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Hahn-Rigaud, Bernadette

Bernadette Hahn-Rigaud

Prof. Dr.

Chairholder OIP
Dept. of Mathematics, IMNG
Optimization and Inverse Problems

Contact

+49 711 685 61605
Email

Pfaffenwaldring 57
70569 Stuttgart
Deutschland
Room: 8.164

Office Hours

on appointment

Subject

Dynamic inverse problems

Inverse problems arise whenever a searched-for quantity cannot be directly observed but has to be extracted indirectly from measured data. Classic examples constitute imaging modalities such as computerized tomography which provide information about the interior of a patient or an object in a non-invasive way.
The classic regularisation theory is based on the assumption that the searched-for quantity is stationary during the acquisition. However, in many applications from medicine to non-destructive testing, this assumption is not satisfied. Therefore, we work on the development and (numerical) analysis of time-dependent regularization methods which incorporate the dynamics of the object.

Imaging and tomography

Developing new methods in imaging and new fields of applications necessitates novel approaches in mathematical modelling, data processing and image reconstruction. A special focus of our group lies in this respect on computerized tomography and magnetic resonance imaging.

Data analysis and image processing

In order to extract features and properties of the studied medium, the reconstruction results are typically subject to further processing steps. Since an ever-increasing amount of data arises in modern applications, special algorithms are required to extract the desired infomation in an efficient and stable way.

