Ronny Ramlau

A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints

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 regularization term is replaced by a one-homogeneous (typically weighted ℓp) penalty on the coefficients (or isometrically transformed coefficients) of such expansions. For (p < 2), the regularized solution will have a sparser expansion with respect to the basis or frame under consideration. The computation of the regularized solution amounts in our setting to a Landweber-fixed-point iteration with a projection applied in each fixed-point iteration step. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse single photon emission computerized tomography (SPECT) problem.

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.

A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data

A level-set based approach for the determination of a piecewise constant density function from data of its Radon transform is presented. Simultaneously, a segmentation of the reconstructed density is obtained. The segmenting contour and the corresponding density are found as minimizers of a Mumford-Shah like functional over the set of admissible contours and - for a fixed contour - over the space of piecewise constant densities which may be discontinuous across the contour. Shape sensitivity analysis is used to find a descent direction for the cost functional which leads to an update formula for the contour in the level-set framework. The descent direction can be chosen with respect to different metrics. The use of an L^2-type and an H^1-type metric is proposed and the corresponding steepest descent flow equations are derived. A heuristic approach for the insertion of additional components of the density is presented. The method is tested for several data sets including synthetic as well as real-world data. It is shown that the method works especially well for large data noise (~10% noise). The choice of the H^1-metric for the determination of the descent direction is found to have positive effect on the number of level-set steps necessary for finding the optimal contours and densities.

TIGRA—an iterative algorithm for regularizing nonlinear ill-posed problems

We report on a new iterative method for regularizing a nonlinear operator equation in Hilbert spaces. The proposed TIGRA algorithm is a combination of Tikhonov regularization and a gradient method for minimizing the Tikhonov functional. Under the assumptions that the operator F is twice continuous Frechet differentiable with a Lipschitz-continuous first derivative and that the solution of the equation F (x) = y fulfils a smoothness condition, we will give a convergence rate result. Finally we present some applications and a numerical result for the reconstruction of the activity function in single-photon-emission computed tomography.

Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators

In this paper we deal with Morozovs discrepancy principle as an a posteriori parameter choice rule for Tikhonov regularization with general convex penalty terms Ψ for nonlinear inverse problems. It is shown that a regularization parameter α fulfilling the discprepancy principle exists, whenever the operator F satisfies some basic conditions, and that for suitable penalty terms the regularized solutions converge to the true solution in the topology induced by Ψ. It is illustrated that for this parameter choice rule it holds α → 0, δq/α → 0 as the noise level δ goes to 0. Finally, we establish convergence rates with respect to the generalized Bregman distance and a numerical example is presented.

Tikhonov replacement functionals for iteratively solving nonlinear operator equations

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 nonlinear operator. Namely, the replacement functional is globally convex and can effectively be minimized by a fixed point iteration. On the basis of the minimizers of the replacement functional, we introduce an iterative algorithm that converges towards a critical point of the Tikhonov functional, and under additional assumptions for the nonlinear operator F, to a global minimizer. Combining our iterative strategy with an appropriate parameter selection rule, we obtain convergence and convergence rates. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse SPECT (single photon emission computerized tomography) problem.

Morozov's Discrepancy Principle for Tikhonov-Regularization of Nonlinear Operators

ABSTRACT We consider Morozovs discrepancy principle for Tikhonov-regularization of nonlinear operator equations. It is shown that minor restrictions to the operator F and the solution x* of the equation already guarantee the existence of a regularization parameter α such that holds, and a convergence rate result is given. Finally we investigate some practically relevant examples, e.g., medical imaging (Single Photon Emission Computed Tomography). It is illustrated that the introduced conditions on F will be met in general by a large class of nonlinear operators.

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.

An adaptive discretization for Tikhonov-Phillips regularization with a posteriori parameter selection

Summary. The aim of this paper is to describe an efficient adaptive strategy for discretizing ill-posed linear operator equations of the first kind: we consider Tikhonov-Phillips regularization \[ x_{\alpha}^{\delta} = \left(A^{\ast}A+\alpha I\right)^{-1}A^{\ast}y^{\delta} \] with a finite dimensional approximation

Regularization by fractional filter methods and data smoothing

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