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53
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.
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
3D distance fields: A survey of techniques and applications
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 2006
"... A distance field is a representation where, at each point within the field, we know the distance from that point to the closest point on any object within the domain. In addition to distance, other properties may be derived from the distance field, such as the direction to the surface, and when the ..."
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Cited by 74 (3 self)
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A distance field is a representation where, at each point within the field, we know the distance from that point to the closest point on any object within the domain. In addition to distance, other properties may be derived from the distance field, such as the direction to the surface, and when the distance field is signed, we may also determine if the point is internal or external to objects within the domain. The distance field has been found to be a useful construction within the areas of computer vision, physics, and computer graphics. This paper serves as an exposition of methods for the production of distance fields, and a review of alternative representations and applications of distance fields. In the course of this paper, we present various methods from all three of the above areas, and we answer pertinent questions such as How accurate are these methods compared to each other? How simple are they to implement?, and What is the complexity and runtime of such methods?
Efficient Algorithms for Solving Static HamiltonJacobi Equations
, 2003
"... Consider the eikonal equation, = 1. If the initial condition is u = 0 on a manifold, then the solution u is the distance to the manifold. We present a new algorithm for solving this problem. More precisely, we present an algorithm for computing the closest point transform to an explicitly described ..."
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Cited by 69 (7 self)
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Consider the eikonal equation, = 1. If the initial condition is u = 0 on a manifold, then the solution u is the distance to the manifold. We present a new algorithm for solving this problem. More precisely, we present an algorithm for computing the closest point transform to an explicitly described manifold on a rectilinear grid in low dimensional spaces. The closest point transform finds the closest point on a manifold and the Euclidean distance to a manifold for all the points in a grid (or the grid points within a specified distance of the manifold). We consider manifolds composed of simple geometric shapes, such as, a set of points, piecewise linear curves or triangle meshes. The algorithm computes the closest point on and distance to the manifold by solving the eikonal equation = 1 by the method of characteristics. The method of characteristics is implemented efficiently with the aid of computational geometry and polygon/polyhedron scan conversion. Thus the method is named the characteristic/scan conversion algorithm. The computed distance is accurate to within machine precision. The computational complexity of the algorithm is linear in both the number of grid points and the complexity of the manifold. Thus it has optimal computational complexity. The algorithm is easily adapted to sharedmemory and distributedmemory concurrent algorithms. Many query problems...
Fast computation of weighted distance functions and geodesics on implicit hypersurfaces
 J. Comput. Phys
"... An algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hypersurfaces is introduced in this paper. The basic idea is to approximate the intrinsic weighted distance by the Euclidean weighted distance computed in a band surrounding the implicit h ..."
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Cited by 63 (8 self)
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An algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hypersurfaces is introduced in this paper. The basic idea is to approximate the intrinsic weighted distance by the Euclidean weighted distance computed in a band surrounding the implicit hypersurface in the embedding space, thereby performing all the computations in a Cartesian grid with classical and efficient numerics. Based on work on geodesics on Riemannian manifolds with boundaries, we bound the error between the two distance functions. We show that this error is of the same order as the theoretical numerical error in computationally optimal, Hamilton–Jacobibased, algorithms for computing distance functions in Cartesian grids. Therefore, we can use these algorithms, modified to deal with spaces with boundaries, and obtain also for the case of intrinsic distance functions on implicit hypersurfaces a computationally efficient technique. The approach can be extended to solve a more general class of Hamilton–Jacobi equations defined on the implicit surface, following the same idea of approximating their solutions by the solutions in the embedding Euclidean space. The framework here introduced thereby allows for the computations to be performed on a Cartesian grid with computationally optimal algorithms, in spite of the fact that the distance and Hamilton–Jacobi equations are intrinsic to the implicit hypersurface. c ○ 2001 Academic Press Key Words: implicit hypersurfaces; distance functions; geodesics; Hamilton– Jacobi equations; fast computations.
EnergyMinimizing Splines in Manifolds
, 2004
"... Variational interpolation in curved geometries has many applications, so there has always been demand for geometrically meaningful and efficiently computable splines in manifolds. We extend the definition of the familiar cubic spline curves and splines in tension, and we show how to compute these on ..."
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Cited by 57 (11 self)
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Variational interpolation in curved geometries has many applications, so there has always been demand for geometrically meaningful and efficiently computable splines in manifolds. We extend the definition of the familiar cubic spline curves and splines in tension, and we show how to compute these on parametric surfaces, level sets, triangle meshes, and point samples of surfaces. This list is more comprehensive than it looks, because it includes variational motion design for animation, and allows the treatment of obstacles via barrier surfaces. All these instances of the general concept are handled by the same geometric optimization algorithm, which minimizes an energy of curves on surfaces of arbitrary dimension and codimension.
A Fast Algorithm for Computing the Closest Point and Distance Transform
, 2000
"... This paper presents a new algorithm for computing the closest point transform to a manifold on a rectilinear grid in low dimensional spaces. The closest point transform nds the closest point on a manifold and the Euclidean distance to a manifold for all the points in a grid, (or the grid points ..."
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Cited by 42 (1 self)
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This paper presents a new algorithm for computing the closest point transform to a manifold on a rectilinear grid in low dimensional spaces. The closest point transform nds the closest point on a manifold and the Euclidean distance to a manifold for all the points in a grid, (or the grid points within a specied distance of the manifold). We consider manifolds composed of simple geometric shapes, such as, a set of points, piecewise linear curves or triangle meshes. The algorithm computes the closest point on and distance to the manifold by solving the Eikonal equation jruj = 1 by the method of characteristics. The method of characteristics is implemented eciently with the aid of computational geometry and polygon/polyhedron scan conversion. The computed distance is accurate to within machine precision. The computational complexity of the algorithm is linear in both the number of grid points and the complexity of the manifold. Thus it has optimal computational complexity. ...
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.
Registration without ICP
 Computer Vision and Image Understanding
, 2002
"... We present a new approach to the geometric alignment of a point cloud to a surface and to related registration problems. The standard algorithm is the familiar ICP algorithm. Here we provide an alternative concept which relies on instantaneous kinematics and on the geometry of the squared distance f ..."
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Cited by 28 (4 self)
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We present a new approach to the geometric alignment of a point cloud to a surface and to related registration problems. The standard algorithm is the familiar ICP algorithm. Here we provide an alternative concept which relies on instantaneous kinematics and on the geometry of the squared distance function of a surface. The proposed algorithm exhibits faster convergence than ICP