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102
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.
Geodesic Active Regions and Level Set Methods for Supervised Texture Segmentation
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2002
"... This paper presents a novel variational framework to deal with frame partition problems in Computer Vision. This framework exploits boundary and regionbased segmentation modules under a curvebased optimization objective function. The task of supervised texture segmentation is considered to demonst ..."
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Cited by 312 (9 self)
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This paper presents a novel variational framework to deal with frame partition problems in Computer Vision. This framework exploits boundary and regionbased segmentation modules under a curvebased optimization objective function. The task of supervised texture segmentation is considered to demonstrate the potentials of the proposed framework. The textured feature space is generated by filtering the given textured images using isotropic and anisotropic filters, and analyzing their responses as multicomponent conditional probability density functions. The texture segmentation is obtained by unifying region and boundarybased information as an improved Geodesic Active Contour Model. The defined objective function is minimized using a gradientdescent method where a level set approach is used to implement the obtained PDE. According to this PDE, the curve propagation towards the final solution is guided by boundary and regionbased segmentation forces, and is constrained by a regularity force. The level set implementation is performed using a fast front propagation algorithm where topological changes are naturally handled. The performance of our method is demonstrated on a variety of synthetic and real textured frames.
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 226 (11 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,
Image Segmentation Using Active Contours: Calculus Of Variations Or Shape Gradients?
 SIAM Applied Mathematics
, 2002
"... We consider the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals. We show that one can go from one class to the other by solving Poisson's or Helmholtz's equation with wellchosen boundary conditions. Using this equiva ..."
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Cited by 99 (30 self)
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We consider the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals. We show that one can go from one class to the other by solving Poisson's or Helmholtz's equation with wellchosen boundary conditions. Using this equivalence, we study the case of a large class of region functionals by standard methods of the calculus of variations and derive the corresponding EulerLagrange equations. We revisit this problem using the notion of shape derivative and show that the same equations can be elegantly derived without going through the unnatural step of converting the region integrals into boundary integrals. We also define a larger class of region functionals based on the estimation and comparison to a prototype of the probability density distribution of image features and show how the shape derivative tool allows us to easily compute the corresponding Gateaux derivatives and EulerLagrange equations. We finally apply this new functional to the problem of regions segmentation in sequences of color images. We briefly describe our numerical scheme and show some experimental results.
Geodesic Active Regions: A new framework to deal with frame partition problems in Computer Vision
, 2002
"... This paper presents a novel variational framework for dealing with frame partition problems in Computer Vision by the propagation of curves. This framework integrates boundary and regionbased frame partition modules under a curvebased energy framework, which aims at finding a set of minimal le ..."
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Cited by 85 (10 self)
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This paper presents a novel variational framework for dealing with frame partition problems in Computer Vision by the propagation of curves. This framework integrates boundary and regionbased frame partition modules under a curvebased energy framework, which aims at finding a set of minimal length curves that preserve three main properties: (i) they are regular and smooth, (ii) they are attracted by the boundary points (boundarybased information), (ii) and they create a partition that is optimal according to the expected region properties of the different hypotheses (regionbased information). The defined objective function is minimized using a gradient descent method. According to the obtained motion equations, the set of initial curves is propagated towards the best partition under the influence of boundary and regionbased forces, and being constrained by a regularity force. The changes of topology are naturally handled thanks to the level set implementation. Furthermore, a coupled multiphase propagation is proposed that imposes the idea of mutually exclusive propagating curves, and increases the robustness as well as the convergence rate. The proposed framework has been validated using three important applications in Computer Vision, the tasks of image and supervised texture segmentation in lowlevel vision and the task of motion estimation and tracking in motion analysis
Coupled Geodesic Active Regions for Image Segmentation: A Level Set Approach
 In European Conference in Computer Vision
, 1999
"... . This paper presents a novel variational method for image segmentation that unifies boundary and regionbased information sources under the Geodesic Active Region framework. A statistical analysis based on the Minimum Description Length criterion and the Maximum Likelihood Principle for the obs ..."
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Cited by 84 (2 self)
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. This paper presents a novel variational method for image segmentation that unifies boundary and regionbased information sources under the Geodesic Active Region framework. A statistical analysis based on the Minimum Description Length criterion and the Maximum Likelihood Principle for the observed density function (image histogram) using a mixture of Gaussian elements, indicates the number of the different regions and their intensity properties. Then, the boundary information is determined using a probabilistic edge detector, while the region information is estimated using the Gaussian components of the mixture model. The defined objective function is minimized using a gradientdescent method where a level set approach is used to implement the resulting PDE system. According to the motion equations, the set of initial curves is propagated toward the segmentation result under the influence of boundary and regionbased segmentation forces, and being constrained by a regul...
