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Regularity properties of solutions to variational problems are established for a broad class of strictly convex splitting type energy densities of the principal form f: [R.sup.2] [right arrow] R, f([[xi].sup.1], [[xi].sup.2]) = [f.sub.1]([[xi].sup.1]) + [f.sub.2]([[xi].sup.2]), with linear growth. We show that, regardless of the corresponding property of [f.sub.2], the assumption (t [member of] R) [Please download the PDF to view the mathematical expression], is sufficient to obtain higher integrability of [[partial derivative].sub.1]u for any finite exponent. We also include a series of variants of our main theorem. In the case f: [R.sup.n] [right arrow] R, similar results hold with the obvious changes in notation. Bibliography: 30 titles., UDC 517.958 1 Introduction In this paper, we discuss variational problems of linear growth with densities which do not belong to the class of [mu]-elliptic energies introduced first in [1]. [...]

We establish the existence of solutions for denoising problems with edge-enhancing diffusion including a large class of diffusion tensors. Bibliography: 5 titles., UDC 517.958 1 Introduction and the Main Result The joint work 1) of the authors with M. Cardenas and J. Weickert on edge-enhancing diffusion inpainting problems motivated the authors to [...]

The TV-regularization method due to Rudin, Osher, and Fatemi is widely used in mathematical image analysis. We consider a nonstationary and iterative variant of this approach and provide a mathematical theory that extends the results of Radmoser et al. to the BV setting. While existence and uniqueness, a maximum-minimum principle, and preservation of the average grey value are not hard to prove, we also establish the convergence to a constant steady state and consider a large family of Lyapunov functionals. These properties allow us to interpret the iterated TV-regularization as a time-discrete scale-space representation of the original image. Bibliography: 19 titles. Dedicated to Professor Nina Uraltseva on the occasion of her 85th birthday, UDC 517.9 1 Introduction The theory of regularization methods for ill-posed problems goes back to Tikhonov [1]. It has found numerous applications in inverse problems and mathematical image analysis, where [...]

We prove the solvability in Sobolev spaces for a class of variational problems related to the TV-model proposed by Rudin, Osher and Fatemi for the denoising of greyscale images. In contrast to their approach, we discuss energy densities with variable growth rates depending on [[absolute value of [nabla].sub.u]] in a rather general form including functionals of (1,p)-growth. Bibliography: 26 titles. Illustrations: 1 figure., UDC 517.9 1 Introduction In 1992, Rudin, Osher and Fatemi [1] proposed to study the variational problem [mathematical expression not reproducible] (1.1) as a model for the restoration of a [...]

We discuss variational approaches towards the denoising of images and towards the image inpainting problem combined with simultaneous denoising. Our techniques are based on variants of the TV-model, but in contrast to this case a complete analytical theory is available in our setting. At the same time, numerical experiments illustrate the advantages of our models in comparison with some established techniques. Bibliography: 50 titles. Illustrations: 1 figure., UDC 517.958 In this paper, we investigate the variational problem J [u] := [[integral] over [OMEGA]-D][(f - u).sup.2] dx + [alpha] [[integral] over [OMEGA]] [??]([absolute value of ([nabla]u)])dx [right arrow] [...]

We continue the analysis of modifications of the total variation image inpainting method formulated on the space BV[([OMEGA]).sup.M] and treat the case of vector-valued images where we do not impose any structure condition on the density F and the dimension of the domain Q is arbitrary. We discuss the existence of generalized solutions of the corresponding variational problem and show the unique solvability of the associated dual variational problem. We establish the uniqueness of the absolutely continuous part [[nabla].sup.a]u of the gradient of BV-solutions u on the domain Q and get the uniqueness of BV-solutions outside the damaged region D. We also prove new density results for functions of bounded variation and for Sobolev functions. Bibliography: 36 titles., UDC 517.9 1 Introduction In this paper, we continue the analysis of some perturbations of the total variation image in-painting model started in [1] from a more theoretical point of [...]

We propose a model for the restoration of images consisting only of completely black or completely white regions with the use of Caccioppoli sets. Bibliography: 15 titles., UDC 517.9 Dedicated to Professor N.N. Uraltseva on occasion of her jubilee The purpose of this short note is to present a technique which might be useful for the restoration [...]

We continue the analysis of some strongly elliptic modifications of the total variation image inpainting model formulated in the space BV([OMEGA]) and investigate the corresponding dual variational problems. Remarkable features are the uniqueness of the dual solution and the uniqueness of the absolutely continuous part [[nabla].sup.a]u of the gradient of BV-solutions u on the whole domain. Additionally, any BV-minimizer u automatically satisfies the inequality 0[less than or equal to]u[less than or equal to]1, which means that u measures the intensity of the grey level. Outside of the damaged region we even have the uniqueness of BV-solutions, whereas on the damaged domain the [L.sup.2]-deviation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of different solutions is governed by the the total variation of the singular part [[nabla].sup.s](u - [upsilon]) of the vector measure [nabla](u - [upsilon]). Moreover, the dual solution is related to the BV-solutions through an equation of stress-strain type. Bibliography: 23 titles., UDC 517.9 Dedicated to Professor N.N. Uraltseva on her jubilee 1 Introduction In this note, we continue the analysis of some perturbations of the total variation image inpainting method started [...]

We investigate modifications of the total variation image inpainting model and discuss the existence as well as the smoothness of solutions to these new variational problems. We emphasize that our variant of a linear growth regularization is completely discussed in the Sobolev space W~l without passing to a relaxed version in the space of functions with bounded variation. Our analysis mainly concentrates on the theoretical background of the problems under consideration. Nevertheless, first computational experiments indicate that our model is numerically comparable with the TV-model with the advantage of a clear underlying analysis presented in the following. Bibliography:22 titles. UDC 517.9 Dedicated to Professor N.N. Uraltseva on her jubilee, 1 Introduction We consider a black and white image described by a function u : [OMEGA] [vector] [0,1], where [OMEGA] denotes abounded Lipschitz domain in [R.sup.2], for example, a rectangle, [...]

We discuss different variants of the so-called total variation image inpainting method collecting existence and regularity results related to the proposed techniques. As a new feature we give an interpretation of a boundary value problem in the class of [L.sup.[infinity]]-data. We use an averaging process and pass to the limit. Bibliography: 20 titles., UDC 517.9 We start with a description of the problem under consideration: suppose that [OMEGA] and D are bounded domains in [R.sup.2] having Lipschitz continuous boundaries. Let the closure [bar.D] [...]