Esther Klann

Regularization by fractional filter methods and data smoothing

This paper is concerned with the regularization of linear ill-posed problems by a combination of data smoothing and fractional filter methods. For the data smoothing, a wavelet shrinkage denoising is applied to the noisy data with known error level δ. For the reconstruction, an approximation to the solution of the operator equation is computed from the data estimate by fractional filter methods. These fractional methods are based on the classical Tikhonov and Landweber method, but avoid, at least partially, the well-known drawback of oversmoothing. Convergence rates as well as numerical examples are presented.

Two-step regularization methods for linear inverse problems

In this paper we investigate reconstruction methods for the treatment of ill-posed inverse problems. These methods are based on a data estimation operator Sλ followed by a classical regularization operator Rα Tα,λ = RαSλ . As a particular example of such a two-step regularization method we investigate in detail the combination of a wavelet shrinkage operator Sλ followed by Tikhonov regularization Rα . The nonlinear shrinkage operator is applied to noisy data and partially recovers the smoothness properties of the exact data. We prove order optimality for the proposed scheme and confirm the theoretical results with an example from medical imaging.

A Mumford-Shah-Like Method for Limited Data Tomography with an Application to Electron Tomography

In this article the Mumford-Shah-like method of [R. Ramlau and W. Ring, J. Comput. Phys., 221 (2007), pp. 539-557] for complete tomographic data is generalized and applied to limited angle and region of interest tomography data. With the Mumford-Shah-like method, one reconstructs a piecewise constant function and simultaneously a segmentation from its (complete) Radon transform data. For limited data, the ability of the Mumford-Shah-like method to find a segmentation, and by that the singularity set of a function, is exploited. The method is applied to generated data from a torso phantom. The results demonstrate the performance of the method in reconstructing the singularity set, the density distribution itself for limited angle data, and also some quantitative information about the density distribution for region of interest data. As a second example limited angle region of interest tomography is considered as a simplified model for electron tomography (ET). For this problem we combine Lambda tomography and the Mumford-Shah-like method. The combined method is applied to simulated ET data.

On fractional Tikhonov regularization

Abstract It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth, i.e., the approximate solution may lack many details that the desired exact solution might possess. Two different approaches, both referred to as fractional Tikhonov methods have been introduced to remedy this shortcoming. This paper investigates the convergence properties of these methods by reviewing results published previously by various authors. We show that both methods are order optimal when the regularization parameter is chosen according to the discrepancy principle. The theory developed suggests situations in which the fractional methods yield approximate solutions of higher quality than Tikhonov regularization in standard form. Computed examples that illustrate the behavior of the methods are presented.

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

Numerical methods for the design of gradient-index optical coatings

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

from measurements

A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data

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Shrinkage versus deconvolution

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Wavelet-based multilevel methods for linear ill-posed problems

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