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An introduction to total variation for image analysis
 in Theoretical Foundations and Numerical Methods for Sparse Recovery, De Gruyter
, 2010
"... These notes address various theoretical and practical topics related to Total Variationbased image reconstruction. They focuse first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize th ..."
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Cited by 44 (3 self)
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These notes address various theoretical and practical topics related to Total Variationbased image reconstruction. They focuse first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize the total variation in a finitedifferences setting, with a series of applications from simple denoising to stereo, or deconvolution issues, and even more exotic uses like the minimization of minimal partition problems.
Existence and uniqueness for dislocation dynamics with nonnegative velocity
, 2004
"... We study the problem of large time existence of solutions for a mathematical model describing dislocation dynamics in crystals. The mathematical model is a geometric and non local eikonal equation which does not preserve the inclusion. Under the assumption that the dislocation line is expanding, we ..."
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Cited by 42 (18 self)
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We study the problem of large time existence of solutions for a mathematical model describing dislocation dynamics in crystals. The mathematical model is a geometric and non local eikonal equation which does not preserve the inclusion. Under the assumption that the dislocation line is expanding, we prove existence and uniqueness of the solution in the framework of discontinuous viscosity solutions. We also show that this solution satisfies some variational properties, which allows to prove that the energy associated to the dislocation dynamics is non increasing. AMS Classification: 35F25, 35D05. Keywords: Dislocation dynamics, eikonal equation, HamiltonJacobi equations,
A level set formulation for Willmore flow
 INTERFACES FREE BOUNDARIES
, 2004
"... A level set formulation of Willmore flow is derived using the gradient flow perspective. Starting from single embedded surfaces and the corresponding gradient flow, the metric is generalized to sets of level set surfaces using the identification of normal velocities and variations of the level set f ..."
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Cited by 41 (10 self)
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A level set formulation of Willmore flow is derived using the gradient flow perspective. Starting from single embedded surfaces and the corresponding gradient flow, the metric is generalized to sets of level set surfaces using the identification of normal velocities and variations of the level set function in time via the level set equation. The approach in particular allows to identify the natural dependent quantities of the derived variational formulation. Furthermore, spatial and temporal discretization are discussed and some numerical simulations are presented.
On the inviscid limit of a model for crack propagation
 MATH. MODELS METH. APPL. SCI.
, 2007
"... We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rateindependent process on the basis of Griffith’s local energy release rate criterion. According to this criterion, the system may stay in a l ..."
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Cited by 39 (9 self)
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We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rateindependent process on the basis of Griffith’s local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.
A variational approach to remove multiplicative noise
, 2006
"... Abstract. This paper focuses on the problem of multiplicative noise removal. We draw our inspiration from the modeling of speckle noise. By using a MAP estimator, we can derive a functional whose minimizer corresponds to the denoised image we want to recover. Although the functional is not convex, w ..."
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Cited by 39 (1 self)
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Abstract. This paper focuses on the problem of multiplicative noise removal. We draw our inspiration from the modeling of speckle noise. By using a MAP estimator, we can derive a functional whose minimizer corresponds to the denoised image we want to recover. Although the functional is not convex, we prove the existence of a minimizer and we show the capability of our model on some numerical examples. We study the associated evolution problem, for which we derive existence and uniqueness results for the solution. We prove the convergence of an implicit scheme to compute the solution. Key words. Calculus of variation, functional analysis, BV, variational approach, multiplicative noise, speckle noise, image restoration. AMS subject classifications. 68U10, 94A08, 49J40, 35A15, 35B45, 35B50. 1. Introduction. Image
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.
The CalderónZygmund theory for elliptic problems with measure data
, 2007
"... Abstract. We consider nonlinear elliptic equations having a measure in the right hand side, of the type div a(x, Du) = µ, and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density propertie ..."
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Cited by 38 (9 self)
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Abstract. We consider nonlinear elliptic equations having a measure in the right hand side, of the type div a(x, Du) = µ, and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable CalderónZygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates.
A Variational Framework for Simultaneous Motion Estimation and Restoration of MotionBlurred Video
, 2007
"... Figure 1. From two real blurred frames (left), we automatically and simultaneously estimate the motion region, the motion vector, and the image intensity of the foreground (middle). Based on this and the background intensity we reconstruct the two frames (right). ..."
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Cited by 36 (1 self)
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Figure 1. From two real blurred frames (left), we automatically and simultaneously estimate the motion region, the motion vector, and the image intensity of the foreground (middle). Based on this and the background intensity we reconstruct the two frames (right).
Secondorder analysis for optimal control problems with pure and mixed state constraints. INRIA Research Report 6199
, 2007
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 36 (20 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. appor t de r ech er ch e
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...