A Fully Global Approach to Image Segmentation via Coupled Curve Evolution Equations
 Journal of Visual Communication and Image Representation
, 2002
"... In this paper, we develop a novel regionbased approach to snakes designed to optimally separate the values of certain image statistics over a known number of region types. Multiple sets of contours deform according to a coupled set of curve evolution equations derived from a single global cost func ..."
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Cited by 82 (13 self)
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In this paper, we develop a novel regionbased approach to snakes designed to optimally separate the values of certain image statistics over a known number of region types. Multiple sets of contours deform according to a coupled set of curve evolution equations derived from a single global cost functional. The resulting active contour model, in contrast to many other edge and region based models, is fully global in that the evolution of each curve depends at all times upon every pixel in the image and is directly coupled to the evolution of every other curve regardless of their mutual proximity. As such evolving contours enjoy a very wide “field of view, ” endowing the algorithm with a robustness to initial contour placement above and beyond the significant improvement exhibited by other region based snakes over earlier edge based snakes. C ○ 2002 Elsevier Science (USA) Key Words: active contours; curve evolution; snakes; segmentation; gradient flows.
Stereoscopic Segmentation
, 2001
"... We cast the problem of multiframe stereo reconstruction of a smooth shape as the global region segmentation of a collection of images of the scene. Dually, the problem of segmenting multiple calibrated images of an object becomes that of estimating the solid shape that gives rise to such images. We ..."
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Cited by 78 (18 self)
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We cast the problem of multiframe stereo reconstruction of a smooth shape as the global region segmentation of a collection of images of the scene. Dually, the problem of segmenting multiple calibrated images of an object becomes that of estimating the solid shape that gives rise to such images. We assume that the radiance has smooth statistics. This assumption covers Lambertian scenes with smooth or constant albedo as well as fine homogeneous textures, which are known challenges to stereo algorithms based on local correspondence. We pose the segmentation problem within a variational framework, and use fast level set methods to approximate the optimal solution numerically. Our algorithm does not work in the presence of strong textures, where traditional reconstruction algorithms do. It enjoys significant robustness to noise under the assumptions it is designed for. 1
Regularized laplacian zero crossings as optimal edge integrators
 International Journal of Computer Vision
, 2001
"... We view the fundamental edge integration problem for object segmentation in a geometric variational framework. First we show that the classical zerocrossings of the image Laplacian edge detector as suggested by Marr and Hildreth, inherently provides optimal edgeintegration with regard to a very na ..."
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Cited by 62 (4 self)
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We view the fundamental edge integration problem for object segmentation in a geometric variational framework. First we show that the classical zerocrossings of the image Laplacian edge detector as suggested by Marr and Hildreth, inherently provides optimal edgeintegration with regard to a very natural geometric functional. This functional accumulates the inner product between the normal to the edge and the gray level imagegradient along the edge. We use this observation to derive new and highly accurate active contours based on this functional and regularized by previously proposed geodesic active contour geometric variational models. 1.
Threedimensional shape knowledge for joint image segmentation and pose estimation
 Pattern Recognition, volume 3663 of LNCS
, 2005
"... In this article we present the integration of 3D shape knowledge into a variational model for level set based image segmentation and tracking. Given a 3D surface model of an object that is visible in the image of one or multiple cameras calibrated to the same world coordinate system, the object co ..."
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Cited by 59 (30 self)
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In this article we present the integration of 3D shape knowledge into a variational model for level set based image segmentation and tracking. Given a 3D surface model of an object that is visible in the image of one or multiple cameras calibrated to the same world coordinate system, the object contour extracted by the segmentation method is applied to estimate the 3D pose parameters of the object. Viceversa, the surface model projected to the image plane helps in a topdown manner to improve the extraction of the contour. While common alternative segmentation approaches, which integrate 2D shape knowledge, face the problem that an object can look very differently from various viewpoints, a 3D free form model ensures that for each view the model can fit the data in the image very well. Moreover, one additionally solves the higher level problem of determining the object pose in 3D space. Due to the variational formulation, the approach clearly states all model assumptions in a single energy functional that is locally minimized by our method. Its performance is demonstrated by experiments with a monocular and a stereo camera system. 1 1