TY - JOUR AB - The problem of phase retrieval has many applications in the field of optical imaging. Motivated by imaging experiments with biological specimens, we primarily consider the setting of low-dose illumination where Poisson noise plays the dominant role. In this paper, we discuss gradient descent algorithms based on different loss functions adapted to data affected by Poisson noise, in particular in the low-dose regime. Starting from the maximum log-likelihood function for the Poisson distribution, we investigate different regularizations and approximations of the problem to design an algorithm that meets the requirements that are faced in applications. In the course of this, we focus on low-count measurements. Based on an improved version of a variance-stabilizing transform for the Poisson distribution, we derive a decision rule for the regularization parameter in an averaged amplitude-based loss function. For all discussed loss functions, we study the convergence of the respective gradient descent algorithms to stationary points and find constant step sizes that guarantee descent of the loss in each iteration. Numerical experiments in the low-dose regime are performed to corroborate the theoretical observations. AU - Diederichs, B. AU - Filbir, F. AU - Römer, P. C1 - 72884 C2 - 56772 CY - Temple Circus, Temple Way, Bristol Bs1 6be, England TI - Wirtinger gradient descent methods for low-dose Poisson phase retrieval. JO - Inverse Probl. VL - 40 IS - 12 PB - Iop Publishing Ltd PY - 2024 SN - 0266-5611 ER - TY - JOUR AB - Ptychography, a special case of the phase retrieval problem, is a popular method in modern imaging. Its measurements are based on the shifts of a locally supported window function. In general, direct recovery of an object from such measurements is known to be an ill-posed problem. Although for some windows the conditioning can be controlled, for a number of important cases it is not possible, for instance for Gaussian windows. In this paper we develop a subspace completion algorithm, which enables stable reconstruction for a much wider choice of windows, including Gaussian windows. The combination with a regularization technique leads to improved conditioning and better noise robustness. AU - Forstner, A.* AU - Krahmer, F.* AU - Melnyk, O. AU - Sissouno, N. C1 - 60343 C2 - 49206 CY - Temple Circus, Temple Way, Bristol Bs1 6be, England TI - Well-conditioned ptychograpic imaging via lost subspace completion. JO - Inverse Probl. VL - 36 IS - 10 PB - Iop Publishing Ltd PY - 2020 SN - 0266-5611 ER - TY - JOUR AB - Magnetic particle imaging (MPI) is a promising new in vivo medical imaging modality in which distributions of super-paramagnetic nanoparticles are tracked based on their response in an applied magnetic field. In this paper we provide a mathematical analysis of the modeled MPI operator in the univariate situation. We provide a Hilbert space setup, in which the MPI operator is decomposed into simple building blocks and in which these building blocks are analyzed with respect to their mathematical properties. In turn, we obtain an analysis of the MPI forward operator and, in particular, of its ill-posedness properties. We further get that the singular values of the MPI core operator decrease exponentially. We complement our analytic results by some numerical studies which, in particular, suggest a rapid decay of the singular values of the MPI operator. AU - Erb, W.* AU - Weinmann, A. AU - Ahlborg, M.* AU - Brandt, C.* AU - Bringout, G.* AU - Buzug, T.M.* AU - Frikel, J.* AU - Kaethner, C.* AU - Knopp, T.* AU - Maerz, T.* AU - Moeddel, M.* AU - Storath, M.* AU - Weber, A.* C1 - 53465 C2 - 44724 TI - Mathematical analysis of the 1D model and reconstruction schemes for magnetic particle imaging. JO - Inverse Probl. VL - 34 IS - 5 PY - 2018 SN - 0266-5611 ER - TY - JOUR AB - Partial differential equation (PDE) models are widely used in engineering and natural sciences to describe spatio-temporal processes. The parameters of the considered processes are often unknown and have to be estimated from experimental data. Due to partial observations and measurement noise, these parameter estimates are subject to uncertainty. This uncertainty can be assessed using profile likelihoods, a reliable but computationally intensive approach. In this paper, we present the integration based approach for the profile likelihood calculation developed by (Chen and Jennrich 2002 J. Comput. Graph. Stat. 11 714-32) and adapt it to inverse problems with PDE constraints. While existing methods for profile likelihood calculation in parameter estimation problems with PDE constraints rely on repeated optimization, the proposed approach exploits a dynamical system evolving along the likelihood profile. We derive the dynamical system for the unreduced estimation problem, prove convergence and study the properties of the integration based approach for the PDE case. To evaluate the proposed method, we compare it with state-of-the-art algorithms for a simple reaction-diffusion model for a cellular patterning process. We observe a good accuracy of the method as well as a significant speed up as compared to established methods. Integration based profile calculation facilitates rigorous uncertainty analysis for computationally demanding parameter estimation problems with PDE constraints. AU - Boiger, R.* AU - Hasenauer, J. AU - Hross, S. AU - Kaltenbacher, B.