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Chair of Optimization and Inverse Problems
Publications

Publications

List of publications

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Papers

2024

M. Nitzsche and B. N. Hahn, Dynamic image reconstruction in MPI with RESESOP-Kaczmarz, 2024.

M. S. Feinler and B. N. Hahn, GAN-based iterative motion estimation in HASTE MRI. 2024.

M. S. Feinler and B. N. Hahn, Learned RESESOP for solving inverse problems with inexact forward operator. 2024.

2023

B. N. Hahn, G. Rigaud, and R. Schmähl, A class of regularizations based on nonlinear isotropic diffusion for inverse problems, IMA Journal of Numerical Analysis, Feb. 2023.

Abstract

Building on the well-known total variation, this paper develops a general regularization technique based on nonlinear isotropic diffusion (NID) for inverse problems with piecewise smooth solutions. The novelty of our approach is to be adaptive (we speak of A-NID), i.e., the regularization varies during the iterates in order to incorporate prior information on the edges, deal with the evolution of the reconstruction and circumvent the limitations due to the nonconvexity of the proposed functionals. After a detailed analysis of the convergence and well-posedness of the method, the latter is validated by simulations performed on synthetic and real data on computerized tomography.

B. N. Hahn, E. T. Quinto, and G. Rigaud, Foreword to special issue of Inverse Problems on modern challenges in imaging, Inverse Problems, vol. 39, no. 3, p. 030401, Feb. 2023.

B. Hahn and B. Wirth, Convex reconstruction of moving particles with inexact motion model, PAMM, vol. 23, no. 2, Sep. 2023.

M. S. Feinler and B. N. Hahn, Retrospective Motion Correction in Gradient Echo MRI by Explicit Motion Estimation Using Deep CNNs. Mar-2023.

S. R. Arridge, M. Burger, B. Hahn, and E. T. Quinto, Tomographic Inverse Problems: Mathematical Challenges and Novel Applications, Oberwolfach Reports, vol. 20, no. 2, pp. 1105–1194, Dec. 2023.

2022

M. Nitzsche, H. Albers, T. Kluth, and B. Hahn, Compensating model imperfections during image reconstruction via Resesop, International Journal on Magnetic Particle Imaging, p. Vol 8 No 1 Suppl 1 (2022), 2022.

B. N. Hahn, M.-L. K. Garrido, C. Klingenberg, and S. Warnecke, Using the Navier-Cauchy equation for motion estimation in dynamic imaging, Inverse Problems and Imaging, vol. 16, no. 5, p. 1179, 2022.

2021

B. N. Hahn, M. L. Kienle-Garrido, and E. T. Quinto, Microlocal properties of dynamic Fourier integral operators, 2021.

B. N. Hahn, Motion compensation strategies in tomography, 2021.

2020

G. Rigaud and B. N. Hahn, Reconstruction algorithm for 3D Compton scattering imaging with incomplete data, Inverse Problems in Science and Engineering, vol. 29, no. 7, pp. 967--989, 2020.

A. P. Polyakova, I. E. Svetov, and B. N. Hahn, The Singular Value Decomposition of the Operators of the Dynamic Ray Transforms Acting on 2D Vector Fields, in Numerical Computations: Theory and Algorithms, Cham, 2020, pp. 446--453.

B. N. Hahn, M. L. Kienle-Garrido, C. Klingenberg, and S. Warnecke, Using the Navier-Cauchy equation for motion estimation in dynamic imaging. 2020.

S. E. Blanke, B. N. Hahn, and A. Wald, Inverse problems with inexact forward operator: iterative regularization and application in dynamic imaging, Inverse Problems, vol. 36, no. 12, p. 124001, 2020.

Abstract

The classic regularization theory for solving inverse problems is built on the assumption that the forward operator perfectly represents the underlying physical model of the data acquisition. However, in many applications, for instance in microscopy or magnetic particle imaging, this is not the case. Another important example represent dynamic inverse problems, where changes of the searched-for quantity during data collection can be interpreted as model uncertainties. In this article, we propose a regularization strategy for linear inverse problems with inexact forward operator based on sequential subspace optimization methods (SESOP). In order to account for local modelling errors, we suggest to combine SESOP with the Kaczmarz’ method. We study convergence and regularization properties of the proposed method and discuss several practical realizations. Relevance and performance of our approach are evaluated at simulated data from dynamic computerized tomography with various dynamic scenarios.

