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83
Fillingin by joint interpolation of vector fields and gray levels
 IEEE Trans. Image Processing
, 2001
"... Abstract—A variational approach for fillingin regions of missing data in digital images is introduced in this paper. The approach is based on joint interpolation of the image graylevels and gradient/isophotes directions, smoothly extending in an automatic fashion the isophote lines into the holes ..."
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Cited by 122 (20 self)
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Abstract—A variational approach for fillingin regions of missing data in digital images is introduced in this paper. The approach is based on joint interpolation of the image graylevels and gradient/isophotes directions, smoothly extending in an automatic fashion the isophote lines into the holes of missing data. This interpolation is computed by solving the variational problem via its gradient descent flow, which leads to a set of coupled second order partial differential equations, one for the graylevels and one for the gradient orientations. The process underlying this approach can be considered as an interpretation of the Gestaltist’s principle of good continuation. No limitations are imposed on the topology of the holes, and all regions of missing data can be simultaneously processed, even if they are surrounded by completely different structures. Applications of this technique include the restoration of old photographs and removal of superimposed text like dates, subtitles, or publicity. Examples of these applications are given. We conclude the paper with a number of theoretical results on the proposed variational approach and its corresponding gradient descent flow. Index Terms—Fillingin, Gestalt principles, image gradients, image graylevels, interpolation, partial differential equations, variational approach. I.
Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models
 SIAM Journal on Applied Mathematics
, 2004
"... Abstract. We show how certain nonconvex optimization problems that arise in image processing and computer vision can be restated as convex minimization problems. This allows, in particular, the finding of global minimizers via standard convex minimization schemes. ..."
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Cited by 93 (5 self)
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Abstract. We show how certain nonconvex optimization problems that arise in image processing and computer vision can be restated as convex minimization problems. This allows, in particular, the finding of global minimizers via standard convex minimization schemes.
Minimizing total variation flow
 Differential and Integral Equations
, 2001
"... (Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect ..."
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Cited by 43 (6 self)
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(Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as t →∞. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts. 1. Introduction. Let Ω be a bounded set in R N with Lipschitzcontinuous boundary ∂Ω. We are interested in the problem ∂u Du = div(
Global conservative solutions of the CamassaHolm equation
"... Abstract. This paper develops a new approach in the analysis of the CamassaHolm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new va ..."
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Cited by 35 (3 self)
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Abstract. This paper develops a new approach in the analysis of the CamassaHolm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global solutions, depending continuously on the initial data. Our solutions are conservative, in the sense that the total energy equals a constant, for almost every time. The nonlinear partial differential equation ut − utxx + 3uux = 2uxuxx + uuxxx, t> 0, x ∈ IR, was derived by Camassa and Holm [CH] as a model for the propagation of shallow water waves, with
On viscosity solutions of fully nonlinear equations with measurable ingredients
 Comm. Pure Appl. Math
, 1996
"... In this paper we study Hölder regularity for the first and second derivatives of continuous viscosity solutions of fully nonlinear equations of the form (1.1) F(D 2 u) = 0. It is well known that viscosity solutions of (1.1) are C 1,α for some 0 < α < 1, and ..."
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Cited by 34 (5 self)
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In this paper we study Hölder regularity for the first and second derivatives of continuous viscosity solutions of fully nonlinear equations of the form (1.1) F(D 2 u) = 0. It is well known that viscosity solutions of (1.1) are C 1,α for some 0 < α < 1, and
A nonlocal anisotropic model for phase transitions  Part II: Asymptotic behaviour of rescaled energies
, 1997
"... We study the asymptotic behaviour as " ! 0, of the nonlocal models for phase transition described by the scaled free energy F " (u) := 1 4" Z \Omega \Theta\Omega J " (x 0 \Gamma x) \Gamma u(x 0 ) \Gamma u(x) \Delta 2 dx 0 dx + 1 " Z \Omega W \Gamma u(x) \Delta dx ; where u is ..."
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Cited by 28 (3 self)
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We study the asymptotic behaviour as " ! 0, of the nonlocal models for phase transition described by the scaled free energy F " (u) := 1 4" Z \Omega \Theta\Omega J " (x 0 \Gamma x) \Gamma u(x 0 ) \Gamma u(x) \Delta 2 dx 0 dx + 1 " Z \Omega W \Gamma u(x) \Delta dx ; where u is a scalar density function, W is a doublewell potential which vanishes at \Sigma1, J is a nonnegative interaction potential and J " (h) := " \GammaN J(h="). We prove that the functionals F " converge in a variational sense to the anisotropic surface energy F (u) := Z Su oe( u ) ; where u is allowed to take the values \Sigma1 only, u is the normal to the interface Su between the phases fu = +1g and fu = \Gamma1g, and oe is the surface tension. This paper concludes the analisys started in [AB].
Variational Restoration and Edge Detection for Color Images
 Journal of Mathematical Imaging and Vision
, 2003
"... Abstract. We propose and analyze extensions of the MumfordShah functional for color images. Our main motivation is the concept of images as surfaces. We also review most of the relevant theoretical background and computer vision literature. Keywords: color, MumfordShah functional, segmentation, va ..."
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Cited by 23 (1 self)
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Abstract. We propose and analyze extensions of the MumfordShah functional for color images. Our main motivation is the concept of images as surfaces. We also review most of the relevant theoretical background and computer vision literature. Keywords: color, MumfordShah functional, segmentation, variational methods.
Connected Components of Sets of Finite Perimeter and Applications to Image Processing
 JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
, 1999
"... This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in R^N, introduced by H. Federer in the more general framework of the theory of currents. We provide a new and simpler proof of the existence and uniqueness of the decomposit ..."
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Cited by 21 (7 self)
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This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in R^N, introduced by H. Federer in the more general framework of the theory of currents. We provide a new and simpler proof of the existence and uniqueness of the decomposition into the socalled Mconnected components. Moreover, we study carefully the structure of the essential boundary of these components and give in particular a reconstruction formula of a set of finite perimeter from the family of the boundaries of its components. In the two dimensional case we show that this notion of connectedness is comparable with the topological one, modulo the choice of a suitable representative in the equivalence class. Our strong motivation for this study is a mathematical justification of all those operations in image processing that involve connectedness and boundaries. As an application, we use this weak notion of connectedness to provide a rigorous mathemati...
The Dirichlet Problem for the Total Variation Flow
, 2001
"... We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use th ..."
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Cited by 21 (7 self)
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We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.
Initial Layers And Uniqueness Of Weak Entropy Solutions To Hyperbolic Conservation Laws
, 2000
"... We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weakstar in L 1 as t ! 0+ and satisfy the entropy inequality in the sense of distributions f ..."
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Cited by 18 (7 self)
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We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weakstar in L 1 as t ! 0+ and satisfy the entropy inequality in the sense of distributions for t ? 0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solutions are always unique and the initial layers do not appear. We also discuss its applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation.