Results 1 - 10 of 1,398

Geodesic Active Contours

by Vicent Caselles, Ron Kimmel, Guillermo Sapiro , 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 ..."
Abstract - Cited by 1425 (47 self) - Add to MetaCart
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

by Luminita A. Vese, Tony F. Chan - 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 2-phase segmentation, developed by ..."
Abstract - Cited by 498 (22 self) - Add to MetaCart
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 2-phase segmentation, developed by the authors earlier in T. Chan and L. Vese (1999. In Scale-Space'99, M. Nilsen et al. (Eds.), LNCS, vol. 1682, pp. 141--151) and T. Chan and L. Vese (2001. IEEE-IP, 10(2):266--277). 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 Four-Color 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.

HOMOGENIZATION AND TWO-SCALE CONVERGENCE

by Gregoire Allaire , 1992
"... Following an idea of G. Nguetseng, the author defines a notion of "two-scale" 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 corrector- ..."
Abstract - Cited by 451 (14 self) - Add to MetaCart
Following an idea of G. Nguetseng, the author defines a notion of "two-scale" 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 corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its "two-scale " 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 so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.

The Geometry of Dissipative Evolution Equations: The Porous Medium Equation

by Felix Otto
"... 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 ..."
Abstract - Cited by 405 (11 self) - Add to MetaCart
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.

A Unified Framework for Hybrid Control: Model and Optimal Control Theory

by Michael S. Branicky, Vivek S. Borkar, Sanjoy K. Mitter - 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 decision-making at the highest. Virtually all control systems today---from flight control to the factory floor---perform computer-coded chec ..."
Abstract - Cited by 305 (9 self) - Add to MetaCart
Complex natural and engineered systems typically possess a hierarchical structure, characterized by continuousvariable dynamics at the lowest level and logical decision-making at the highest. Virtually all control systems today---from flight control to the factory floor---perform computer-coded checks and issue logical as well as continuous-variable 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 continuous-variable commands and makes logical decisions. We first identify the phenomena that arise in real-world hybrid systems. Then, we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems, evolving on continuous-variable 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 near-optimal (precise) controls and derive "generalized quasi-variational 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.
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Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order

by Etienne Pardoux - in Stochastic Analysis and Related Topics VI: The Geilo Workshop , 1996
"... The aim of this set of lectures is to present the theory of backward stochastic differential equations, in short BSDEs, and its connections with viscosity solutions of systems of semi– linear second order partial differential equations of parabolic and elliptic type, in short PDEs. Linear BSDEs have ..."
Abstract - Cited by 260 (15 self) - Add to MetaCart
The aim of this set of lectures is to present the theory of backward stochastic differential equations, in short BSDEs, and its connections with viscosity solutions of systems of semi– linear second order partial differential equations of parabolic and elliptic type, in short PDEs. Linear BSDEs have appeared long time ago, both as the equations for the adjoint process in
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Global Minimum for Active Contour Models: A Minimal Path Approach

by Laurent D. Cohen, Ron Kimmel , 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 ..."
Abstract - Cited by 238 (70 self) - Add to MetaCart
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 topology-based saddle search routine. We show examples of our method applied to real aerial and medical images.

Weighted ENO Schemes for Hamilton-Jacobi Equations

by Guang-shan Jiang, Danping Peng - SIAM J. Sci. Comput , 1997
"... In this paper, we present a weighted ENO (essentially non-oscillatory) scheme to approximate the viscosity solution of the Hamilton-Jacobi 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 ..."
Abstract - Cited by 229 (0 self) - Add to MetaCart
In this paper, we present a weighted ENO (essentially non-oscillatory) scheme to approximate the viscosity solution of the Hamilton-Jacobi 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, Hamilton-Jacobi equation, shape from shading, level set. AMS(MOS) subject classification. 35L99, 65M06. 1 Introduction The Hamilton-Jacobi equation: OE t +H(x; t; OE; DOE) = 0; OE(x; 0) = OE 0 (x) (1.1) 1 Research supported by ONR N00014-92-J-1890. Email: gsj@math.ucla.edu. 2 Research supported by NSF DMS-94 04942. Email: dpeng@math.ucla.edu. where x 2 R d ...
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The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality

by Gerhard Huisken, Tom Ilmanen - J. DIFFERENTIAL GEOM , 1998
"... In this paper we develop the theory of weak solutions for the inverse mean curvature flow of hypersurfaces in a Riemannian manifold, and apply it to prove the Riemannian version of the Penrose inequality for the total mass of an asymptotically flat 3-manifold of nonnegative scalar curvature, announc ..."
Abstract - Cited by 201 (0 self) - Add to MetaCart
In this paper we develop the theory of weak solutions for the inverse mean curvature flow of hypersurfaces in a Riemannian manifold, and apply it to prove the Riemannian version of the Penrose inequality for the total mass of an asymptotically flat 3-manifold of nonnegative scalar curvature, announced in [HI1]. Let M be a smooth Riemannian manifold of dimension n 2 with metric g = (g ij ). A classical solution of the inverse mean curvature flow is a smooth family x : N \Theta [0; T ] !M

Mean-field backward stochastic differential equations and related patial differential equations

by Rainer Buckdahn, Juan Li, Shige Peng , 2007
"... In [5] the authors obtained Mean-Field backward stochastic differential equations (BSDE) associated with a Mean-field stochastic differential equation (SDE) in a natural way as limit of some highly dimensional system of forward and backward SDEs, corresponding to a large number of “particles” (or “a ..."
Abstract - Cited by 181 (14 self) - Add to MetaCart
In [5] the authors obtained Mean-Field backward stochastic differential equations (BSDE) associated with a Mean-field stochastic differential equation (SDE) in a natural way as limit of some highly dimensional system of forward and backward SDEs, corresponding to a large number of “particles” (or “agents”). The objective of the present paper is to deepen the investigation of such Mean-Field BSDEs by studying them in a more general framework, with general driver, and to discuss comparison results for them. In a second step we are interested in partial differential equations (PDE) whose solutions can be stochastically interpreted in terms of Mean-Field BSDEs. For this we study a Mean-Field BSDE in a Markovian framework, associated with a Mean-Field forward equation. By combining classical BSDE methods, in particular that of “backward semigroups” introduced by Peng [14], with specific arguments for Mean-Field BSDEs we prove that this Mean-Field BSDE describes the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated to Mean-Field BSDEs
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