2019

Mathematisches Forschungsinstitut Oberwolfach, Tomographic Inverse Problems: Theory and Applications, Workshop Reports, 2019.

T. Kluth, B. N. Hahn, and C. Brandt, Spatio-temporal concentration reconstruction using motion priors in magnetic particle imaging, in Proc. Int. Workshop Magnetic Particle Imaging, 2019.

Abstract

In this presentation we address the dynamic concentration reconstruction problem in magnetic particle imaging. The tradeoff between the signal-to-noise ratio in the processed data used for the reconstruction and the concentration dynamics of the tracer requires sophisticated reconstruction techniques to avoid motion artifacts. We incorporate motion priors in the concentration reconstruction and as an extra we can obtain an estimate for a flow field.

B. N. Hahn and M.-L. Kienle Garrido, An efficient reconstruction approach for a class of dynamic imaging operators, Inverse Problems, vol. 35, no. 9, p. 094005, 2019.

Abstract

This article deals with the problem of recovering an image from tomographic data corrupted by the motion of an object. Using generalized Radon transforms as modeling for the dynamic imaging operators, we present a novel approach for the reconstruction process. The proposed method, motivated by results on microlocal analysis, approximates the solution via a filtered backprojection-type algorithm which has the advantage of being quick to implement. The numerical results applied on dynamic photoacoustic tomography (PAT) data corroborate the efficiency and relevance of the proposed method.

2018

T. Schuster, B. Hahn, and M. Burger, Dynamic inverse problems: modelling—regularization—numerics, Inverse Problems, vol. 34, no. 4, p. 040301, Mar. 2018.

G. Rigaud and B. N. Hahn, 3D Compton scattering imaging and contour reconstruction for a class of Radon transforms, Inverse Problems, vol. 34, no. 7, p. 075004, 2018.

Abstract

Compton scattering imaging is a nascent concept arising from the current development of high-sensitive energy detectors and is devoted to exploit the scattering radiation to image the electron density of the studied medium. Such detectors are able to collect incoming photons in terms of energy. This paper introduces potential 3D modalities in Compton scattering imaging (CSI). The associated measured data are modeled using a class of generalized Radon transforms. The study of this class of operators leads to build a filtered backprojection kind algorithm preserving the contours of the sought-for function and offering a fast approach to partially solve the associated inverse problems. Simulation results including Poisson noise demonstrate the potential of this new imaging concept as well as the proposed image reconstruction approach.

2017

B. N. Hahn, Motion Estimation and Compensation Strategies in Dynamic Computerized Tomography, Sensing and Imaging, vol. 18, no. 10, pp. 1–20, 2017.

Abstract

A main challenge in computerized tomography consists in imaging moving objects. Temporal changes during the measuring process lead to inconsistent data sets, and applying standard reconstruction techniques causes motion artefacts which can severely impose a reliable diagnostics. Therefore, novel reconstruction techniques are required which compensate for the dynamic behavior. This article builds on recent results from a microlocal analysis of the dynamic setting, which enable us to formulate efficient analytic motion compensation algorithms for contour extraction. Since these methods require information about the dynamic behavior, we further introduce a motion estimation approach which determines parameters of affine and certain non-affine deformations directly from measured motion-corrupted Radon-data. Our methods are illustrated with numerical examples for both types of motion.

B. N. Hahn, A motion artefact study and locally deforming objects in computerized tomography, Inverse Problems, vol. 33, no. 11, p. 114001, 2017.

Abstract

Movements of the object during the data collection in computerized tomography can introduce motion artefacts in the reconstructed image. They can be reduced by employing information about the dynamic behaviour within the reconstruction step. However, inaccuracies concerning the movement are inevitable in practice. In this article, we give an explicit characterization of what is visible in an image obtained by a reconstruction algorithm with incorrect motion information. Then, we use this result to study in detail the situation of locally deforming objects, i.e. individual parts of the object have a different dynamic behaviour. In this context, we prove that additional artefacts arise due to the global nature of the Radon transform, even if the motion is exactly known. Based on our analysis, we propose a numerical scheme to reduce these artefacts in the reconstructed image. All our results are illustrated by numerical examples.

