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This paper focuses on identifying vertex characteristics in 2D images using topological asymptotic analysis. Vertex characteristics include both the location and the type of the vertex, with the latter defined by the number of lines forming it and the corresponding angles. This problem is crucial for computer vision tasks, such as distinguishing between fore- and background objects in 3D scenes. We compute the second-order topological derivative of a Mumford-Shah type functional with respect to inclusion shapes representing various vertex types. This derivative assigns a likelihood to each pixel that a particular vertex type appears there. Numerical tests demonstrate the effectiveness of the proposed approach.

We consider solving a probably ill-conditioned linear operator equation, where the operator is not modeled by physical laws but is specified via training pairs (consisting of images and data) of the input-output relation of the operator. We derive a stable method for computing the operator, which consists of first a Gram-Schmidt orthonormalization of images and a principal component analysis of the data. This two-step algorithm provides a spectral decomposition of the linear operator. Moreover, we show that both Gram-Schmidt and principal component analysis can be written as a deep neural network, which relates this procedure to de-and encoder networks. Therefore, we call the two-step algorithm a linear algebra network. Finally, we provide numerical simulations showing the strategy is feasible for reconstructing spectral functions and for solving operator equations without explicitly exploiting the physical model.

We present a method to reconstruct the dielectric susceptibility (scattering potential) of an inhomogeneous scattering medium, based on the solution to the inverse scattering problem with internal sources. We employ the theory of reproducing kernel Hilbert spaces, together with regularization to recover the susceptibility of two- and three-dimensional scattering media. Numerical examples illustrate the effectiveness of the proposed reconstruction method., Comment: 11 Pages, 6 figures, 1 table

We investigate the transport of intensity equation (TIE) and the transport of phase equation (TPE) for solving the phase retrieval problem. Both the TIE and the TPE are derived from the paraxial Helmholtz equation and relate phase information to the intensity. The TIE is usually favored since the TPE is nonlinear. The main contribution of this paper is a discussion of preferential use of either one of the two equations or potential benefits of a hybrid use. Moreover, we discuss the solution of the TPE with the method of characteristics and with viscosity methods. Both the TIE and the viscosity method are numerically implemented with finite element methods.

Diffraction tomography is an inverse scattering technique used to reconstruct the spatial distribution of the material properties of a weakly scattering object. The object is exposed to radiation, typically light or ultrasound, and the scattered waves induced from different incident field angles are recorded. In conventional diffraction tomography, the incident wave is assumed to be a monochromatic plane wave, an unrealistic simplification in practical imaging scenarios. In this article, we extend conventional diffraction tomography by introducing the concept of customized illumination scenarios, with a pronounced emphasis on imaging with focused beams. We present a new forward model that incorporates a generalized incident field and extends the classical Fourier diffraction theorem to the use of this incident field. This yields a new two-step reconstruction process which we comprehensively evaluate through numerical experiments.

Neural networks with quadratic decision functions have been introduced as alternatives to standard neural networks with affine linear ones. They are advantageous when the objects or classes to be identified are compact and of basic geometries like circles, ellipses etc. In this paper we investigate the use of such ansatz functions for classification. In particular we test and compare the algorithm on the MNIST dataset for classification of handwritten digits and for classification of subspecies. We also show, that the implementation can be based on the neural network structure in the software Tensorflow and Keras, respectively.

We investigate generalized versions of the Iteratively Regularized Landweber Method, initially introduced in [Appl. Math. Optim., 38(1):45-68, 1998], to address linear and nonlinear ill-posed problems. Our approach is inspired by the data-driven perspective emphasized in the introduction by Aspri et al. [Numer. Funct. Anal. Optim., 41(10):1190-1227, 2020]. We provide a rigorous analysis establishing convergence and stability results and present numerical outcomes for linear operators, with the Radon transform serving as a prototype.

We consider the problem of blob detection for uncertain images, such as images that have to be inferred from noisy measurements. Extending recent work motivated by astronomical applications, we propose an approach that represents the uncertainty in the position and size of a blob by a region in a three-dimensional scale space. Motivated by classic tube methods such as the taut-string algorithm, these regions are obtained from level sets of the minimizer of a total variation functional within a high-dimensional tube. The resulting non-smooth optimization problem is challenging to solve, and we compare various numerical approaches for its solution and relate them to the literature on constrained total variation denoising. Finally, the proposed methodology is illustrated on numerical experiments for deconvolution and models related to astrophysics, where it is demonstrated that it allows to represent the uncertainty in the detected blobs in a precise and physically interpretable way.

