Thorsten Hohage

Simon Maretzke, Matthias Bartels, Martin Krenkel et al.

Like many other advanced imaging methods, x-ray phase contrast imaging and tomography require mathematical inversion of the observed data to obtain real-space information. While an accurate forward model describing the generally nonlinear image formation from a given object to the observations is often available, explicit inversion formulas are typically not known. Moreover, the measured data might be insufficient for stable image reconstruction, in which case it has to be complemented by suitable a priori information. In this work, regularized Newton methods are presented as a general framework for the solution of such ill-posed nonlinear imaging problems. For a proof of principle, the approach is applied to x-ray phase contrast imaging in the near-field propagation regime. Simultaneous recovery of the phase- and amplitude from a single near-field diffraction pattern without homogeneity constraints is demonstrated for the first time. The presented methods further permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval and tomographic inversion. We demonstrate the potential of this approach by three-dimensional imaging of a colloidal crystal at 95 nm isotropic resolution.

Frank Werner, Thorsten Hohage

In this paper we study a Tikhonov-type method for ill-posed nonlinear operator equations $\gdag = F(\udag)$ where $\gdag$ is an integrable, non-negative function. We assume that data are drawn from a Poisson process with density $t\gdag$ where $t>0$ may be interpreted as an exposure time. Such problems occur in many photonic imaging applications including positron emission tomography, confocal fluorescence microscopy, astronomic observations, and phase retrieval problems in optics. Our approach uses a Kullback-Leibler-type data fidelity functional and allows for general convex penalty terms. We prove convergence rates of the expectation of the reconstruction error under a variational source condition as $t\to\infty$ both for an a priori and for a Lepski{\uı}-type parameter choice rule.

Thorsten Hohage, Frederic Weidling

We describe a general strategy for the verification of variational source condition by formulating two sufficient criteria describing the smoothness of the solution and the degree of ill-posedness of the forward operator in terms of a family of subspaces. For linear deterministic inverse problems we show that variational source conditions are necessary and sufficient for convergence rates slower than the square root of the noise level. A similar result is shown for linear inverse problems with white noise. If the forward operator can be written in terms of the functional calculus of a Laplace-Beltrami operator, variational source conditions can be characterized by Besov spaces. This is discussed for a number of prominent inverse problems.

Thorsten Hohage, Frank Werner

We study Newton type methods for inverse problems described by nonlinear operator equations $F(u)=g$ in Banach spaces where the Newton equations $F'(u_n;u_{n+1}-u_n) = g-F(u_n)$ are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the well-known iteratively regularized Gauss-Newton method (IRGNM). We prove convergence and convergence rates as the noise level tends to 0 both for an a priori stopping rule and for a Lepski{\uı}-type a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback-Leibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the far-field pattern and a phase retrieval problem. The performence of the proposed method for these problems is illustrated in numerical examples.

Frederic Weidling, Thorsten Hohage

This paper is concerned with the inverse problem to recover the scalar, complex-valued refractive index of a medium from measurements of scattered time-harmonic electromagnetic waves at a fixed frequency. The main results are two variational source conditions for near and far field data, which imply logarithmic rates of convergence of regularization methods, in particular Tikhonov regularization, as the noise level tends to $0$. Moreover, these variational source conditions imply conditional stability estimates which improve and complement known stability estimates in the literature.

Claudia König, Frank Werner, Thorsten Hohage

This paper is concerned with exponentially ill-posed operator equations with additive impulsive noise on the right hand side, i.e. the noise is large on a small part of the domain and small or zero outside. It is well known that Tikhonov regularization with an $L^1$ data fidelity term outperforms Tikhonov regularization with an $L^2$ fidelity term in this case. This effect has recently been explained and quantified for the case of finitely smoothing operators. Here we extend this analysis to the case of infinitely smoothing forward operators under standard Sobolev smoothness assumptions on the solution, i.e. exponentially ill-posed inverse problems. It turns out that high order polynomial rates of convergence in the size of the support of large noise can be achieved rather than the poor logarithmic convergence rates typical for exponentially ill-posed problems. The main tools of our analysis are Banach spaces of analytic functions and interpolation-type inequalities for such spaces. We discuss two examples, the (periodic) backwards heat equation and an inverse problem in gradiometry.

