =-=[18]-=-. Calculating the kernel requires the solution to the Eikonal equation on the image manifold. The second application is the implementation 13sof face recognition [1] by geometric invariants [4, 5] wit... |

5 | Efficient beltrami flow using a short time kernel - Spira, Kimmel, et al. |

3 | R.: An Ecient Solution to the Eikonal Equation on Parametric Manifolds - Spira, Kimmel - 2004 |

1 | Fast voronoi diagrams and osets on triangulated surfaces - Kimmel, Sethian - 1999 |