Gerd Teschke

A compressive Landweber iteration for solving ill-posed inverse problems

In this paper we shall be concerned with the construction of an adaptive Landweber iteration for solving linear ill-posed and inverse problems. Classical Landweber iteration schemes provide in combination with suitable regularization parameter rules order optimal regularization schemes. However, for many applications the implementation of Landwebers method is numerically very intensive. Therefore we propose an adaptive variant of Landwebers iteration that may reduce the computational expense significantly, i.e. leading to a compressed version of Landwebers iteration. We borrow the concept of adaptivity that was primarily developed for well-posed operator equations (in particular, for elliptic PDEs) essentially exploiting the concept of wavelets (frames), Besov regularity, best N-term approximation and combine it with classical iterative regularization schemes. As the main result of this paper we define an adaptive variant of Landwebers iteration. In combination with an adequate refinement/stopping rule (a priori as well as a posteriori principles) we prove that the proposed procedure is a regularization method which converges in norm for exact and noisy data. The proposed approach is verified in the field of computerized tomography imaging.

An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vector-valued regimes and an application to color image inpainting

This paper is concerned with nonlinear inverse problems where data and solution are vector valued and, moreover, where the solution is assumed to have a sparse expansion with respect to a preassigned frame. We especially focus on such problems where the different channels of the solution exhibit a common or so-called joint sparsity pattern encoding special characteristics of the function under consideration (e.g. a coupling of non-vanishing channel components). Quite recently, an iterative strategy for linear inverse problems with such joint sparsity constraints was presented. Here, we develop an iterative approach for nonlinear inverse problems for which we show norm convergence and regularization properties. The focus throughout the paper is in the context of color image inpainting/recolorization.

Generalized Sampling: Stable Reconstructions, Inverse Problems and Compressed Sensing over the Continuum

Abstract The purpose of this paper is to report on recent approaches to reconstruction problems based on analog, or in other words, infinite-dimensional, image and signal models. We describe three main contributions to this problem. First, linear reconstructions from sampled measurements via so-called generalized sampling (GS). Second, the extension of generalized sampling to inverse and ill-posed problems. And third, the combination of generalized sampling with sparse recovery techniques. This final contribution leads to a theory and set of methods for infinite-dimensional compressed sensing, or as we shall also refer to it, compressed sensing over the continuum.

On Some Iterative Concepts for Image Restoration

Publisher Summary This chapter discusses several iterative strategies for solving inverse problems in the context of signal and image processing. The chapter focuses on problems for which it is reasonable to assume that the solution has a sparse expansion with respect to a wavelet basis or frame. In each case, a variational formulation of the problem is presented, and an iteration scheme for which iterates approximate the solution is constructed. Surrogate functionals are applied; the corresponding strategy is shown to converge in norm and to regularize the problem. The chapter begins with the concrete problem of simultaneously denoising, decomposing, and deblurring a given image. The associated variational formulation of the problem contains terms that promote sparsity and smoothness. The process to transform problem is presented in the chapter, such as the basic method of Daubechies and all. In a second example, a natural extension to vector-valued inverse problems is discussed. Potential applications include seismic or astrophysical data decomposition/reconstruction and color image reconstruction. The illustration presented contains audio data coding. In the linear case, and under fairly general assumptions on the constraint, it is proved that weak convergence of the iterative scheme always holds. In certain cases (i.e, for special families of convex constraints), this weak convergence implies norm convergence. The presented technique covers a wide range of problems. The chapter also discusses image restoration problems in which Besov- or bounded variation (BV) constraints are involved. The chapter concludes with a sketch design of hybrid wavelet-partial differential equation (PDE) image restoration schemes (i.e, with variational problems that contain wavelet and BV constraints).

The application of joint sparsity and total variation minimization algorithms to a real-life art restoration problem

On March 11, 1944, the famous Eremitani Church in Padua (Italy) was destroyed in an Allied bombing along with the inestimable frescoes by Andrea Mantegna et al. contained in the Ovetari Chapel. In the last 60xa0years, several attempts have been made to restore the fresco fragments by traditional methods, but without much success. One of the authors contributed to the development of an efficient pattern recognition algorithm to map the original position and orientation of the fragments, based on comparisons with an old gray level image of the fresco prior to the damage. This innovative technique allowed for the partial reconstruction of the frescoes. Unfortunately, the surface covered by the colored fragments is only 77xa0m2, while the original area was of several hundreds. This means that we can reconstruct only a fraction (less than 8%) of this inestimable artwork. In particular the original color of the blanks is not known. This begs the question of whether it is possible to estimate mathematically the original colors of the frescoes by making use of the potential information given by the available fragments and the gray level of the pictures taken before the damage. Moreover, is it possible to estimate how faithful such a restoration is? In this paper we retrace the development of the recovery of the frescoes as an inspiring and challenging real-life problem for the development of new mathematical methods. Then we shortly review two models recently studied independently by the authors for the recovery of vector valued functions from incomplete data, with applications to the recolorization problem. The models are based on the minimization of a functional which is formed by the discrepancy with respect to the data and additional regularization constraints. The latter refer to joint sparsity measures with respect to frame expansions, in particular wavelet or curvelet expansions, for the first functional and functional total variation for the second. We establish relations between these two models. As a major contribution of this work we perform specific numerical test on the real-life problem of the A. Mantegna’s frescoes and we compare the results due to the two methods.

Coorbit Theory, Multi- -Modulation Frames, and the Concept of Joint Sparsity for Medical Multichannel Data Analysis

This paper is concerned with the analysis and decomposition of medical multichannel data. We present a signal processing technique that reliably detects and separates signal components such as mMCG, fMCG, or MMG by involving the spatiotemporal morphology of the data provided by the multisensor geometry of the so-called multichannel superconducting quantum interference device (SQUID) system. The mathematical building blocks are coorbit theory, multi--modulation frames, and the concept of joint sparsity measures. Combining the ingredients, we end up with an iterative procedure (with component-dependent projection operations) that delivers the individual signal components.

Sparse Representations and Efficient Sensing of Data (Dagstuhl Seminar 11051)

This report documents the program and the outcomes of Dagstuhl Seminar 11051 ``Sparse Representations and Efficient Sensing of Data. nThe scope of the seminar was twofold. nFirst, we wanted to elaborate the state of the art in nthe field of sparse data representation and corresponding efficient data sensing methods. nSecond, we planned to explore and analyze the impact of methods in computational science disciplines that serve these fields, and the possible resources allocated for industrial applications.

Coorbit Theory, Multi- Open image in new window -Modulation Frames, and the Concept of Joint Sparsity for Medical Multichannel Data Analysis

This paper is concerned with the analysis and decomposition of medical multichannel data. We present a signal processing technique that reliably detects and separates signal components such as mMCG, fMCG, or MMG by involving the spatiotemporal morphology of the data provided by the multisensor geometry of the so-called multichannel superconducting quantum interference device (SQUID) system. The mathematical building blocks are coorbit theory, multi--modulation frames, and the concept of joint sparsity measures. Combining the ingredients, we end up with an iterative procedure (with component-dependent projection operations) that delivers the individual signal components.