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We consider linear inverse problems where the solution is assumed to fulfill some general homogeneous convex constraint. We develop an algorithm that amounts to a projected Landweber iteration and that provides and iterative approach to the solution of this inverse problem. For relatively moderate assumptions on the constraint we can always prove weak(More)
Inspired by papers of Vese–Osher [OV02] and Osher–Solé–Vese [OSV02] we present a wavelet–based treatment of variational problems arising in the field of image processing. In particular, we follow their approach and discuss a special class of variational functionals that induce a decomposition of images into oscillating and cartoon components and possibly an(More)
In this paper, we consider nonlinear inverse problems where the solution is assumed to have a sparse expansion with respect to a preassigned basis or frame. We develop a scheme which allows to minimize a Tikhonov functional where the usual quadratic regu-larization term is replaced by a one–homogeneous (typically weighted p) penalty on the coefficients (or(More)
We shall be concerned with the construction of Tikhonov–based iteration schemes for solving nonlinear operator equations. In particular, we are interested in algorithms for the computation of a minimizer of the Tikhonov functional. To this end, we introduce a replacement functional, that has much better properties than the classical Tikhonov functional with(More)
This paper is concerned with the construction of an iterative algorithm to solve non-linear inverse problems with an 1 constraint. One extensively studied method to obtain a solution of such an 1 penalized problem is iterative soft-thresholding. Regrettably, such iteration schemes are computationally very intensive. A subtle alternative to iterative(More)
Generalized sampling is new framework for sampling and reconstruction in infinite-dimensional Hilbert spaces. Given measurements (inner products) of an element with respect to one basis, it allows one to reconstruct in another, arbitrary basis, in a way that is both convergent and numerically stable. However, generalized sampling is thus far only valid for(More)
Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since it might be difficult to obtain directional information by means of wavelets, several new representation systems were proposed in the past, including ridgelets, curvelets and, more recently,(More)
| In this paper we describe a method for classifying material properties from measurements of the Barkhausen eeect, which originates from a fast magnetiza-tion of ferromagnetic materials using alternating currents. We use wavelet analysis to develop a tool box for evaluating Barkhausen measurements. The described wavelet techniques allow to detect extremely(More)
In this paper we shall be concerned with compressive sampling strategies and sparse recovery principles for linear inverse and ill-posed problems. As the main result, we provide compressed measurement models for ill-posed problems and recovery accuracy estimates for sparse approximations of the solution of the underlying inverse problem. The main(More)