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557
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.
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 316 (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.
The Geometry of Dissipative Evolution Equations: The Porous Medium Equation
"... We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show the time asymptotic behavior can be easily understood in this framework. We use the intuition and the ..."
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Cited by 224 (10 self)
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We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior. Contents 1 The porous medium equation as a gradient flow 2 1.1 The porous medium equation . . . . . . . . . . . . . . . . . . 2 1.2 Abstract gradient flow . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Two interpretations of the porous medium equation as gradient flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 A physical argument in favor of the new gradient flow 6 3 A mathematical argument in favor of new gradient flow 9 3.1 Self similar solutions and asymptotic behaviour . . . . . . . . 9 3.2 A new asymptotic result . . . . . . . . . . . . . . . . . . . . . 10 3.3 The asymptotic result express...
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 196 (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 Unified Framework for Hybrid Control: Model and Optimal Control Theory
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 1998
"... Complex natural and engineered systems typically possess a hierarchical structure, characterized by continuousvariable dynamics at the lowest level and logical decisionmaking at the highest. Virtually all control systems todayfrom flight control to the factory floorperform computercoded chec ..."
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Cited by 183 (8 self)
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Complex natural and engineered systems typically possess a hierarchical structure, characterized by continuousvariable dynamics at the lowest level and logical decisionmaking at the highest. Virtually all control systems todayfrom flight control to the factory floorperform computercoded checks and issue logical as well as continuousvariable control commands. The interaction of these different types of dynamics and information leads to a challenging set of "hybrid" control problems. We propose a very general framework that systematizes the notion of a hybrid system, combining differential equations and automata, governed by a hybrid controller that issues continuousvariable commands and makes logical decisions. We first identify the phenomena that arise in realworld hybrid systems. Then, we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems, evolving on continuousvariable state spaces and subject to continuous controls and discrete transitions. The model captures the identified phenomena, subsumes previous models, yet retains enough structure on which to pose and solve meaningful control problems. We develop a theory for synthesizing hybrid controllers for hybrid plants in an optimal control framework. In particular, we demonstrate the existence of optimal (relaxed) and nearoptimal (precise) controls and derive "generalized quasivariational inequalities" that the associated value function satisfies. We summarize algorithms for solving these inequalities based on a generalized Bellman equation, impulse control, and linear programming.
HOMOGENIZATION AND TWOSCALE CONVERGENCE
, 1992
"... Following an idea of G. Nguetseng, the author defines a notion of "twoscale" convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in L2(f) are proven to be relatively compact with respect to this new type of convergence. A correctortype theor ..."
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Cited by 176 (11 self)
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Following an idea of G. Nguetseng, the author defines a notion of "twoscale" convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in L2(f) are proven to be relatively compact with respect to this new type of convergence. A correctortype theorem (i.e., which permits, in some cases, replacing a sequence by its "twoscale " limit, up to a strongly convergent remainder in L2(12)) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the socalled energy method of Tartar. The power and simplicity of the twoscale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear secondorder elliptic equations.
Weighted ENO Schemes for HamiltonJacobi Equations
 SIAM J. Sci. Comput
, 1997
"... In this paper, we present a weighted ENO (essentially nonoscillatory) scheme to approximate the viscosity solution of the HamiltonJacobi equation: OE t +H(x 1 ; \Delta \Delta \Delta ; x d ; t; OE; OE x1 ; \Delta \Delta \Delta ; OE xd ) = 0: This weighted ENO scheme is constructed upon and has the ..."
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Cited by 151 (0 self)
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In this paper, we present a weighted ENO (essentially nonoscillatory) scheme to approximate the viscosity solution of the HamiltonJacobi equation: OE t +H(x 1 ; \Delta \Delta \Delta ; x d ; t; OE; OE x1 ; \Delta \Delta \Delta ; OE xd ) = 0: This weighted ENO scheme is constructed upon and has the same stencil nodes as the 3 rd order ENO scheme but can be as high as 5 th order accurate in the smooth part of the solution. In addition to the accuracy improvement, numerical comparisons between the two schemes also demonstrate that, the weighted ENO scheme is more robust than the ENO scheme. Key words. ENO, weighted ENO, HamiltonJacobi equation, shape from shading, level set. AMS(MOS) subject classification. 35L99, 65M06. 1 Introduction The HamiltonJacobi equation: OE t +H(x; t; OE; DOE) = 0; OE(x; 0) = OE 0 (x) (1.1) 1 Research supported by ONR N0001492J1890. Email: gsj@math.ucla.edu. 2 Research supported by NSF DMS94 04942. Email: dpeng@math.ucla.edu. where x 2 R d ...
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.
Conformal Curvature Flows: From Phase Transitions to Active Vision
, 1995
"... In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel edgedetection paradigm in which the feature of interest may be consid ..."
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Cited by 117 (30 self)
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In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel edgedetection paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus an edgeseeking curve is attracted very naturally and efficiently to the desired feature. Comparison with the AllenCahn model clarifies some of the choices made in these models, and suggests inhomogeneous models which may in return be useful in phase transitions. We also consider some 3D active surface models based on these ideas. The justification of this model rests on the careful study of the viscosity solutions of evolution equations derived from a levelset approach. Key words: Active vision, antiphase boundary, visual tracking, edge detection, segmentation, gradient flows, Riemannian metrics, viscosity solutions, geometric heat equ...
A Variational Method In Image Recovery
 SIAM J. Numer. Anal
, 1997
"... This paper is concerned with a classical denoising and deblurring problem in image recovery. Our approach is based on a variational method. By using the LegendreFenchel transform, we show how the nonquadratic criterion to be minimized can be split into a sequence of halfquadratic problems easier t ..."
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Cited by 101 (22 self)
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This paper is concerned with a classical denoising and deblurring problem in image recovery. Our approach is based on a variational method. By using the LegendreFenchel transform, we show how the nonquadratic criterion to be minimized can be split into a sequence of halfquadratic problems easier to solve numerically. First we prove an existence and uniqueness result, and then we describe the algorithm for computing the solution and we give a proof of convergence. Finally, we present some experimental results for synthetic and real images.