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142
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 1073 (43 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.
Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow
, 1999
"... In this paper, we develop methods to rapidly remove rough features from irregularly triangulated data intended to portray a smooth surface. The main task is to remove undesirable noise and uneven edges while retaining desirable geometric features. The problem arises mainly when creating highfidelit ..."
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Cited by 436 (25 self)
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In this paper, we develop methods to rapidly remove rough features from irregularly triangulated data intended to portray a smooth surface. The main task is to remove undesirable noise and uneven edges while retaining desirable geometric features. The problem arises mainly when creating highfidelity computer graphics objects using imperfectlymeasured data from the real world. Our approach contains three novel features: an implicit integration method to achieve efficiency, stability, and large timesteps; a scaledependent Laplacian operator to improve the diffusion process; and finally, a robust curvature flow operator that achieves a smoothing of the shape itself, distinct from any parameterization. Additional features of the algorithm include automatic exact volume preservation, and hard and soft constraints on the positions of the points in the mesh. We compare our method to previous operators and related algorithms, and prove that our curvature and Laplacian operators have several mathematicallydesirable qualities that improve the appearance of the resulting surface. In consequence, the user can easily select the appropriate operator according to the desired type of fairing. Finally, we provide a series of examples to graphically and numerically demonstrate the quality of our results.
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 218 (26 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...
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 168 (24 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...
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 higher ..."
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Cited by 159 (11 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 145 (4 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,...
Essentially nonoscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws
, 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
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Cited by 125 (17 self)
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In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di erence schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically selfcontained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code
HamiltonJacobi Skeletons
, 1999
"... The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shapefromshading and for recent dynamic theories of shape. Its numerical simulation can be delicate, ..."
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Cited by 119 (12 self)
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The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shapefromshading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front and is typically based on level set methods. However, there are more classical approaches rooted in Hamiltonian physics which have yet to be widely used by the computer vision community. In this paper we review the Hamiltonian formulation, which offers specific advantages when it comes to the detection of singularities or shocks. We specialize to the case of Blum's grass fire flow and measure the average outward ux of the vector field that underlies the Hamiltonian system. This measure has very different limiting behaviors depending upon whether the region over which it is computed shrinks to a singular point or a nonsingular one. Hence, it is an effective way to distinguish between these two cases. We combine the ux measurement with a homotopy preserving thinning process applied in a discrete lattice. This leads to a robust and accurate algorithm for computing skeletons in 2D as well as 3D, which has low computational complexity. We illustrate the approach with several computational examples.
Controllers for Reachability Specifications for Hybrid Systems
 Automatica
, 1999
"... The problem of systematically synthesizing hybrid controllers which satisfy multiple control objectives is considered. We present a technique, based on the principles of optimal control, for determining the class of least restrictive controllers that satisfies the most important objective (which we ..."
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Cited by 117 (37 self)
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The problem of systematically synthesizing hybrid controllers which satisfy multiple control objectives is considered. We present a technique, based on the principles of optimal control, for determining the class of least restrictive controllers that satisfies the most important objective (which we refer to as safety). The system performance with respect to lower priority objectives (which we refer to as efficiency) can then be optimized within this class. We motivate our approach by showing how the proposed synthesis technique simplifies to well known results from supervisory control and pursuit evasion games when restricted to purely discrete and purely continuous systems respectively. We then illustrate the application of this technique to two examples, one hybrid (the steam boiler benchmark problem), and one primarily continuous (a flight vehicle management system with discrete flight modes). 1 Introduction Hybrid systems, or systems that involve the interaction of discrete and co...
Analysis of Planar Shapes Using Geodesic Paths on Shape Spaces
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2004
"... For analyzing shapes of planar, closed curves, we propose di#erential geometric representations of curves using their direction functions and curvature functions. Shapes are represented as elements of infinitedimensional spaces and their pairwise di#erences are quantified using the lengths of ge ..."
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Cited by 112 (17 self)
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For analyzing shapes of planar, closed curves, we propose di#erential geometric representations of curves using their direction functions and curvature functions. Shapes are represented as elements of infinitedimensional spaces and their pairwise di#erences are quantified using the lengths of geodesics connecting them on these spaces. We use a Fourier basis to represent tangents to the shape spaces and then use a gradientbased shooting method to solve for the tangent that connects any two shapes via a geodesic.