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196
Fillingin by joint interpolation of vector fields and gray levels
 IEEE TRANS. IMAGE PROCESSING
, 2001
"... 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 missi ..."
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Cited by 157 (24 self)
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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.
Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models
 SIAM JOURNAL ON APPLIED MATHEMATICS
, 2006
"... 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 153 (6 self)
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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.
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 94 (7 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 72 (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
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 66 (9 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(
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) ..."
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Cited by 60 (4 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].
Global dissipative solutions of the CamassaHolm equation
 Anal. Appl. (Singap
"... Abstract. This paper is devoted to the continuation of solutions to the CamassaHolm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L ∞ space, containing a nonlocal source term ..."
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Cited by 39 (3 self)
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Abstract. This paper is devoted to the continuation of solutions to the CamassaHolm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L ∞ space, containing a nonlocal source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data ū ∈ H 1 (IR), and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking. The CamassaHolm equation ut − utxx + 3uux = 2uxuxx + uuxxx t> 0, x ∈ IR, (0.1) models the propagation of water waves in the shallow water regime, when the wavelength is considerably larger than the average water depth. Here u(t, x) represents the water’s free surface over
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 u ..."
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Cited by 39 (9 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.
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 36 (8 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...