Kamil S. Kazimierski

Regularization methods in Banach spaces

Regularization methods aimed at finding stable approximate solutions are a necessary tool to tackle inverse and ill-posed problems. Usually the mathematical model of an inverse problem consists of an operator equation of the first kind and often the associated forward operator acts between Hilbert spaces. However, for numerous problems the reasons for using a Hilbert space setting seem to be based rather on conventions than on an approprimate and realistic model choice, so often a Banach space setting would be closer to reality. Furthermore, sparsity constraints using general Lp-norms or the BV-norm have recently become very popular. Meanwhile the most well-known methods have been investigated for linear and nonlinear operator equations in Banach spaces. Motivated by these facts the authors aim at collecting and publishing these results in a monograph.

Minimization of Tikhonov Functionals in Banach Spaces

Tikhonov functionals are known to be well suited for obtaining regularized solutions of linear operator equations. We analyze two iterative methods for finding the minimizer of norm-based Tikhonov functionals in Banach spaces. One is the steepest descent method, whereby the iterations are directly carried out in the underlying space, and the other one performs iterations in the dual space. We prove strong convergence of both methods.

Accelerated Landweber iteration in Banach spaces

We investigate a method of accelerated Landweber type for the iterative regularization of nonlinear ill-posed operator equations in Banach spaces. Based on an auxiliary algorithm with a simplified choice of the step-size parameter we present a convergence and stability analysis of the algorithm under consideration. We will close our discussion with the presentation of a numerical example.

Modified Landweber Iteration in Banach Spaces—Convergence and Convergence Rates

We introduce and discuss an iterative method of modified Landweber type for regularization of nonlinear operator equations in Banach spaces. Under smoothness and convexity assumptions on the solution space we present convergence and stability results. Furthermore, we will show that under the so-called approximate source conditions convergence rates may be achieved by a proper a-priori choice of the parameter of the presented algorithm. We will illustrate these theoretical results with a numerical example.

Tikhonov regularization in Lp applied to inverse medium scattering

This paper presents Tikhonov- and iterated soft-shrinkage regularization methods for nonlinear inverse medium scattering problems. Motivated by recent sparsity-promoting reconstruction schemes for inverse problems, we assume that the contrast of the medium is supported within a small subdomain of a known search domain and minimize Tikhonov functionals with sparsity-promoting penalty terms based on Lp-norms. Analytically, this is based on scattering theory for the Helmholtz equation with the refractive index in Lp, 1 < p < ∞, and on crucial continuity and compactness properties of the contrast-to-measurement operator. Algorithmically, we use an iterated soft-shrinkage scheme combined with the differentiability of the forward operator in Lp to approximate the minimizer of the Tikhonov functional. The feasibility of this approach together with the quality of the obtained reconstructions is demonstrated via numerical examples.

Minimization of the Tikhonov functional in Banach spaces smooth and convex of power type by steepest descent in the dual

For Tikhonov functionals of the form Ψ(x)=‖Ax−y‖Yr+α‖x‖Xq we investigate a steepest descent method in the dual of the Banach space X. We show convergence rates for the proposed method and present numerical tests.

Spatio-temporal TGV denoising for ASL perfusion imaging

&NA; In arterial spin labeling (ASL) a perfusion weighted image is achieved by subtracting a label image from a control image. This perfusion weighted image has an intrinsically low signal to noise ratio and numerous measurements are required to achieve reliable image quality, especially at higher spatial resolutions. To overcome this limitation various denoising approaches have been published using the perfusion weighted image as input for denoising. In this study we propose a new spatio‐temporal filtering approach based on total generalized variation (TGV) regularization which exploits the inherent information of control and label pairs simultaneously. In this way, the temporal and spatial similarities of all images are used to jointly denoise the control and label images. To assess the effect of denoising, virtual ground truth data were produced at different SNR levels. Furthermore, high‐resolution in‐vivo pulsed ASL data sets were acquired and processed. The results show improved image quality, quantitative accuracy and robustness against outliers compared to seven state of the art denoising approaches. HighlightsImproved CBF analysis for ASL by dedicated TGV denoising.Reduced number of averages for high resolution ASL perfusion imaging.ASL denoising benchmarking.

A sparsity regularization and total variation based computational framework for the inverse medium problem in scattering

Abstract We present a fast computational framework for the inverse medium problem in scattering, i.e. we look at discretization, reconstruction and numerical performance. The Helmholtz equation in two and three dimensions is used as a physical model of scattering including point sources and plane waves as incident fields as well as near and far field measurements. For the reconstruction of the medium, we set up a rapid variational regularization scheme and indicate favorable choices of the various parameters. The underlying paradigm is, roughly speaking, to minimize the discrepancy between the reconstruction and measured data while, at the same time, taking into account various structural a-priori information via suitable penalty terms. In particular, the involved penalty terms are designed to promote information expected in real-world environments. To this end, a combination of sparsity promoting terms, total variation, and physical bounds of the inhomogeneous medium, e.g. positivity constraints, is employed in the regularization penalty. A primal-dual algorithm is used to solve the minimization problem related to the variational regularization. The computational feasibility, performance and efficiency of the proposed approach is demonstrated for synthetic as well as experimentally measured data.

Norm sensitivity of sparsity regularization with respect to p

We consider the Tikhonov functional incorporating a lp-norm as a penalty term. The minimizer of this functional is investigated in regard to its sensitivity with respect to the parameter p. To quantify the sensitivity of the minimizer, we determine the derivative with respect to p. The underlying idea is based on the implicit function theorem. We show that the minimizer of the Tikhonov functional is differentiable with respect to p for any 1 < p < 2. For p = 1, the derivative exists only under additional assumptions.

Variational Decompression of Image Data From DjVu Encoded Files

The DjVu file format and image compression techniques are widely used in the archival of digital documents. Its key ingredients are the separation of the document into fore- and background layers and a binary switching mask, followed by a lossy, transform-based compression of the former and a dictionary-based compression of the latter. The lossy compression of the layers is based on a wavelet decomposition and bit truncation, which leads, in particular at higher compression rates, to severe compression artifacts in the standard decompression of the layers. The aim of this paper is to break ground for the variational decompression of DjVu files. To this aim, we provide an in-depth analysis and discussion of the compression standard with a particular focus on modeling data constraints for decompression. This allows to carry out DjVu decompression as regularized inversion of the compression procedure. As particular example, we evaluate the performance of such a framework using total variation and total generalized variation regularization. Furthermore, we provide routines for obtaining the necessary data constraints from a compressed DjVu file and for the forward and adjoint transformation operator involved in DjVu compression.