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226
A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2002
"... We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is also a generalization of an active contour model without edges based 2phase segmentation, developed by ..."
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Cited by 498 (22 self)
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We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is also a generalization of an active contour model without edges based 2phase segmentation, developed by the authors earlier in T. Chan and L. Vese (1999. In ScaleSpace'99, M. Nilsen et al. (Eds.), LNCS, vol. 1682, pp. 141151) and T. Chan and L. Vese (2001. IEEEIP, 10(2):266277). The multiphase level set formulation is new and of interest on its own: by construction, it automatically avoids the problems of vacuum and overlap; it needs only log n level set functions for n phases in the piecewise constant case; it can represent boundaries with complex topologies, including triple junctions; in the piecewise smooth case, only two level set functions formally suffice to represent any partition, based on The FourColor Theorem. Finally, we validate the proposed models by numerical results for signal and image denoising and segmentation, implemented using the Osher and Sethian level set method.
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.
Filling Holes In Complex Surfaces Using Volumetric Diffusion
, 2001
"... We address the problem of building watertight 3D models from surfaces that contain holesfor example, sets of range scans that observe most but not all of a surface. We specifically address situations in which the holes are too geometrically and topologically complex to fill using triangulation al ..."
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Cited by 172 (2 self)
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We address the problem of building watertight 3D models from surfaces that contain holesfor example, sets of range scans that observe most but not all of a surface. We specifically address situations in which the holes are too geometrically and topologically complex to fill using triangulation algorithms. Our solution begins by constructing a signed distance function, the zero set of which defines the surface. Initially, this function is defined only in the vicinity of observed surfaces. We then apply a diffusion process to extend this function through the volume until its zero set bridges whatever holes may be present. If additional information is available, such as knownempty regions of space inferred from the lines of sight to a 3D scanner, it can be incorporated into the diffusion process. Our algorithm is simple to implement, is guaranteed to produce manifold noninterpenetrating surfaces, and is efficient to run on large datasets because computation is limited to areas near holes. By showing results for complex range scans, we demonstrate that our algorithm produces holefree surfaces that are plausible, visually acceptable, and usually close to the intended geometry.
Fast surface reconstruction using the level set method
 In VLSM ’01: Proceedings of the IEEE Workshop on Variational and Level Set Methods
, 2001
"... In this paper we describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from scattered data ..."
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Cited by 151 (12 self)
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In this paper we describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from scattered data set. The data set might consist of points, curves and/or surface patches. A weighted minimal surfacelike model is constructed and its variational level set formulation is implemented with optimal efficiency. The reconstructed surface is smoother than piecewise linear and has a natural scaling in the regularization that allows varying flexibility according to the local sampling density. As is usual with the level set method we can handle complicated topology and deformations, as well as noisy or highly nonuniform data sets easily. The method is based on a simple rectangular grid, although adaptive and triangular grids are also possible. Some consequences, such as hole filling capability, are demonstrated, as well as the viability and convergence of our new fast tagging algorithm.
Ordered Upwind Methods for Static HamiltonJacobi Equations: Theory and Algorithms
, 2003
"... We develop a family of fast methods for approximating the solutions to a wide class of static Hamilton–Jacobi PDEs; these fast methods include both semiLagrangian and fully Eulerian versions. Numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear ..."
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Cited by 136 (9 self)
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We develop a family of fast methods for approximating the solutions to a wide class of static Hamilton–Jacobi PDEs; these fast methods include both semiLagrangian and fully Eulerian versions. Numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations. Our techniques, which we refer to as “Ordered Upwind Methods” (OUMs), use partial information about the characteristic directions to decouple these nonlinear systems, greatly reducing the computational labor. Our techniques are considered in the context of controltheoretic and frontpropagation problems. We begin by discussing existing OUMs, focusing on those designed for isotropic problems. We then introduce a new class of OUMs which decouple systems for general (anisotropic) problems. We prove convergence of one such scheme to the viscosity solution of the corresponding Hamilton–Jacobi PDE. Next, we introduce a set of finitedifferences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems. The performance of the methods is analyzed, and computational experiments are performed using test problems from computational geometry and seismology.
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
Level set surface editing operators
 SIGGRAPH
, 2002
"... Figure 1: Surfaces edited with level set operators. Left: A damaged Greek bust model is repaired with a new nose, chin and sharpened hair. Right: A new model is constructed from models of a griffin and dragon (small figures), producing a twoheaded, winged dragon. We present a level set framework fo ..."