Neural networks functions are supposed to be able to encode the desired solution of an inverse problem very efficiently. In this paper, we consider the problem of solving linear inverse problems with neural network coders. First we establish some correspondences of this formulation with existing concepts in regularization theory, in particular with state space regularization, operator decomposition and iterative regularization methods. A Gauss-Newton's method is suitable for solving encoded linear inverse problems, which is supported by a local convergence result. The convergence studies, however, are not complete, and are based on a conjecture on linear independence of activation functions and its derivatives.

We consider the problem of elastic diffraction tomography, which consists in reconstructing elastic properties (i.e. mass density and elastic Lam\'e parameters) of a weakly scattering medium from full-field data of scattered waves outside the medium. Elastic diffraction tomography refers to the elastic inverse scattering problem after linearization using a firstorder Born approximation. In this paper, we prove the Fourier diffraction theorem, which relates the 2D Fourier transform of scattered waves with the Fourier transform of the scatterer in the 3D spatial Fourier domain. Elastic wave mode separation is performed, which decomposes a wave into four modes. A new two-step inversion process is developed, providing information on the modes first and secondly on the elastic parameters. Finally, we discuss reconstructions with plane wave excitation experiments for different tomographic setups and with different plane wave excitation frequencies, respectively., Comment: 23 pages 5 figures

The method of common lines is a well-established reconstruction technique in cryogenic electron microscopy (cryo-EM), which can be used to extract the relative orientations of an object given tomographic projection images from different directions. In this paper, we deal with an analogous problem in optical diffraction tomography. Based on the Fourier diffraction theorem, we show that rigid motions of the object, i.e., rotations and translations, can be determined by detecting common circles in the Fourier-transformed data. We introduce two methods to identify common circles. The first one is motivated by the common line approach for projection images and detects the relative orientation by parameterizing the common circles in the two images. The second one assumes a smooth motion over time and calculates the angular velocity of the rotational motion via an infinitesimal version of the common circle method. Interestingly, using the stereographic projection, both methods can be reformulated as common line methods, but these lines are, in contrast to those used in cryo-EM, not confined to pass through the origin and allow for a full reconstruction of the relative orientations. Numerical proof-of-the-concept examples demonstrate the performance of our reconstruction methods., Comment: 35 pages, 13 figures

Context. Blob detection is a common problem in astronomy. One example is in stellar population modelling, where the distribution of stellar ages and metallicities in a galaxy is inferred from observations. In this context, blobs may correspond to stars born in-situ versus those accreted from satellites, and the task of blob detection is to disentangle these components. A difficulty arises when the distributions come with significant uncertainties, as is the case for stellar population recoveries inferred from modelling spectra of unresolved stellar systems. There is currently no satisfactory method for blob detection with uncertainties. Aims. We introduce a method for uncertainty-aware blob detection developed in the context of stellar population modelling of integrated-light spectra of stellar systems. Methods. We develop theory and computational tools for an uncertainty-aware version of the classic Laplacian-of-Gaussians method for blob detection, which we call ULoG. This identifies significant blobs considering a variety of scales. As a prerequisite to apply ULoG to stellar population modelling, we introduce a method for efficient computation of uncertainties for spectral modelling. This method is based on the truncated Singular Value Decomposition and Markov Chain Monte Carlo sampling (SVD-MCMC). Results. We apply the methods to data of the star cluster M54. We show that the SVD-MCMC inferences match those from standard MCMC, but are a factor 5-10 faster to compute. We apply ULoG to the inferred M54 age/metallicity distributions, identifying between 2 or 3 significant, distinct populations amongst its stars., Comment: Revision submitted to A&A, comments welcome

We consider the inverse problem of quantitative reconstruction of properties (e.g., bulk modulus, density) of visco-acoustic materials based on measurements of responding waves after stimulation of the medium. Numerical reconstruction is performed by an iterative minimization algorithm. Firstly, we investigate the robustness of the algorithm with respect to attenuation model uncertainty, that is, when different attenuation models are used to simulate synthetic observation data and for the inversion, respectively. Secondly, to handle data-sets with multiple reflections generated by wall boundaries around the domain, we perform inversion using complex frequencies, and show that it offers a robust framework that alleviates the difficulties of multiple reflections. To illustrate the efficiency of the algorithm, we perform numerical simulations of ultrasound imaging experiments to reconstruct a synthetic breast sample that contains an inclusion of high-contrast properties. We perform experiments in two and three dimensions, where the latter also serves to demonstrate the numerical feasibility in a large-scale configuration., Comment: 30 pages, 13 figures

