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298
Active Contours without Edges
, 2001
"... In this paper, we propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, MumfordShah functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by gradient. We minimize an energy ..."
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Cited by 804 (36 self)
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In this paper, we propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, MumfordShah functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by gradient. We minimize an energy which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a "meancurvature flow"like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. We will give a numerical algorithm using finite differences. Finally, we will present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.
Shape modeling with front propagation: A level set approach
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1995
"... Abstract Shape modeling is an important constituent of computer vision as well as computer graphics research. Shape models aid the tasks of object representation and recognition. This paper presents a new approach to shape modeling which retains some of the attractive features of existing methods ..."
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Cited by 631 (17 self)
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Abstract Shape modeling is an important constituent of computer vision as well as computer graphics research. Shape models aid the tasks of object representation and recognition. This paper presents a new approach to shape modeling which retains some of the attractive features of existing methods and overcomes some of their limitations. Our techniques can be applied to model arbitrarily complex shapes, which include shapes with significant protrusions, and to situations where no a priori assumption about the object’s topology is made. A single instance of our model, when presented with an image having more than one object of interest, has the ability to split freely to represent each object. This method is based on the ideas developed by Osher and Sethian to model propagating solidhiquid interfaces with curvaturedependent speeds. The interface (front) is a closed, nonintersecting, hypersurface flowing along its gradient field with constant speed or a speed that depends on the curvature. It is moved by solving a “HamiltonJacob? ’ type equation written for a function in which the interface is a particular level set. A speed term synthesizpd from the image is used to stop the interface in the vicinity of object boundaries. The resulting equation of motion is solved by employing entropysatisfying upwind finite difference schemes. We present a variety of ways of computing evolving front, including narrow bands, reinitializations, and different stopping criteria. The efficacy of the scheme is demonstrated with numerical experiments on some synthesized images and some low contrast medical images. Index Terms Shape modeling, shape recovery, interface motion, level sets, hyperbolic conservation laws, HamiltonJacobi
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 326 (21 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.
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 199 (65 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.
A NonOscillatory Eulerian Approach to Interfaces in Multimaterial Flows (The Ghost Fluid Method)
, 2000
"... While Eulerian schemes work well for most gas flows, they have been shown to admit nonphysical oscillations near some material interfaces. In contrast,... ..."
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Cited by 176 (40 self)
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While Eulerian schemes work well for most gas flows, they have been shown to admit nonphysical oscillations near some material interfaces. In contrast,...
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 141 (22 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.
Level set methods: An overview and some recent results
 J. Comput. Phys
, 2001
"... The level set method was devised by Osher and Sethian in [64] as a simple and versatile method for computing and analyzing the motion of an interface Γ in two or three dimensions. Γ bounds a (possibly multiply connected) region Ω. The goal is to compute and analyze the subsequent motion of Γ under a ..."
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Cited by 136 (12 self)
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The level set method was devised by Osher and Sethian in [64] as a simple and versatile method for computing and analyzing the motion of an interface Γ in two or three dimensions. Γ bounds a (possibly multiply connected) region Ω. The goal is to compute and analyze the subsequent motion of Γ under a velocity field �v. This velocity can depend on position, time, the geometry of the interface and the external physics. The interface is captured for later time as the zero level set of a smooth (at least Lipschitz continuous) function ϕ(�x,t), i.e., Γ(t)={�xϕ(�x,t)=0}. ϕ is positive inside Ω, negative outside Ω andiszeroonΓ(t). Topological merging and breaking are well defined and easily performed. In this review article we discuss recent variants and extensions, including the motion of curves in three dimensions, the Dynamic Surface Extension method, fast methods for steady state problems, diffusion generated motion and the variational level set approach. We also give a user’s guide to the level set dictionary and technology, couple the method to a wide variety of problems involving external physics, such as compressible and incompressible (possibly reacting) flow, Stefan problems, kinetic crystal growth, epitaxial growth of thin films,
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 136 (18 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
Implicit, Nonparametric Shape Reconstruction from Unorganized Points Using A Variational Level Set Method
 Computer Vision and Image Understanding
, 1998
"... In this paper we consider a fundamental visualization problem which arises in computer vision, computer graphics and numerical simulation. The problem is to find a curve in two dimensions, or a surface in three dimensions which can be regarded as the shape represented by a set of unorganized points, ..."
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Cited by 130 (20 self)
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In this paper we consider a fundamental visualization problem which arises in computer vision, computer graphics and numerical simulation. The problem is to find a curve in two dimensions, or a surface in three dimensions which can be regarded as the shape represented by a set of unorganized points, and/or curves, and/or surface patches. We do not assume any knowledge of the ordering, connectivity or topology of the data sets or of the true shape. Only the location of each point or general Hausdorff distance to the data set is known. The key idea in our approach is to find an implicit nonparametric representation of the curve or surface on a fixed rectangular grid. With this representation of surfaces we can easily (a) find the closest point and distance from any point to the surface (useful in illumination and many other applications), (b) find the intersection curve of two surfaces which is guaranteed to lie on both surfaces in our representation, and (c) perform any Boolean operatio...
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 121 (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.