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AuthorSchönlieb, Carola-Bibiane (3)Langer, Andreas (2)Mouhot, Clément (2)Aki, Gonca L. (1)Antonelli, Paolo (1)View MoreJournalJournal of Mathematical Imaging and Vision (3)Archive for Rational Mechanics and Analysis (2)Acta Applicandae Mathematicae (1)Acta Mathematica (1)Annales Henri Poincaré (1)View MoreKAUST Grant Number

KUK-I1-007-43 (11)

Publisher
Springer Nature (11)

SubjectDenoising (2)Staircasing (2)Total variation (2)Adaptive dynamics (1)Approximation bound (1)View MoreTypeArticle (11)Year (Issue Date)2016 (1)2013 (1)2012 (2)2011 (4)2010 (3)Item AvailabilityMetadata Only (11)

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A convergent overlapping domain decomposition method for total variation minimization

Fornasier, Massimo; Langer, Andreas; Schönlieb, Carola-Bibiane (Numerische Mathematik, Springer Nature, 2010-06-22) [Article]

In this paper we are concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to the data and a total variation constraint. To our knowledge, this is the first successful attempt of addressing such a strategy for the nonlinear, nonadditive, and nonsmooth problem of total variation minimization. We provide several numerical experiments, showing the successful application of the algorithm for the restoration of 1D signals and 2D images in interpolation/inpainting problems, respectively, and in a compressed sensing problem, for recovering piecewise constant medical-type images from partial Fourier ensembles. © 2010 Springer-Verlag.

A Combined First and Second Order Variational Approach for Image Reconstruction

Papafitsoros, K.; Schönlieb, C. B. (Journal of Mathematical Imaging and Vision, Springer Nature, 2013-05-10) [Article]

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler's elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images-a known disadvantage of the ROF model-while being simple and efficiently numerically solvable. ©Springer Science+Business Media New York 2013.

Bregmanized Domain Decomposition for Image Restoration

Langer, Andreas; Osher, Stanley; Schönlieb, Carola-Bibiane (Journal of Scientific Computing, Springer Nature, 2012-05-22) [Article]

Computational problems of large-scale data are gaining attention recently due to better hardware and hence, higher dimensionality of images and data sets acquired in applications. In the last couple of years non-smooth minimization problems such as total variation minimization became increasingly important for the solution of these tasks. While being favorable due to the improved enhancement of images compared to smooth imaging approaches, non-smooth minimization problems typically scale badly with the dimension of the data. Hence, for large imaging problems solved by total variation minimization domain decomposition algorithms have been proposed, aiming to split one large problem into N > 1 smaller problems which can be solved on parallel CPUs. The N subproblems constitute constrained minimization problems, where the constraint enforces the support of the minimizer to be the respective subdomain. In this paper we discuss a fast computational algorithm to solve domain decomposition for total variation minimization. In particular, we accelerate the computation of the subproblems by nested Bregman iterations. We propose a Bregmanized Operator Splitting-Split Bregman (BOS-SB) algorithm, which enforces the restriction onto the respective subdomain by a Bregman iteration that is subsequently solved by a Split Bregman strategy. The computational performance of this new approach is discussed for its application to image inpainting and image deblurring. It turns out that the proposed new solution technique is up to three times faster than the iterative algorithm currently used in domain decomposition methods for total variation minimization. © Springer Science+Business Media, LLC 2012.

Long Time Evolution of Populations under Selection and Vanishing Mutations

Raoul, Gaël (Acta Applicandae Mathematicae, Springer Nature, 2011-02-08) [Article]

In this paper, we consider a long time and vanishing mutations limit of an integro-differential model describing the evolution of a population structured with respect to a continuous phenotypic trait. We show that the asymptotic population is a steady-state of the evolution equation without mutations, and satisfies an evolutionary stability condition. © 2011 Springer Science+Business Media B.V.

Fractional Diffusion Limit for Collisional Kinetic Equations

Mellet, Antoine; Mischler, Stéphane; Mouhot, Clément (Archive for Rational Mechanics and Analysis, Springer Nature, 2010-08-20) [Article]

This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation. © 2010 Springer-Verlag.

On Landau damping

Mouhot, Clément; Villani, Cédric (Acta Mathematica, Springer Nature, 2011-09) [Article]

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp "deflection" estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions. © 2011 Institut Mittag-Leffler.

Two-phase semilinear free boundary problem with a degenerate phase

Matevosyan, Norayr; Petrosyan, Arshak (Calculus of Variations and Partial Differential Equations, Springer Nature, 2010-10-16) [Article]

We study minimizers of the energy functional ∫D[{pipe}∇u{pipe}2 + λ(u+)p]dx for p ∈ (0, 1) without any sign restriction on the function u. The distinguished feature of the problem is the lack of nondegeneracy in the negative phase. The main result states that in dimension two the free boundaries Γ+ = ∂{u > 0} ∩ D andΓ- = ∂{u < 0} ∩ D are C1,α-regular, provided 1 - ∈0 < p < 1. The proof is obtained by a careful iteration of the Harnack inequality to obtain a nontrivial growth estimate in the negative phase, compensating for the apriori unknown nondegeneracy. © 2010 Springer-Verlag.

Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem

Lellmann, Jan; Lenzen, Frank; Schnörr, Christoph (Journal of Mathematical Imaging and Vision, Springer Nature, 2012-11-09) [Article]

We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation methods for finite-dimensional problems. While for the latter several optimality bounds are known, to our knowledge no such bounds exist in the infinite-dimensional setting. We provide such a bound by analyzing a probabilistic rounding method, showing that it is possible to obtain an integral solution of the original partitioning problem from a solution of the relaxed problem with an a priori upper bound on the objective. The approach has a natural interpretation as an approximate, multiclass variant of the celebrated coarea formula. © 2012 Springer Science+Business Media New York.

The Quantum Hydrodynamics System in Two Space Dimensions

Antonelli, Paolo; Marcati, Pierangelo (Archive for Rational Mechanics and Analysis, Springer Nature, 2011-09-16) [Article]

In this paper we study global existence of weak solutions for the quantum hydrodynamics system in two-dimensional energy space. We do not require any additional regularity and/or smallness assumptions on the initial data. Our approach replaces the WKB formalism with a polar decomposition theory which is not limited by the presence of vacuum regions. In this way we set up a self consistent theory, based only on particle density and current density, which does not need to define velocity fields in the nodal regions. The mathematical techniques we use in this paper are based on uniform (with respect to the approximating parameter) Strichartz estimates and the local smoothing property. © 2011 Springer-Verlag.

Infimal Convolution Regularisation Functionals of BV and L<sup>p</sup> Spaces

Burger, Martin; Papafitsoros, Konstantinos; Papoutsellis, Evangelos; Schönlieb, Carola-Bibiane (Journal of Mathematical Imaging and Vision, Springer Nature, 2016-02-03) [Article]

We study a general class of infimal convolution type regularisation functionals suitable for applications in image processing. These functionals incorporate a combination of the total variation seminorm and Lp norms. A unified well-posedness analysis is presented and a detailed study of the one-dimensional model is performed, by computing exact solutions for the corresponding denoising problem and the case p=2. Furthermore, the dependency of the regularisation properties of this infimal convolution approach to the choice of p is studied. It turns out that in the case p=2 this regulariser is equivalent to the Huber-type variant of total variation regularisation. We provide numerical examples for image decomposition as well as for image denoising. We show that our model is capable of eliminating the staircasing effect, a well-known disadvantage of total variation regularisation. Moreover as p increases we obtain almost piecewise affine reconstructions, leading also to a better preservation of hat-like structures.

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