In this paper, we study the mathematical imaging problem of diffraction tomography (DT), which is an inverse scattering technique used to find material properties of an object by illuminating it with probing waves and recording the scattered waves. Conventional DT relies on the Fourier diffraction theorem, which is applicable under the condition of weak scattering. However, if the object has high contrasts or is too large compared to the wavelength, it tends to produce multiple scattering, which complicates the reconstruction. We give a survey on diffraction tomography and compare the reconstruction of low and high contrast objects. We also implement and compare the reconstruction using the full waveform inversion method which, contrary to the Born and Rytov approximations, works with the total field and is more robust to multiple scattering., Comment: 32 pages, 21 figures

In this paper we study shallow neural network functions which are linear combinations of compositions of activation and quadratic functions, replacing standard affine linear functions, often called neurons. We show the universality of this approximation and prove convergence rates results based on the theory of wavelets and statistical learning. We show for simple test cases that this ansatz requires a smaller numbers of neurons than standard affine linear neural networks. Moreover, we investigate the efficiency of this approach for clustering tasks with the MNIST data set. Similar observations are made when comparing deep (multi-layer) networks.

In this paper we discuss a deterministic form of ensemble Kalman inversion as a regularization method for linear inverse problems. By interpreting ensemble Kalman inversion as a low-rank approximation of Tikhonov regularization, we are able to introduce a new sampling scheme based on the Nystr\"om method that improves practical performance. Furthermore, we formulate an adaptive version of ensemble Kalman inversion where the sample size is coupled with the regularization parameter. We prove that the proposed scheme yields an order optimal regularization method under standard assumptions if the discrepancy principle is used as a stopping criterion. The paper concludes with a numerical comparison of the discussed methods for an inverse problem of the Radon transform.

In this paper, we study the mathematical imaging problem of optical diffraction tomography (ODT) for the scenario of a microscopic rigid particle rotating in a trap created, for instance, by acoustic or optical forces. Under the influence of the inhomogeneous forces the particle carries out a time-dependent smooth, but complicated motion described by a set of affine transformations. The rotation of the particle enables one to record optical images from a wide range of angles, which largely eliminates the "missing cone problem" in optics. This advantage, however, comes at the price that the rotation axis in this scenario is not fixed, but continuously undergoes some variations, and that the rotation angles are not equally spaced, which is in contrast to standard tomographic reconstruction assumptions. In the present work, we assume that the time-dependent motion parameters are known, and that the particle's scattering potential is compatible with making the first order Born or Rytov approximation. We prove a Fourier diffraction theorem and derive novel backprojection formulae for the reconstruction of the scattering potential, which depends on the refractive index distribution inside the object, taking its complicated motion into account. This provides the basis for solving the ODT problem with an efficient non-uniform discrete Fourier transform.

In this paper, we consider visualization of displacement fields via optical flow methods in elastographic experiments consisting of a static compression of a sample. We propose an elastographic optical flow method (EOFM) which takes into account experimental constraints, such as appropriate boundary conditions, the use of speckle information, as well as the inclusion of structural information derived from knowledge of the background material. We present numerical results based on both simulated and experimental data from an elastography experiment in order to demonstrate the relevance of our proposed approach., Comment: 12 pages, 8 figures

We study the problem of regularization of inverse problems adopting a purely data driven approach, by using the similarity to the method of regularization by projection. We provide an application of a projection algorithm, utilized and applied in frames theory, as a data driven reconstruction procedure in inverse problems, generalizing the algorithm proposed by the authors in Inverse Problems 36 (2020), n. 12, 125009, based on an orthonormalization procedure for the training pairs. We show some numerical experiments, comparing the different methods.