2016

B. N. Hahn and E. T. Quinto, Detectable singularities from dynamic Radon data, SIAM J. Imaging Sciences, vol. 9, no. 3, pp. 1195–1225, 2016.

Abstract

In this paper, we use microlocal analysis to understand what X-ray tomographic data acquisition does to singularities of an object which changes during the measuring process. Depending on the motion model, we study which singularities are detected by the measured data. In particular, this analysis shows that, due to the dynamic behavior, not all singularities might be detected, even if the radiation source performs a complete turn around the object. Thus, they cannot be expected to be (stably) visible in any reconstruction. On the other hand, singularities could be added (or masked) as well. To understand this precisely, we provide a characterization of visible and added singularities by analyzing the microlocal properties of the forward and reconstruction operators. We illustrate the characterization using numerical examples.

B. N. Hahn, Null space and resolution in dynamic computerized tomography, Inverse Problems, vol. 32, no. 2, p. 025006, 2016.

Abstract

One major challenge in computerized tomography is to image objects which change during the data acquisition and hence lead to inconsistent data sets. Motion artefacts in the reconstructions can be reduced by applying specially adapted algorithms which take the dynamic behaviour into account. Within this article, we analyse the achievable resolution in the dynamic setting in case of two-dimensional affine deformations. To this end, we characterize the null space of the operator describing the dynamic case, using its singular value decomposition and a necessary dynamic consistency condition. This shows that independent of any reconstruction method, the specimen's dynamics results in a loss of resolution compared to the stationary setting. Our theoretical results are illustrated at a numerical example.

2015

B. N. Hahn, Dynamic linear inverse problems with moderate movements of the object: Ill-posedness and regularization, Inverse Problems & Imaging, vol. 9, no. 2, pp. 395–413, 2015.

Abstract

One challenge for modern imaging methods is the investigation of objects which change during the data acquisition. This occurs in non-destructive testing as well as in medical applications, e.g. on account of patient or organ movements. Due to the object's deformations, the respective imaging modality is described by a dynamic inverse problem. In this paper, a classification scheme for linear dynamic problems depending on the object's motion is provided. Based on this scheme, we study the class in detail where the dynamic problem is still related to the operator in the static case, and where we call the deformations moderate. We proof important properties of the dynamic operator, derive a singular value decomposition and develop suitable regularization methods. The application of these methods to specific problems is illustrated at two examples including dynamic computerized tomography.

D. Gerth, B. N. Hahn, and R. Ramlau, The method of the approximate inverse for atmospheric tomography, Inverse Problems, vol. 31, no. 6, p. 065002, 2015.

Abstract

In this paper, we propose a new approach for the atmospheric tomography based on the method of the approximate inverse. The image quality of Earth-bound telescopes is severely degraded by turbulences of the atmosphere of the Earth. In order to receive sharp images, the incoming light is corrected for these distortions using deformable mirrors. In atmospheric tomography, the turbulence profile is reconstructed so that the shape of the mirrors can be adjusted in an optimal way. We perform this reconstruction step by applying the method of the approximate inverse, a non-iterative regularization method which requires the solution of an adjoint problem, to Multi-Conjugate Adaptive Optics. We show that the approximate inverse leads to efficient algorithms and give numerical examples. The adjoint problem, which can be precomputed, is solved with an iterative Kaczmarz-type algorithm. In this first study of the new algorithm, tip/tilt effects and spot elongation are not included.

2014

B. N. Hahn, Reconstruction of dynamic objects with affine deformations in computerized tomography, Journal of Inverse and Ill-posed Problems, vol. 22, no. 3, pp. 323–339, 2014.

Abstract

The data acquisition in computerized tomography takes a certain amount of time since the x-ray source has to be rotated around the specimen. An object that changes during the scanning causes inconsistent data sets. To avoid the motion artefacts in reconstructions, the algorithm has to take the dynamic behavior of the specimen into account. In this context, some a priori information about the movement is required. A reconstruction method is proposed that compensates for the motion with a special focus on affine deformations. It also permits the combination of reconstruction and image analysis tools to extract features of the object without motion artefacts. The algorithm is validated with a numerical example from medical imaging.

B. N. Hahn, Efficient algorithms for linear dynamic inverse problems with known motion, Inverse Problems, vol. 30, no. 3, p. 035008, 2014.