* C1 - 50171 C2 - 42110 CY - Bristol TI - Integration based profile likelihood calculation for PDE constrained parameter estimation problems. JO - Inverse Probl. VL - 32 IS - 12 PB - Iop Publishing Ltd PY - 2016 SN - 0266-5611 ER - TY - JOUR AB - The Mumford–Shah model is a very powerful variational approach for edge preserving regularization of image reconstruction processes. However, it is algorithmically challenging because one has to deal with a non-smooth and non-convex functional. In this paper, we propose a new efficient algorithmic framework for Mumford–Shah regularization of inverse problems in imaging. It is based on a splitting into specific subproblems that can be solved exactly. We derive fast solvers for the subproblems which are key for an efficient overall algorithm. Our method neither requires a priori knowledge of the gray or color levels nor of the shape of the discontinuity set. We demonstrate the wide applicability of the method for different modalities. In particular, we consider the reconstruction from Radon data, inpainting, and deconvolution. Our method can be easily adapted to many further imaging setups. The relevant condition is that the proximal mapping of the data fidelity can be evaluated a within reasonable time. In other words, it can be used whenever classical Tikhonov regularization is possible. AU - Hohm, K.* AU - Storath, M.* AU - Weinmann, A. C1 - 47376 C2 - 39265 TI - An algorithmic framework for Mumford-Shah regularization of inverse problems in imaging. JO - Inverse Probl. VL - 31 IS - 11 PY - 2015 SN - 0266-5611 ER - TY - JOUR AB - We propose a new algorithmic approach to the non-smooth and non-convex Potts problem (also called piecewise-constant Mumford-Shah problem) for inverse imaging problems. We derive a suitable splitting into specific subproblems that can all be solved efficiently. Our method does not require a priori knowledge on the gray levels nor on the number of segments of the reconstruction. Further, it avoids anisotropic artifacts such as geometric staircasing. We demonstrate the suitability of our method for joint image reconstruction and segmentation from limited data in x-ray and photoacoustic tomography. For instance, our method is able to reconstruct the Shepp-Logan phantom from $7$ angular views only. We demonstrate the practical applicability in an experiment with real PET data. AU - Storath, M.* AU - Weinmann, A. AU - Frikel, J. AU - Unser, M.* C1 - 32611 C2 - 35159 TI - Joint image reconstruction and segmentation using the Potts model. JO - Inverse Probl. VL - 31 IS - 2 PY - 2015 SN - 0266-5611 ER - TY - JOUR AB - The filtered backprojection algorithm (FBP) in limited angle tomography reliably reconstructs only specific features of the original object and creates additional artifacts in the reconstruction. While the former is well understood mathematically, the added artifacts have not been studied very much in the literature. In our paper Inverse Problems 29125007 we mathematically explain why additional artifacts are created by the FBP and Lambda-CT algorithms for a limited angular range, and we derive an artifact reduction strategy using microlocal analysis. AU - Frikel, J. AU - Quinto, E.T.* C1 - 28930 C2 - 33583 TI - How to characterize and decrease artifacts in limited angle tomography using microlocal analysis. JO - Inverse Probl. PY - 2014 SN - 0266-5611 ER - TY - JOUR AB - The reconstruction of images from data modeled by circular or spherical mean Radon transforms plays an important role in thermoacoustic and photoacoustic tomography. We consider a modification of a summability-type approximate reconstruction method described in earlier work and show that in the limit it leads to exact reconstruction. Among the consequences of this development are certain two- and three-dimensional inversion-type formulas in which the detectors lie on ellipses or ellipsoids respectively. AU - Ansorg, M.* AU - Filbir, F. AU - Madych, W.R.* AU - Seyfried, R. C1 - 11799 C2 - 30816 TI - Summability kernels for circular and spherical mean data. JO - Inverse Probl. VL - 29 IS - 1 PB - IOP Publishing PY - 2013 SN - 0266-5611 ER - TY - JOUR AB - We consider the reconstruction problem for limited angle tomography using filtered backprojection (FBP) and lambda tomography. We use microlocal analysis to explain why the well-known streak artifacts are present at the end of the limited angular range. We explain how to mitigate the streaks and prove that our modified FBP and lambda operators are standard pseudodifferential operators, and so they do not add artifacts. We provide reconstructions to illustrate our mathematical results. AU - Frikel, J. AU - Quinto, E.T.* C1 - 28301 C2 - 33088 TI - Characterization and reduction of artifacts in limited angle tomography. JO - Inverse Probl. VL - 29 IS - 12 PB - IOP Publishing PY - 2013 SN - 0266-5611 ER - TY - JOUR AB - The inversion of the one-dimensional Radon transform on the rotation group SO(3) is an ill-posed inverse problem which applies to x-ray tomography with polycrystalline materials. This paper presents a novel approach to the numerical inversion of the one-dimensional Radon transform on SO(3). Based on a Fourier slice theorem the discrete inverse Radon transform of a function sampled on the product space S-2 x S-2 of two two-dimensional spheres is determined as the solution of a minimization problem, which is iteratively solved using fast Fourier techniques for S-2 and SO(3). The favorable complexity and stability of the algorithm based on these techniques has been confirmed with numerical tests. (Preprint) AU - Hielscher, R. AU - Potts, D.* AU - Prestin, J.* AU - Schaeben, H.* AU - Schmalz, M.* C1 - 162 C2 - 25962 TI - The Radon transform on SO(3): A Fourier slice theorem and numerical inversion. JO - Inverse Probl. VL - 24 IS - 2 PB - Institute of Physics PY - 2008 SN - 0266-5611 ER -