In this paper, we consider the problem of estimating the internal displacement field of an object which is being subjected to a deformation, from Optical Coherence Tomography (OCT) images before and after compression. For the estimation of the internal displacement field we propose a novel algorithm, which utilizes particular speckle information to enhance the quality of the motion estimation. We present numerical results based on both simulated and experimental data in order to demonstrate the usefulness of our approach, in particular when applied for quantitative elastography, when the material parameters are estimated in a second step based on the internal displacement field., Comment: 31 pages

In this paper, we provide assembly instructions for an easy to build experimental setup in order to gain practical experience with tomography. In view of this, this paper can be seen as a complementary work to excellent mathematical textbooks at an undergraduate level concerned with the basic mathematical principles of tomography. Since the setup uses light for tomographic imaging, the investigated objects need to be light transparent, like origami figures. In case the reader wants to experiment with computational tomographic reconstructions without assembling the device, we provide a database of several objects together with their tomographic measurements and a publicly available software. Moreover, recent advances in Cryo-imaging enabled three-dimensional high-resolution visualization of single particles, such as for instance viruses. To exemplify, and experience, single particle Cryo-electron microscopy we provide an advanced assembly, which we use to generate data simulating a Cryo-recording. We also discuss some of the major practical difficulties for reconstructing particles from Cryo-microscopic data., Comment: 13 pages, 14 figures

Quantitative imaging of sub-surface Earth's properties in elastic media is performed from Distributed Acoustic Sensing data. A new misfit functional based upon the reciprocity-gap is designed, taking cross-correlations of displacement and strain, and these products further associate an observation with a simulation. In comparison with other misfit functionals, this one has the advantage to only require little a-priori information on the exciting sources. In particular, the misfit criterion enables the use of data from regional earthquakes (teleseismic events can be included as well), followed by exploration data to perform a multi-resolution reconstruction. The data from regional earthquakes contain the low-frequency content which is missing in the exploration ones, allowing for the recovery of the long spatial wavelength, even with very few sources. These data are used to build prior models for the subsequent reconstruction from the higher-frequency exploration data. This gives the elastic Full Reciprocity-gap Waveform Inversion method, and we demonstrate its performance with a pilot experiment for elastic isotropic reconstruction., Comment: 21 pages, 6 figures

In this paper we propose a variational regularization method for denoising and inpainting of diffusion tensor magnetic resonance images. We consider these images as manifold-valued Sobolev functions, i.e. in an infinite dimensional setting, which are defined appropriately. The regularization functionals are defined as double integrals, which are equivalent to Sobolev semi-norms in the Euclidean setting. We extend the analysis of Ciak, Melching and Scherzer "Regularization with Metric Double Integrals of Functions with Values in a Set of Vectors", in: Journal of Mathematical Imaging and Vision (2019) concerning stability and convergence of the variational regularization methods by a uniqueness result, apply them to diffusion tensor processing, and validate our model in numerical examples with synthetic and real data., Comment: 10 figures

In this paper, we perform non-linear minimization using the Hybridizable Discontinuous Galerkin method (HDG) for the discretization of the forward problem, and implement the adjoint-state method for the computation of the functional derivatives. Compared to continuous and discontinuous Galerkin discretizations, HDG reduces the computational cost by working with the numerical traces, hence removing the degrees of freedom that are inside the cells. It is particularly attractive for large-scale time-harmonic quantitative inverse problems which make repeated use of the forward discretization as they rely on an iterative minimization procedure. HDG is based upon two levels of linear problems: a global system to find the numerical traces, followed by local systems to construct the volume solution. This technicality requires a careful derivation of the adjoint-state method, that we address in this paper. We work with the acoustic wave equations in the frequency domain and illustrate with a three-dimensional experiment using partial reflection-data, where we further employ the features of DG-like methods to efficiently handle the topography with p-adaptivity., Comment: 24 pages, 8 figures

Single molecule localization microscopy is a recently developed superresolution imaging technique to visualize structural properties of single cells. The basic principle consists in chemically attaching fluorescent dyes to the molecules, which after excitation with a strong laser may emit light. To achieve superresolution, signals of individual fluorophores are separated in time. In this paper we follow the physical and chemical literature and derive mathematical models describing the propagation of light emitted from dyes in single molecule localization microscopy experiments via Maxwell's equations. This forms the basis of formulating inverse problems related to single molecule localization microscopy. We also show that the current status of reconstruction methods is a simplification of more general inverse problems for Maxwell's equations as discussed here., Comment: 38 pages, 10 figures

In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equations in the presence of small inclusions. As a byproduct, we derive a topological derivative based algorithm for the reconstruction of piecewise smooth functions. This algorithm can be used for edge detection in imaging, topological optimization, and for inverse problems, such as Quantitative Photoacoustic Tomography, for which we demonstrate the effectiveness of our asymptotic expansion method numerically.