Abstract

An inverse problem is called dynamic if the object changes during the data acquisition process. This occurs e.g. in medical applications when fast moving organs like the lungs or the heart are imaged. Most regularization methods are based on the assumption that the object is static during the measuring procedure. Hence, their application in the dynamic case often leads to serious motion artefacts in the reconstruction. Therefore, an algorithm has to take into account the temporal changes of the investigated object. In this paper, a reconstruction method that compensates for the motion of the object is derived for dynamic linear inverse problems. The algorithm is validated at numerical examples from computerized tomography.

2013

B. N. Hahn, A. K. Louis, M. Maisl, and C. Schorr, Combined reconstruction and edge detection in dimensioning, Meas. Sci. Technol, vol. 24, no. 12, p. 125601, 2013.

Abstract

The result of tomographic examination is a series of two-dimensional (2D) or three-dimensional (3D) images, on which a diagnosis is based. Automatic evaluations of these images are rather common in nondestructive testing, in medical analysis this may partially be the case in the future. Typically the two tasks, image reconstruction and image evaluation, are treated separately. This paper describes an approach where the two steps are combined in just one method. By joint optimization of the two steps, the results are much superior to the separate treatment of the two tasks, as the comparisons in Louis (2008 SIAM J. Imaging Sci. 1 188–208) show. By constructing corresponding reconstruction filters, the algorithms are of filtered backprojection type, hence the computing time essentially remains the same as for the reconstruction of the density itself. We consider the reconstruction problem in x-ray tomography for the fan-beam geometry with flat detectors and edge detection. This method is then extended to filtered backprojection with Feldkamp-type kernels for the 3D cone-beam case. We calculate special reconstruction kernels which work also in the case of very noisy real data, and we present numerical examples from the measured data.

2012

B. N. Hahn and A. K. Louis, Reconstruction in the three-dimensional parallel scanning geometry with application in synchrotronbased x-ray tomography, Inverse Problems, vol. 28, no. 4, p. 045013, 2012.

Abstract

Common reconstruction approaches in the three-dimensional parallel scanning geometry, that occurs for example in synchrotron-based x-ray tomography, use layerwise 2D procedures. A finite number of layers of the investigated object are reconstructed by a 2D method, before a 3D visualization is obtained by interpolation between these layers. In this paper, a fully three-dimensional reconstruction algorithm for the parallel scanning geometry is presented. It allows us to calculate arbitrary slices or even the complete 3D density function directly without causing any interpolation errors. Furthermore, the 3D method enables the fast and stable combination of reconstruction and image analysis step to extract features of the investigated object, for example, derivatives used in canny edge detectors. The application to real data illustrates the stability and advantageous regularization properties of the new approach.

B. N. Hahn, Reconstruction of dynamic objects in computerized tomography, Oberwolfach Reports, vol. 9, pp. 3069-3071B, 2012.

Talks

List of Talks

2019

08/2019: Dynamic tomography: Modelling, Analysis and Algorithms, Modern Challenges in Imaging - In the Footsteps of Allan Cormack, Tufts University, US
07/2019: Iterative motion compensation in tomography, Applied Inverse Problem Conference, Grenoble
07/2019: An efficient reconstruction approach for a class of dynamic imaging operators, Applied Inverse Problem Conference, Grenoble
07/2019: An iterative regularization method for dynamic inverse problems, 9th International Congress on Industrial and Applied Mathematics, Valencia
01/2019: Oberwolfach Workshop Tomographic inverse problems: Theory and applications

2018

07 / 2018: "Microlocal analysis of dynamic imaging problems", Conference on Mathematics of Wave Phenomena, Karlsruhe Institute of Technology
07 / 2018: "Modelling and algorithms in dynamic imaging", IFIP TC 7 Conference on System Modelling and Optimization, Universität Duisburg-Essen
07/2018: "Dynamic Imaging - Analysis and Algorithms", Graduiertenkollegseminar π 3: Parameter Identification, Universität Bremen
06 / 2018: "Algorithms for motion compensation", SIAM Conference on Imaging Science, Bologna, Italien
05 / 2018: "Analysis and algorithms in dynamic tomography", Inverse Problems: Modeling and Simulation, Malta
04 / 2018: BMBF-Thementag Big Data/Data Science, Bonn
03 / 2018: "Time-dependent imaging problems in tomography and localization microscopy", GAMM-Jahrestagung, TU München
01 / 2018: Plenarvortrag "Time-dependent inverse problems in imaging", Fifth Conference on Mathematical Image Analysis, Berlin
01/2018: "Motion compensation in tomography", Seminar "Experimentelle Radiologie“, Universitätsklinikum Würzburg