Thorsten Hohage, Stefan Langer

We study the construction and updating of spectral preconditioners for regularized Newton methods and their application to electromagnetic inverse medium scattering problems. Moreover, we show how a Lepskiĭ-type stopping rule can be implemented efficiently for these methods. In numerical examples, the proposed method compares favorably with other iterative regularization method in terms of work-precision diagrams for exact data. For data perturbed by random noise, the Lepskiĭ-type stopping rule performs considerably better than the commonly used discrepancy principle.

Thorsten Hohage, Lothar Nannen

We consider the numerical solution of scalar wave equations in domains which are the union of a bounded domain and a finite number of infinite cylindrical waveguides. The aim of this paper is to provide a new convergence analysis of both the Perfectly Matched Layer (PML) method and the Hardy space infinite element method in a unified framework. We treat both diffraction and resonance problems. The theoretical error bounds are compared with errors in numerical experiments.

Lothar Nannen, Thorsten Hohage, Achim Schädle et al.

A construction of prismatic Hardy space infinite elements to discretize wave equations on unbounded domains $Ω$ in $H^1_{loc}(Ω)$, $H_{loc}(curl;Ω)$ and $H_{loc}(div;Ω)$ is presented. As our motivation is to solve Maxwell's equations we take care that these infinite elements fit into the discrete de Rham diagram, i.e. they span discrete spaces, which together with the exterior derivative form an exact sequence. Resonance as well as scattering problems are considered in the examples. Numerical tests indicate super-algebraic convergence in the number of additional unknowns per degree of freedom on the coupling boundary that are required to realize the Dirichlet to Neumann map.

Benjamin Sprung, Thorsten Hohage

We study the convergence of variationally regularized solutions to linear ill-posed operator equations in Banach spaces as the noise in the right hand side tends to $0$. The rate of this convergence is determined by abstract smoothness conditions on the solution called source conditions. For non-quadratic data fidelity or penalty terms such source conditions are often formulated in the form of variational inequalities. Such variational source conditions (VSCs) as well as other formulations of such conditions in Banach spaces have the disadvantage of yielding only low-order convergence rates. A first step towards higher order VSCs has been taken by Grasmair (2013) who obtained convergence rates up to the saturation of Tikhonov regularization. For even higher order convergence rates, iterated versions of variational regularization have to be considered. In this paper we introduce VSCs of arbitrarily high order which lead to optimal convergence rates in Hilbert spaces. For Bregman iterated variational regularization in Banach spaces with general data fidelity and penalty terms, we derive convergence rates under third order VSC. These results are further discussed for entropy regularization with elliptic pseudodifferential operators where the VSCs are interpreted in terms of Besov spaces and the optimality of the rates can be demonstrated. Our theoretical results are confirmed in numerical experiments.

Julian Eckhardt, Ralf Hiptmair, Thorsten Hohage et al.

By introducing a shape manifold as a solution set to solve inverse obstacle scattering problems we allow the reconstruction of general, not necessarily star-shaped curves. The bending energy is used as a stabilizing term in Tikhonov regularization to gain independence of the parametrization. Moreover, we discuss how self-intersections can be avoided by penalization with the Möbius energy and prove the regularizing property of our approach as well as convergence rates under variational source conditions. In the second part of the paper the discrete setting is introduced, and we describe a numerical method for finding the minimizer of the Tikhonov functional on a shape-manifold. Numerical examples demonstrate the feasibility of reconstructing non-star-shaped obstacles.

Thorsten Hohage, Milad Karimi, Björn Müller

Holographic coherent X-ray imaging enables nanoscale imaging of biological cells and tissues, rendering both phase and absorption contrast, i.e. real and imaginary parts of the refractive index. Unlike the standard model, which assumes a perfectly coherent incident beam, we consider partial coherence characterized by a known covariance operator. In addition, we assume time-resolved intensity measurements, granting access not only to expected intensities but also to their correlations. We investigate the information content of these correlations and analytically demonstrate that, under a symmetry-breaking condition on the sample and the illumination area, both phase and absorption contrast can be uniquely recovered in both the full and the linearized models. A key challenge in numerical reconstruction is the substantial increase in data dimensionality caused by computing intensity correlations during preprocessing. We propose a novel approach that leverages a low-rank assumption on the incident beam covariance operator, bypassing explicit correlation computation while still exploiting its full information. Numerical experiments demonstrate its feasibility, yielding accurate simultaneous reconstructions of phase and absorption contrast.