A recent result by Lasica, Moll and Mucha about the $\ell^1$-anisotropic Rudin-Osher-Fatemi model in $\mathbb{R}^2$ asserts that the solution is piecewise constant on a rectilinear grid, if the datum is. By means of a new proof we extend this result to $\mathbb{R}^n$. The core of our proof consists in showing that averaging operators associated to certain rectilinear grids map subgradients of the $\ell^1$-anisotropic total variation seminorm to subgradients.

We present a family of non-local variational regularization methods for solving tomographic problems, where the solutions are functions with range in a closed subset of the Euclidean space, for example if the solution only attains values in an embedded sub-manifold. Recently, in Ciak, Melching and Scherzer "Regularization with Metric Double Integrals of Functions with Values in a Set of Vectors", in: Journal of Mathematical Imaging and Vision (2019), such regularization methods have been investigated analytically and their efficiency has been tested for basic imaging tasks such as denoising and inpainting. In this paper we investigate solving complex vector tomography problems with non-local variational methods both analytically and numerically., Comment: 6 Figures

We study linear inverse problems under the premise that the forward operator is not at hand but given indirectly through some input-output training pairs. We demonstrate that regularization by projection and variational regularization can be formulated by using the training data only and without making use of the forward operator. We study convergence and stability of the regularized solutions in view of T. I. Seidman. "Nonconvergence Results for the Application of Least-Squares Estimation to Ill-Posed Problems". Journal of Optimization Theory and Applications 30.4 (1980), pp. 535-547, who showed that regularization by projection is not convergent in general, by giving some insight on the generality of Seidman's nonconvergence example. Moreover, we show, analytically and numerically, that regularization by projection is indeed capable of learning linear operators, such as the Radon transform.

The tangential condition was introduced in [Hanke et al., 95] as a sufficient condition for convergence of the Landweber iteration for solving ill-posed problems. In this paper we present a series of time dependent benchmark inverse problems for which we can verify this condition.

Optical and acoustical trapping has been established as a tool for holding and moving microscopic particles suspended in a liquid in a contact-free and non-invasive manner. Opposed to standard microscopic imaging where the probe is fixated, this technique allows imaging in a more natural environment. This paper provides a method for estimating the movement of a transparent particle which is maneuvered by tweezers, assuming that the inner structure of the probe is not subject to local movements. The mathematical formulation of the motion estimation shows some similarities to Cryo-EM single particle imaging, where the recording orientations of the probe need to be estimated.

This paper is concerned with robust preconditioning of wave equations constrained linear inverse problems from boundary observation data. The main result of this paper is a concept for regularization parameter robust preconditioning. Analogous concepts have been developed for control problems based on elliptic partial equations before.

We study the seismic inverse problem for the recovery of subsurface properties in acoustic media. In order to reduce the ill-posedness of the problem, the heterogeneous wave speed parameter to be recovered is represented using a limited number of coefficients associated with a basis of eigenvectors of a diffusion equation, following the regularization by discretization approach. We compare several choices for the diffusion coefficient in the partial differential equations, which are extracted from the field of image processing. We first investigate their efficiency for image decomposition (accuracy of the representation with respect to the number of variables and denoising). Next, we implement the method in the quantitative reconstruction procedure for seismic imaging, following the Full Waveform Inversion method, where the difficulty resides in that the basis is defined from an initial model where none of the actual structures is known. In particular, we demonstrate that the method is efficient for the challenging reconstruction of media with salt-domes. We employ the method in two and three-dimensional experiments and show that the eigenvector representation compensates for the lack of low frequency information, it eventually serves us to extract guidelines for the implementation of the method., Comment: 45 pages, 37 figures

Recently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov's algorithm and the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), respectively. In this paper we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residuum of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. We prove that the proposed flows for minimising the residuum of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions. In particular we show that in comparison to convex analysis results the rates can be significantly higher, which is possible by constraining the investigations to the particular convex energy functional, which is the squared norm of the residuum.

We derive and analyse a new variant of the iteratively regularized Landweber iteration, for solving linear and nonlinear ill-posed inverse problems. The method takes into account training data, which are used to estimate the interior of a black box, which is used to define the iteration process. We prove convergence and stability for the scheme in infinite dimensional Hilbert spaces. These theoretical results are complemented by several numerical experiments for solving linear inverse problems for the Radon transform and a nonlinear inverse problem for Schlieren tomography.

