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C.– Semiconcave functions, Hamilton Jacobi equations and optimal control Birkäuser
, 2004
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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.
Total generalized variation
 SIAM Journal on Imaging Sciences
"... The novel concept of total generalized variation of a function u is introduced and some of its essential properties are proved. Differently from the bounded variation seminorm, the new concept involves higher order derivatives of u. Numerical examples illustrate the high quality of this functional ..."
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Cited by 107 (18 self)
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The novel concept of total generalized variation of a function u is introduced and some of its essential properties are proved. Differently from the bounded variation seminorm, the new concept involves higher order derivatives of u. Numerical examples illustrate the high quality of this functional as a regularization term for mathematical imaging problems. In particular this functional selectively regularizes on different regularity levels and, as a side effect, does not lead to a staircasing effect.
A mass transportation approach to quantitative isoperimetric inequalities
 Invent. Math
, 2010
"... Abstract. A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov’s proof of the isoperimetric ..."
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Cited by 72 (22 self)
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Abstract. A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov’s proof of the isoperimetric inequality and the BrenierMcCann Theorem. A sharp quantitative version of the BrunnMinkowski inequality for convex sets is proved as a corollary. 1.
Existence and convergence for quasistatic evolution in brittle fracture
 Comm. Pure Appl. Math
"... This paper investigates the mathematical well–posedness of the variational model of quasi–static growth for a brittle crack proposed by Francfort & Marigo in [14]. The starting point is a time discretized version of that evolution which results in a sequence of minimization problems of Mumford & ..."
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Cited by 71 (13 self)
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This paper investigates the mathematical well–posedness of the variational model of quasi–static growth for a brittle crack proposed by Francfort & Marigo in [14]. The starting point is a time discretized version of that evolution which results in a sequence of minimization problems of Mumford & Shah type functionals. The natural weak setting is that of special functions of bounded variation, and the main difficulty in showing existence of the time– continuous quasi–static growth is to pass to the limit as the time–discretization step tends to 0. This is performed with the help of a jump transfer theorem which permits, under weak convergence assumptions for a sequence {un} of SBV –functions to its BV –limit u, to transfer the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence {un}. In particular, it is shown that the notion of minimizer of a Mumford & Shah type functional for its own jump set is stable under weak convergence assumptions. Furthermore, our analysis justifies numerical methods used for computing the time–continuous quasi–static evolution.
A MULTISCALE IMAGE REPRESENTATION USING HIERARCHICAL (BV, L²) DECOMPOSITIONS
 MULTISCALE MODEL. SIMUL.
, 2004
"... We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 { + v0, where [u0,v0] is the minimizer of a Jfunctional, J(f, λ0; X, Y) = infu+v= ..."
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Cited by 71 (10 self)
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We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 { + v0, where [u0,v0] is the minimizer of a Jfunctional, J(f, λ0; X, Y) = infu+v=f ‖u‖X + λ0‖v ‖ p} Y. Such minimizers are standard tools for image manipulations
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(
FAST DUAL MINIMIZATION OF THE VECTORIAL TOTAL VARIATION NORM AND APPLICATIONS TO COLOR IMAGE PROCESSING
, 2008
"... Abstract. We propose a regularization algorithm for color/vectorial images which is fast, easy to code and mathematically wellposed. More precisely, the regularization model is based on the dual formulation of the vectorial Total Variation (VTV) norm and it may be regarded as the vectorial extensio ..."
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Cited by 53 (2 self)
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Abstract. We propose a regularization algorithm for color/vectorial images which is fast, easy to code and mathematically wellposed. More precisely, the regularization model is based on the dual formulation of the vectorial Total Variation (VTV) norm and it may be regarded as the vectorial extension of the dual approach defined by Chambolle in [13] for grayscale/scalar images. The proposed model offers several advantages. First, it minimizes the exact VTV norm whereas standard approaches use a regularized norm. Then, the numerical scheme of minimization is straightforward to implement and finally, the number of iterations to reach the solution is low, which gives a fast regularization algorithm. Finally, and maybe more importantly, the proposed VTV minimization scheme can be easily extended to many standard applications. We apply this L 1 vectorial regularization algorithm to the following problems: color inverse scale space, color denoising with the chromaticitybrightness color representation, color image inpainting, color wavelet shrinkage, color image decomposition, color image deblurring, and color denoising on manifolds. Generally speaking, this VTV minimization scheme can be used in problems that required vector field (color, other feature vector) regularization while preserving discontinuities. Keywords: Vectorvalued TV norm, dual formulation, BV space, image denoising, ROF model, inverse scale space, chromaticitybrightness color representation, image decomposition, image inpainting, image deblurring, wavelet shrinkage, denoising on manifold. 1.
Stable image reconstruction using total variation minimization
 SIAM Journal on Imaging Sciences
, 2013
"... This article presents nearoptimal guarantees for accurate and robust image recovery from undersampled noisy measurements using total variation minimization, and our results may be the first of this kind. In particular, we show that from O(s log(N)) nonadaptive linear measurements, an image can be ..."
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Cited by 48 (2 self)
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This article presents nearoptimal guarantees for accurate and robust image recovery from undersampled noisy measurements using total variation minimization, and our results may be the first of this kind. In particular, we show that from O(s log(N)) nonadaptive linear measurements, an image can be reconstructed to within the best sterm approximation of its gradient, up to a logarithmic factor. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of a suitably incoherent matrix. 1