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Cited by 103 (10 self)
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Figure 1: Surfaces edited with level set operators. Left: A damaged Greek bust model is repaired with a new nose, chin and sharpened hair. Right: A new model is constructed from models of a griffin and dragon (small figures), producing a twoheaded, winged dragon. We present a level set framework for implementing editing operators for surfaces. Level set models are deformable implicit surfaces where the deformation of the surface is controlled by a speed function in the level set partial differential equation. In this paper we define a collection of speed functions that produce a set of surface editing operators. The speed functions describe the velocity at each point on the evolving surface in the direction of the surface normal. All of the information needed to deform a surface is encapsulated in the speed function, providing a simple, unified computational framework. The user combines predefined building blocks to create the desired speed function. The surface editing operators are quickly computed and may be applied both regionally and globally. The level set framework offers several advantages. 1) By construction, selfintersection cannot occur, which guarantees the generation of physicallyrealizable, simple, closed surfaces. 2) Level set models easily change topological genus, and 3) are free of the edge connectivity and mesh quality problems associated with mesh models. We present five examples of surface editing operators: blending, smoothing, sharpening, openings/closings and embossing. We demonstrate their effectiveness on several scanned objects and scanconverted models.
Rapid and Accurate Computation of the Distance Function Using Grids
 J. Comput. Phys
, 2002
"... We present two fast and simple algorithms for approximating the distance function for given isolated points on uniform grids. The algorithms are then generalized to compute the distance to piecewise linear objects. By incorporating the geometry of Huygens ’ principle in the reverse order with the cl ..."
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Cited by 53 (3 self)
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We present two fast and simple algorithms for approximating the distance function for given isolated points on uniform grids. The algorithms are then generalized to compute the distance to piecewise linear objects. By incorporating the geometry of Huygens ’ principle in the reverse order with the classical viscosity solution theory for the eikonal equation ∇u=1, the algorithms become almost purely algebraic and yield very accurate approximations. The generalized closest point formulation used in the second algorithm provides a framework for further extension to compute the distance accurately to smooth geometric objects on different grid geometries, without the construction of the Voronoi diagrams. This method provides a fast and simple translator of data commonly given in computational geometry to the volumetric representation used in level set methods. c ○ 2002 Elsevier Science (USA) 1.
Level set based shape prior segmentation
 In Proc. CVPR’05
, 2005
"... We propose a level set based variational approach that incorporates shape priors into ChanVese’s model [3] for the shape prior segmentation problem. In our model, besides the level set function for segmentation, as in Cremers’ work [5], we introduce another labelling level set function to indicate ..."
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Cited by 52 (2 self)
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We propose a level set based variational approach that incorporates shape priors into ChanVese’s model [3] for the shape prior segmentation problem. In our model, besides the level set function for segmentation, as in Cremers’ work [5], we introduce another labelling level set function to indicate the regions on which the prior shape should be compared. Our model can segment an object, whose shape is similar to the given prior shape, from a background where there are several objects. Moreover, we provide a proof for a fast solution principle, which was mentioned [7] and similar to the one proposed in [19], for minimizing ChanVese’s segmentation model without length term. We extend the principle to the minimization of our prescribed functionals. 1.
Approximation with Active Bspline Curves and Surfaces
, 2002
"... An active contour model for parametric curve and surface approximation is presented. The active curve or surface adapts to the model shape to be approximated in an optimization algorithm. The quasiNewton optimization procedure in each iteration step minimizes a quadratic function which is built up ..."
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Cited by 49 (6 self)
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An active contour model for parametric curve and surface approximation is presented. The active curve or surface adapts to the model shape to be approximated in an optimization algorithm. The quasiNewton optimization procedure in each iteration step minimizes a quadratic function which is built up with help of local quadratic approximants of the squared distance function of the model shape and an internal energy which has a smoothing and regularization effect. The approach completely avoids the parametrization problem. We also show how to use a similar strategy for the solution of variational problems for curves on surfaces. Examples are the geodesic path connecting two points on a surface and interpolating or approximating spline curves on surfaces. Finally we indicate how the latter topic leads to the variational design of smooth motions which interpolate or approximate given positions. 1.