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USER’S GUIDE TO VISCOSITY SOLUTIONS OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS
, 1992
"... The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking argume ..."
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Cited by 1403 (15 self)
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The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a selfcontained exposition of the basic theory of viscosity solutions.
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 1188 (37 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.
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 493 (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.
Simultaneous Structure and Texture Image Inpainting
, 2003
"... An algorithm for the simultaneous fillingin of texture and structure in regions of missing image information is presented in this paper. The basic idea is to first decompose the image into the sum of two functions with different basic characteristics, and then reconstruct each one of these function ..."
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Cited by 220 (13 self)
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An algorithm for the simultaneous fillingin of texture and structure in regions of missing image information is presented in this paper. The basic idea is to first decompose the image into the sum of two functions with different basic characteristics, and then reconstruct each one of these functions separately with structure and texture fillingin algorithms. The first function used in the decomposition is of bounded variation, representing the underlying image structure, while the second function captures the texture and possible noise. The region of missing information in the bounded variation image is reconstructed using image inpainting algorithms, while the same region in the texture image is filledin with texture synthesis techniques. The original image is then reconstructed adding back these two subimages. The novel contribution of this paper is then in the combination of these three previously developed components, image decomposition with inpainting and texture synthesis, which permits the simultaneous use of fillingin algorithms that are suited for different image characteristics. Examples on real images show the advantages of this proposed approach.
Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing
 JOURNAL OF SCIENTIFIC COMPUTING
, 2002
"... This paper is devoted to the modeling of real textured images by functional minimization and partial differential equations. Following the ideas of Yves Meyer in a total variation minimization framework of L. Rudin, S. Osher and E. Fatemi, we decompose a given (possible textured) image f into a su ..."
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Cited by 201 (25 self)
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This paper is devoted to the modeling of real textured images by functional minimization and partial differential equations. Following the ideas of Yves Meyer in a total variation minimization framework of L. Rudin, S. Osher and E. Fatemi, we decompose a given (possible textured) image f into a sum of two functions u + v, where u E BV is a function of bounded variation (a cartoon or sketchy approximation of f), while v is a function representing the texture or noise. To model v we use the space of oscillating functions introduced by Yves Meyer, which is in some sense the dual of the BV space. The new algorithm is very simple, making use of differential equations and is easily solved in practice. Finally, we implement the method by finite differences, and we present various numerical results on real textured images, showing the obtained decomposition u + v, but we also show how the method can be used for texture discrimination and texture segmentation.
The BrunnMinkowski inequality
 Bull. Amer. Math. Soc. (N.S
, 2002
"... Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains ..."
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Cited by 184 (9 self)
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Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains the relationship between the BrunnMinkowski inequality and other inequalities in geometry and analysis, and some applications. 1.
Differential equations methods for the MongeKantorovich mass transfer problem
 Mem. Amer. Math. Soc
, 1999
"... We demonstrate that a solution to the classical Monge–Kantorovich problem of optimally rearranging the measure µ + = f + dx onto µ − = f − dy can be constructed by studying the pLaplacian equation −div(Dup  p−2 Dup) = f + − f − in the limit as p → ∞. The idea is to show up → u, where u satisfie ..."
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Cited by 166 (8 self)
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We demonstrate that a solution to the classical Monge–Kantorovich problem of optimally rearranging the measure µ + = f + dx onto µ − = f − dy can be constructed by studying the pLaplacian equation −div(Dup  p−2 Dup) = f + − f − in the limit as p → ∞. The idea is to show up → u, where u satisfies Du  ≤ 1, −div(aDu) = f + − f − for some density a ≥ 0, and then to build a flow by solving an ODE involving a, Du, f + and f −. Contents 1.
A tour of the theory of absolutely minimizing functions
"... Abstract. These notes are intended to be a rather complete and selfcontained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowledge of the subject. In particular, we improve known results regarding ..."
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Cited by 144 (10 self)
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Abstract. These notes are intended to be a rather complete and selfcontained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowledge of the subject. In particular, we improve known results regarding existence via arguments that are simpler than those that can be found in the literature. We present a proof of the main known uniqueness result which is largely selfcontained and does not rely on the theory of viscosity solutions. A unifying idea in our approach is the use of cone functions. This elementary geometric device renders the theory versatile and transparent. A number of tools and issues routinely encountered in the theory of elliptic partial differential equations are illustrated here in an especially clean manner, free from burdensome technicalities indeed, usually free from partial differential equations themselves. These include a priori continuity estimates, the Harnack inequality, Perron’s method for proving existence results, uniqueness and regularity questions, and some basic tools of viscosity solution theory. We believe that our presentation provides a unified summary of the existing theory as well as new results of interest to experts and researchers and, at the same time, a source which can be used for introducing students to some significant analytical tools.
Contractions in the 2Wasserstein Length Space and Thermalization of Granular Media
, 2004
"... An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical ..."
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Cited by 122 (35 self)
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An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinitedimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even nonconvexity) — if uniformly controlled — will quantify contractivity (limit expansivity) of the flow.
A topology preserving level set method for geometric deformable models
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2003
"... Active contour and surface models, also known as deformable models, are powerful image segmentation techniques. Geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior, parameterization independence, and ease of implement ..."
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Cited by 120 (7 self)
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Active contour and surface models, also known as deformable models, are powerful image segmentation techniques. Geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior, parameterization independence, and ease of implementation. However, a long claimed advantage of geometric deformable models—the ability to automatically handle topology changes—turns out to be a liability in applications where the object to be segmented has a known topology that must be preserved. In this paper, we present a new class of geometric deformable models designed using a novel topologypreserving level set method, which achieves topology preservation by applying the simple point concept from digital topology. These new models maintain the other advantages of standard geometric deformable models including subpixel accuracy and production of nonintersecting curves or surfaces. Moreover, since the topologypreserving constraint is enforced efficiently through local computations, the resulting algorithm incurs only nominal computational overhead over standard geometric deformable models. Several experiments on simulated and real data are provided to demonstrate the performance of this new deformable model algorithm.