Regularization properties of Mumford–Shah-type functionals with perimeter and norm constraints for linear ill-posed problems

In this paper we consider the simultaneous reconstruction and segmentation of a function f from measurements g = Kf, where K is a linear operator. Assuming that the inversion of K is illposed, regularization methods have to be used for the inversion process in case of inexact data. We propose using a Mumford–Shah-type functional for the stabilization of the inversion. Restricting our analysis to the recovery of piecewise constant functions, we investigate the existence of minimizers, their stability, and the regularization properties of our approach. Finally, we present a numerical example from single photon emission computed tomography (SPECT).FWF, T 529-N18, Mumford-Shah models for tomography I

Wavelet methods for a weighted sparsity penalty for region of interest tomography

We consider region of interest (ROI) tomography of piecewise constant functions. Additionally, an algorithm is developed for ROI tomography of piecewise constant functions using a Haar wavelet basis. A weighted ℓp–penalty is used with weights that depend on the relative location of wavelets to the region of interest. We prove that the proposed method is a regularization method, i.e., that the regularized solutions converge to the exact piecewise constant solution if the noise tends to zero. Tests on phantoms demonstrate the effectiveness of the method.FWF, T 529-N18, Mumford-Shah models for tomography IINSF, 1311558, Tomography and Microlocal AnalysisFWF, W 1214, Doktoratskolleg "Computational Mathematics

Regularisierung von linearen schlechtgestellten Problemen in zwei Schritten: Kombination von Datenglättungs- und Rekonstruktionsverfahren

This thesis is a contribution to the field of "ill-posed inverse problems". During the last ten years a new development in this field has taken place: Besides operator-adapted methods for the solution of inverse problems also methods adjusted to smoothness propertiesof functions are studied. The intention of this thesis is to present and analyze "two-step methods" for the solution of linear ill-posed problems. It is the fundamental idea of a two-step method to perform first a data estimation step of probably noisy data and then to perform a reconstruction step to solve the inverse problem using the data estimate. Besides the general description of two-step methods two realizations are analyzed. On the one hand classical regularization methods like the ones proposed by Tikhonov or Landweber are interpreted as two-step methods. On the other hand the combination of wavelet shrinkage and classical regularization methods is analyzed. This yields an order optimal method which is, by the use of wavelet shrinkage, adapted to smoothness properties of functions in Sobolev and Besov spaces and, by the use of the singular system, adapted to the operator under consideration

Regularization of Linear Ill-posed Problems in Two Steps: Combination of Data Smoothing and Reconstruction Methods

This thesis is a contribution to the field of ill-posed inverse problems . During the last ten years a new development in this field has taken place: Besides operator-adapted methods for the solution of inverse problems also methods adjusted to smoothness propertiesof functions are studied. The intention of this thesis is to present and analyze two-step methods for the solution of linear ill-posed problems. It is the fundamental idea of a two-step method to perform first a data estimation step of probably noisy data and then to perform a reconstruction step to solve the inverse problem using the data estimate. Besides the general description of two-step methods two realizations are analyzed. On the one hand classical regularization methods like the ones proposed by Tikhonov or Landweber are interpreted as two-step methods. On the other hand the combination of wavelet shrinkage and classical regularization methods is analyzed. This yields an order optimal method which is, by the use of wavelet shrinkage, adapted to smoothness properties of functions in Sobolev and Besov spaces and, by the use of the singular system, adapted to the operator under consideration

A Mumford–Shah-type approach to simultaneous reconstruction and segmentation for emission tomography problems with Poisson statistics

We propose a variational model to simultaneous reconstruction and segmentation in emission tomography. As in the original Mumford–Shah model [27] we use the contour length as penalty term to preserve edge information whereas a different data fidelity term is used to measure the information discrimination between the computed tomography data of the reconstructed object and the observed (or simulated) data. As data fidelity term we use the Kullback–Leibler divergence which originates from the Poisson distribution present in emission tomography. In this paper we focus on piecewise constant reconstructions which is a reasonable assumption in medical imaging. The segmenting contour as well as the corresponding reconstructions are found as minimizers of a Mumford–Shah-type functional over the space of piecewise constant functions. The numerical scheme is implemented by evolving the level-set surface according to the shape derivative of the functional. The method is validated for simulated data with different levels of noise.EC/FP7/600209/EU/International Post-Doc Initiative of the Technische Universität Berlin/IPODIFWF, T 529-N18, Mumford-Shah models for tomography IIFWF, W 1214, Doktoratskolleg "Computational Mathematics