Total variation flow, total variation regularization and the taut string algorithm are known to be equivalent filters for one-dimensional discrete signals. In addition, the filtered signal simultaneously minimizes a large number of convex functionals in a certain neighbourhood of the data. In this article we study the question to what extent this situation remains true in a more general setting, namely for data given on the vertices of a finite oriented graph and the total variation being $J(f) = \sum_{i,j} |f(v_i) - f(v_j)|$. Relying on recent results on invariant $\varphi$-minimal sets we prove that the minimizer to the corresponding Rudin-Osher-Fatemi (ROF) model on the graph has the same universal minimality property as in the one-dimensional setting. Interestingly, this property is lost, if $J$ is replaced by the discrete isotropic total variation. Next, we relate the ROF minimizer to the solution of the gradient flow for $J$. It turns out that, in contrast to the one-dimensional setting, these two problems are not equivalent in general, but conditions for equivalence are available.

We present an approach for variational regularization of inverse and imaging problems for recovering functions with values in a set of vectors. We introduce regularization functionals, which are derivative-free double integrals of such functions. These regularization functionals are motivated from double integrals, which approximate Sobolev semi-norms of intensity functions. These were introduced in Bourgain, Br\'ezis and Mironescu, "Another Look at Sobolev Spaces". In: Optimal Control and Partial Differential Equations-Innovations and Applications, IOS press, Amsterdam, 2001. For the proposed regularization functionals we prove existence of minimizers as well as a stability and convergence result for functions with values in a set of vectors.

We consider the flow of multiple particles in a Bingham fluid in an anti-plane shear flow configuration. The limiting situation in which the internal and applied forces balance and the fluid and particles stop flowing, that is, when the flow settles, is formulated as finding the optimal ratio between the total variation functional and a linear functional. The minimal value for this quotient is referred to as the critical yield number or, in analogy to Rayleigh quotients, generalized eigenvalue. This minimum value can in general only be attained by discontinuous, hence not physical, velocities. However, we prove that these generalized eigenfunctions, whose jumps we refer to as limiting yield surfaces, appear as rescaled limits of the physical velocities. Then, we show the existence of geometrically simple minimizers. Furthermore, a numerical method for the minimization is then considered. It is based on a nonlinear finite difference discretization, whose consistency is proven, and a standard primal-dual descent scheme. Finally, numerical examples show a variety of geometric solutions exhibiting the properties discussed in the theoretical sections., Comment: 29 pages, 10 figures, to appear in Applied Mathematics & Optimization

In the context of convex variational regularization it is a known result that, under suitable differentiability assumptions, source conditions in the form of variational inequalities imply range conditions, while the converse implication only holds under an additional restriction on the operator. In this article we prove the analogous result for polyconvex regularization. More precisely, we show that the variational inequality derived by Kirisits, Scherzer (2017) implies that the derivative of the regularization functional must lie in the range of the dual-adjoint of the derivative of the operator. In addition, we show how to adapt the restriction on the operator in order to obtain the converse implication.

In a recent paper by A. Chambolle et al. [Geometric properties of solutions to the total variation denoising problem. Inverse Problems 33, 2017] it was proven that if the subgradient of the total variation at the noise free data is not empty, the level-sets of the total variation denoised solutions converge to the level-sets of the noise free data with respect to the Hausdorff distance. The condition on the subgradient corresponds to the source condition introduced by Burger and Osher [Convergence rates of convex variational regularization. Inverse Problems 20, 2004], who proved convergence rates results with respect to the Bregman distance under this condition. We generalize the result of Chambolle et al. to total variation regularization of general linear inverse problems under such a source condition. As particular applications we present denoising in bounded and unbounded, convex and non convex domains, deblurring and inversion of the circular Radon transform. In all these examples the convergence result applies. Moreover, we illustrate the convergence behavior through numerical examples., Comment: 26 pages, 3 Figures, to appear in Inverse Problems

We consider a problem of quantitative static elastography, the estimation of the Lam\'e parameters from internal displacement field data. This problem is formulated as a nonlinear operator equation. To solve this equation, we investigate the Landweber iteration both analytically and numerically. The main result of this paper is the verification of a nonlinearity condition in an infinite dimensional Hilbert space context. This condition guarantees convergence of iterative regularization methods. Furthermore, numerical examples for recovery of the Lam\'e parameters from displacement data simulating a static elastography experiment are presented., Comment: 29 pages