Alexey Agaltsov, Thorsten Hohage, Roman Novikov

This paper is concerned with the inverse problem to recover a compactly supported Schr{ö}dinger potential given the differential scattering cross section, i.e. the modulus, but not the phase of the scattering amplitude. To compensate for the missing phase information we assume additional measurements of the differential cross section in the presence of known background objects. We propose an iterative scheme for the numerical solution of this problem and prove that it converges globally of arbitrarily high order depending on the smoothness of the unknown potential as the energy tends to infinity. At fixed energy, however, the proposed iteration does not converge to the true solution even for exact data. Nevertheless, numerical experiments show that it yields remarkably accurate approximations with small computational effort even for moderate energies. At small noise levels it may be worth to improve these approximations by a few steps of a locally convergent iterative regularization method, and we demonstrate to which extent this reduces the reconstruction error.

Hans-Georg Raumer, Carsten Spehr, Thorsten Hohage et al.

This article discusses aeroacoustic imaging methods based on correlation measurements in the frequency domain. Standard methods in this field assume that the estimated correlation matrix is superimposed with additive white noise. In this paper we present a mathematical model for the measurement process covering arbitrarily correlated noise. The covariance matrix of correlation data is given in terms of fourth order moments. The aim of this paper is to explore the use of such additional information on the measurement data in imaging methods. For this purpose a class of weighted data spaces is introduced, where each data space naturally defines an associated beamforming method with a corresponding point spread function. This generic class of beamformers contains many well-known methods such as Conventional Beamforming, (Robust) Adaptive Beamforming or beamforming with shading. This article examines in particular weightings that depend on the noise (co)variances. In a theoretical analysis we prove that the beamformer, weighted by the full noise covariance matrix, has minimal variance among all beamformers from the described class. Application of the (co)variance weighted methods on synthetic and experimental data show that the resolution of the results is improved and noise effects are reduced.

Frederic Weidling, Benjamin Sprung, Thorsten Hohage

This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We show order optimal rates of convergence for finitely smoothing operators and for the backwards heat equation for a range of Besov spaces using variational source conditions. We also derive order optimal rates for a white noise model with the help of variational source conditions and concentration inequalities for sharp negative Besov norms of the noise.

Thorsten Hohage, Philip Miller

This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We focus on $B^0_{p,1}$ penalty terms which yield estimators that are sparse with respect to a wavelet frame. Our framework includes among others, the Radon transform and some nonlinear inverse problems in differential equations with distributed measurements. Using variational source conditions it is shown that such estimators achieve minimax-optimal rates of convergence for finitely smoothing operators in certain Besov balls both for deterministic and for statistical noise models.

Simon Maretzke, Thorsten Hohage

Propagation-based X-ray phase contrast enables nanoscale imaging of biological tissue by probing not only the attenuation, but also the real part of the refractive index of the sample. Since only intensities of diffracted waves can be measured, the main mathematical challenge consists in a phase-retrieval problem in the near-field regime. We treat an often used linearized version of this problem known as contract transfer function model. Surprisingly, this inverse problem turns out to be well-posed assuming only a compact support of the imaged object. Moreover, we establish bounds on the Lipschitz stability constant. In general this constant grows exponentially with the Fresnel number of the imaging setup. However, both for homogeneous objects, characterized by a fixed ratio of the induced refractive phase shifts and attenuation, and in the case of measurements at two distances, a much more favorable algebraic dependence on the Fresnel number can be shown. In some cases we establish order optimality of our estimates.

Thorsten Hohage, Frederic Weidling

This paper is concerned with the classical inverse scattering problem to recover the refractive index of a medium given near or far field measurements of scattered time-harmonic acoustic waves. It contains the first rigorous proof of (logarithmic) rates of convergence for Tikhonov regularization under Sobolev smoothness assumptions for the refractive index. This is achieved by combining two lines of research, conditional stability estimates via geometrical optics solutions and variational regularization theory.