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631
Geodesic Active Contours
, 1997
"... A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both in ..."
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Cited by 1425 (47 self)
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A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical “snakes ” based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.
Computing Geodesic Paths on Manifolds
 Proc. Natl. Acad. Sci. USA
, 1998
"... The Fast Marching Method [8] 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. In this paper we extend the Fast Marching Method to triangulated domains with the same computational complexity. A ..."
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Cited by 294 (28 self)
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The Fast Marching Method [8] 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. In this paper we extend the Fast Marching Method to triangulated domains with the same computational complexity. As an application, we provide an optimal time algorithm for computing the geodesic distances and thereby extracting shortest paths on triangulated manifolds. 1 Introduction Sethian`s Fast Marching Method [8], 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 in the domain. The technique hinges on producing numerically consistent approximations to the operators in the Eikonal equation that select the correct viscosity solution; this is done through the use of upwind nite dierence operators. The structure of this upwinding is then used to systematically construct the solution to the Eik...
Geodesic Active Contours and Level Sets for the Detection and Tracking of Moving Objects
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2000
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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 266 (26 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...
Global Minimum for Active Contour Models: A Minimal Path Approach
, 1997
"... A new boundary detection approach for shape modeling is presented. It detects the global minimum of an active contour model’s energy between two end points. Initialization is made easier and the curve is not trapped at a local minimum by spurious edges. We modify the “snake” energy by including the ..."
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Cited by 238 (70 self)
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A new boundary detection approach for shape modeling is presented. It detects the global minimum of an active contour model’s energy between two end points. Initialization is made easier and the curve is not trapped at a local minimum by spurious edges. We modify the “snake” energy by including the internal regularization term in the external potential term. Our method is based on finding a path of minimal length in a Riemannian metric. We then make use of a new efficient numerical method to find this shortest path. It is shown that the proposed energy, though based only on a potential integrated along the curve, imposes a regularization effect like snakes. We explore the relation between the maximum curvature along the resulting contour and the potential generated from the image. The method is capable to close contours, given only one point on the objects’ boundary by using a topologybased saddle search routine. We show examples of our method applied to real aerial and medical images.
The Fast Construction of Extension Velocities in Level Set Methods
 Journal of Computational Physics
, 1997
"... Level set techniques are numerical techniques for tracking the evolution of interfaces. They rely on two central embeddings; rst the embedding of the interface as the zero level set of a higher dimensional function, and second, the embedding (or extension) of the interface's velocity to this hi ..."
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Cited by 218 (12 self)
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Level set techniques are numerical techniques for tracking the evolution of interfaces. They rely on two central embeddings; rst the embedding of the interface as the zero level set of a higher dimensional function, and second, the embedding (or extension) of the interface's velocity to this higher dimensional level set function. This paper applies Sethian's Fast Marching Method, which is a very fast technique for solving the Eikonal and related equations, to the problem of building fast and appropriate extension velocities for the neighboring level sets. Our choice and construction of extension velocities serves several purposes. First, it provides a way of building velocities for neighboring level sets in the cases where the velocity is de ned only on the front itself. Second, it provides a subgrid resolution in some cases not present in the standard level set approach. Third, it provides a way to update an interface according to a given velocity eld prescribed on the front in suc...
Fast Marching Methods
 SIAM Review
, 1998
"... Fast Marching Methods are numerical schemes for computing solutions to the nonlinear Eikonal equation and related static HamiltonJacobi equations. Based on entropysatisfying upwind schemes and fast sorting techniques, they yield consistent, accurate, and highly efficient algorithms. They are opti ..."
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Cited by 215 (5 self)
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Fast Marching Methods are numerical schemes for computing solutions to the nonlinear Eikonal equation and related static HamiltonJacobi equations. Based on entropysatisfying upwind schemes and fast sorting techniques, they yield consistent, accurate, and highly efficient algorithms. They are optimal in the sense that the computational complexity of the algorithms is O(N log N ), where N is the total number of points in the domain. The schemes are of use in a variety of applications, including problems in shape offsetting, computing distances from complex curves and surfaces, shapefromshading, photolithographic development, computing rst arrivals in seismic travel times, construction of shortest geodesics on surfaces, optimal path planning around obstacles, and visibility and reection calculations. In this paper, we review the development of these techniques, including the theoretical and numerical underpinnings, provide details of the computational schemes including higher order versions,...
A Hybrid Particle Level Set Method for Improved Interface Capturing
 J. Comput. Phys
, 2002
"... In this paper, we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field. Our method uses Lagrangian marker particles to rebuild the level set in regions which are underresolved. This is ofte ..."
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Cited by 215 (25 self)
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In this paper, we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field. Our method uses Lagrangian marker particles to rebuild the level set in regions which are underresolved. This is often the case for flows undergoing stretching and tearing. The overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method. Our method compares favorably with volume of fluid methods in the conservation of mass and purely Lagrangian schemes for interface resolution. The method is presented in three spatial dimensions.
Simulating Water and Smoke with an Octree Data Structure
, 2004
"... We present a method for simulating water and smoke on an unrestricted octree data structure exploiting mesh refinement techniques to capture the small scale visual detail. We propose a new technique for discretizing the Poisson equation on this octree grid. The resulting linear system is symmetric ..."
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Cited by 210 (18 self)
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We present a method for simulating water and smoke on an unrestricted octree data structure exploiting mesh refinement techniques to capture the small scale visual detail. We propose a new technique for discretizing the Poisson equation on this octree grid. The resulting linear system is symmetric positive definite enabling the use of fast solution methods such as preconditioned conjugate gradients, whereas the standard approximation to the Poisson equation on an octree grid results in a nonsymmetric linear system which is more computationally challenging to invert. The semiLagrangian characteristic tracing technique is used to advect the velocity, smoke density, and even the level set making implementation on an octree straightforward. In the case of smoke, we have multiple refinement criteria including object boundaries, optical depth, and vorticity concentration. In the case of water, we refine near the interface as determined by the zero isocontour of the level set function.
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.