2017

09 / 2017: "Biomedical image reconstruction and analysis", Symposium Image Analysis and Modeling, Center for Computational and Theoretical Biology, Würzburg
08 / 2017: "Numerical methods for motion artifact reduction", Workshop Computational Imaging and Tomography, Innsbruck, Österreich
05 / 2017: Plenarvortrag "Challenges in dynamic tomography", Meeting on Tomography and Applications, Mailand, Italien
03 / 2017: "The Radon transform for dynamic objects", 100 years of the Radon transform, Linz, Österreich
01/2017: "Bewegungsartefakte in der Computertomographie", Universitätsklinikum Würzburg, Medizinische Physik

2016

12 / 2016: "Dynamic Imaging", Antrittsvorlesung an der Universität Würzburg
11 / 2016: "Inverse problems in dynamic imaging", Centre for Medical Image Computing, UCL, London
07 / 2016: "Artefact reduction in dynamic tomography", International Conference on Sensing and Imaging, Taiyuan, China
05 / 2016: "Singular feature extraction in dynamic imaging", Inverse Problems: Modeling and Simulation, Fethiye, Türkei
05 / 2016: "Computertomographie mit sich bewegendem Objekt", Lehrstuhl für Röntgenmikroskopie, Universität Würzburg

2015

09 / 2015: Plenarvortrag: "A time-dependent regularization method for motion compensation", Dynamical Inverse Problems and Imaging, Münster
08 / 2015: "Image reconstruction from motion corrupted CT-data", Scientific Computing Seminar, Technical University of Denmark
07 / 2015: "Object motion in tomographic imaging", Computational and Analytical Aspects of Image reconstruction, ICERM, Brown University, Providence, RI, USA
05 / 2015: "Reconstruction of singularities in time-dependent computerized tomography", Applied Inverse Problem Conference, Helsinki, Finnland
05 / 2015: "Efficient algorithms and resolution properties for time-dependent inverse problems", Challenges in Inverse Problems, Montpellier, Frankreich

2014

08 / 2014: "Algorithms for locally changing objects in computerized tomography", Inverse Problems - from Theory to Application, Bristol, UK
08 / 2014: "Imaging moving objects", Mathematics and Algorithms in Tomography, Oberwolfach
05 / 2014: "Challenges in time-dependent computerized tomography", Inverse Problems: Modeling and Simulation, Fethiye, Türkei
04 / 2014: "Dynamic inverse problems", Theory and application Seminar "Inverse Problems“, University of Helsinki, Finnland
02 / 2014: Poster: "Efficient algorithms in dynamic computerized tomography", Advanced X-ray Tomography: Experiment, Modeling, and Algorithms, Lorentz Center, Leiden, Niederlande

2013

07 / 2013: "Motion compensation in dynamic inverse problems with application in computerized tomography", Applied Inverse Problem Conference, Daejeon, Korea

2012

10 / 2012: "Reconstruction of dynamic objects in computerized tomography", Computational Inverse Problems, Oberwolfach
10 / 2012: Chinese - German Symposium on Inverse Problems, Saarbrücken
09 / 2012: "Reconstruction in dynamic computerized tomography", DMV Jahrestagung, Saarbrücken
06 / 2012: "Reconstruction of dynamic objects in computerized tomography", Johann Radon Institute for Computational and Applied Mathematics, Linz
04 / 2012: "Reconstruction of dynamic objects", Computational Inverse Problems, Wien

2010

11 / 2010: "Feature-Rekonstruktion in der 3D-Parallelgeometrie", Oberseminar Angewandte Mathematik, Saarbrücken
04 / 2010: Mathematics and Algorithms in Tomography, Oberwolfach

2011

07 / 2011: 7th International Congress on Industrial and Applied Mathematics, Vancouver

Prof. Bernadette Hahn

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Prof. Bernadette Hahn-Rigaud

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