Publications

2024
1. M. Nitzsche and B. N. Hahn, Dynamic image reconstruction in MPI with RESESOP-Kaczmarz. 2024.
  - BibTeX
  BibTeX
  @misc{nitzsche2024dynamic, archiveprefix = {arXiv}, author = {Nitzsche, Marius and Hahn, Bernadette N}, eprint = {2401.14202}, primaryclass = {math.OC}, title = {Dynamic image reconstruction in MPI with RESESOP-Kaczmarz}, year = 2024 }
2. M. S. Feinler and B. N. Hahn, GAN-based iterative motion estimation in HASTE MRI. 2024.
  - BibTeX
  - Link
  BibTeX
  @misc{feinler2024ganbasediterativemotionestimation, archiveprefix = {arXiv}, author = {Feinler, Mathias S. and Hahn, Bernadette N.}, eprint = {2404.07576}, primaryclass = {math.NA}, title = {GAN-based iterative motion estimation in HASTE MRI}, url = {https://arxiv.org/abs/2404.07576}, year = 2024 }
  Link
  https://arxiv.org/abs/2404.07576
2023
1. 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.
  BibTeX
  @article{10.1093/imanum/drad002, 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.}}, author = {Hahn, Bernadette N and Rigaud, Gaël and Schmähl, Richard}, doi = {10.1093/imanum/drad002}, eprint = {https://academic.oup.com/imajna/advance-article-pdf/doi/10.1093/imanum/drad002/49332831/drad002.pdf}, issn = {0272-4979}, journal = {IMA Journal of Numerical Analysis}, month = {02}, note = {drad002}, title = {A class of regularizations based on nonlinear isotropic diffusion for inverse problems}, url = {https://doi.org/10.1093/imanum/drad002}, year = 2023 }
  Link
  https://doi.org/10.1093/imanum/drad002
2. 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.
  - BibTeX
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  BibTeX
  @article{Hahn_2023, author = {Hahn, Bernadette N and Quinto, Eric Todd and Rigaud, Gaël}, doi = {10.1088/1361-6420/acb569}, issn = {1361-6420}, journal = {Inverse Problems}, month = {02}, number = 3, pages = 030401, publisher = {IOP Publishing}, title = {Foreword to special issue of Inverse Problems on modern challenges in imaging}, url = {http://dx.doi.org/10.1088/1361-6420/acb569}, volume = 39, year = 2023 }
  Link
  http://dx.doi.org/10.1088/1361-6420/acb569
3. B. Hahn and B. Wirth, Convex reconstruction of moving particles with inexact motion model, PAMM, vol. 23, no. 2, Sep. 2023.
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  BibTeX
  @article{Hahn_2023, author = {Hahn, Bernadette and Wirth, Benedikt}, doi = {10.1002/pamm.202300054}, journal = {{PAMM}}, month = {09}, number = 2, publisher = {Wiley}, title = {Convex reconstruction of moving particles with inexact motion model}, url = {https://doi.org/10.1002%2Fpamm.202300054}, volume = 23, year = 2023 }
  Link
  https://doi.org/10.1002%2Fpamm.202300054
4. M. S. Feinler and B. N. Hahn, Retrospective Motion Correction in Gradient Echo MRI by Explicit Motion Estimation Using Deep CNNs. 2023.
  - BibTeX
  BibTeX
  @misc{feinler2023retrospective, archiveprefix = {arXiv}, author = {Feinler, Mathias S. and Hahn, Bernadette N.}, eprint = {2303.17239}, primaryclass = {cs.CV}, title = {Retrospective Motion Correction in Gradient Echo MRI by Explicit Motion Estimation Using Deep CNNs}, year = 2023 }
5. 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.
  - BibTeX
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  BibTeX
  @article{Arridge_2023, author = {Arridge, Simon R. and Burger, Martin and Hahn, Bernadette and Quinto, Eric Todd}, doi = {10.4171/owr/2023/21}, issn = {1660-8941}, journal = {Oberwolfach Reports}, month = {12}, number = 2, pages = {1105–1194}, publisher = {European Mathematical Society - EMS - Publishing House GmbH}, title = {Tomographic Inverse Problems: Mathematical Challenges and Novel Applications}, url = {http://dx.doi.org/10.4171/OWR/2023/21}, volume = 20, year = 2023 }
  Link
  http://dx.doi.org/10.4171/OWR/2023/21
2022
1. 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.
  - BibTeX
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  BibTeX
  @article{https://doi.org/10.18416/ijmpi.2022.2203062, author = {Nitzsche, Marius and Albers, Hannes and Kluth, Tobias and Hahn, Bernadette}, copyright = {Creative Commons Attribution 4.0 International}, doi = {10.18416/IJMPI.2022.2203062}, journal = {International Journal on Magnetic Particle Imaging}, language = {en}, pages = {Vol 8 No 1 Suppl 1 (2022)}, publisher = {International Journal on Magnetic Particle Imaging}, title = {Compensating model imperfections during image reconstruction via Resesop}, url = {https://journal.iwmpi.org/index.php/iwmpi/article/view/464}, year = 2022 }
  Link
  https://journal.iwmpi.org/index.php/iwmpi/article/view/464
2. 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.
  - BibTeX
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  BibTeX
  @article{Hahn_2022, author = {Hahn, Bernadette N. and Garrido, Melina-Loren Kienle and Klingenberg, Christian and Warnecke, Sandra}, doi = {10.3934/ipi.2022018}, issn = {1930-8345}, journal = {Inverse Problems and Imaging}, number = 5, pages = 1179, publisher = {American Institute of Mathematical Sciences (AIMS)}, title = {Using the Navier-Cauchy equation for motion estimation in dynamic imaging}, url = {http://dx.doi.org/10.3934/ipi.2022018}, volume = 16, year = 2022 }
  Link
  http://dx.doi.org/10.3934/ipi.2022018
2021
1. B. N. Hahn, M. L. Kienle-Garrido, and E. T. Quinto, Microlocal properties of dynamic Fourier integral operators, 2021.
  - BibTeX
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  BibTeX
  @article{hahn2021microlocal, author = {Hahn, B.N. and Kienle-Garrido, M. L. and Quinto, E.T.}, doi = {10.1007/978-3-030-57784-1_4}, title = {Microlocal properties of dynamic Fourier integral operators}, url = {http://dx.doi.org/10.1007/978-3-030-57784-1_4}, year = 2021 }
  Link
  http://dx.doi.org/10.1007/978-3-030-57784-1_4
2. B. N. Hahn, Motion compensation strategies in tomography, 2021.
  - BibTeX
  - Link
  BibTeX
  @article{hahn2021motion, author = {Hahn, B.N.}, doi = {10.1007/978-3-030-57784-1_3}, title = {Motion compensation strategies in tomography}, url = {http://dx.doi.org/10.1007/978-3-030-57784-1_3}, year = 2021 }
  Link
  http://dx.doi.org/10.1007/978-3-030-57784-1_3
2020
1. 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.
  - BibTeX
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  BibTeX
  @article{Rigaud_2020, author = {Rigaud, G. and Hahn, B. N.}, doi = {10.1080/17415977.2020.1815723}, journal = {Inverse Problems in Science and Engineering}, number = 7, pages = {967--989}, publisher = {Informa {UK} Limited}, title = {Reconstruction algorithm for 3D Compton scattering imaging with incomplete data}, url = {https://doi.org/10.1080/17415977.2020.1815723}, volume = 29, year = 2020 }
  Link
  https://doi.org/10.1080/17415977.2020.1815723
2. 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.
  Abstract
  The problem of dynamic 2D vector tomography is considered. Object motion is a combination of rotation and shifting. Properties of the dynamic ray transform operators are investigated. Singular value decomposition of the operators is constructed with usage of classic orthogonal polynomials.
  BibTeX
  @inproceedings{10.1007/978-3-030-40616-5_42, abstract = {The problem of dynamic 2D vector tomography is considered. Object motion is a combination of rotation and shifting. Properties of the dynamic ray transform operators are investigated. Singular value decomposition of the operators is constructed with usage of classic orthogonal polynomials.}, address = {Cham}, author = {Polyakova, Anna P. and Svetov, Ivan E. and Hahn, Bernadette N.}, booktitle = {Numerical Computations: Theory and Algorithms}, editor = {Sergeyev, Yaroslav D. and Kvasov, Dmitri E.}, isbn = {978-3-030-40616-5}, pages = {446--453}, publisher = {Springer International Publishing}, title = {The Singular Value Decomposition of the Operators of the Dynamic Ray Transforms Acting on 2D Vector Fields}, url = {https://link.springer.com/chapter/10.1007/978-3-030-40616-5_42}, year = 2020 }
  Link
  https://link.springer.com/chapter/10.1007/978-3-030-40616-5_42
3. B. N. Hahn, M. L. Kienle-Garrido, C. Klingenberg, and S. Warnecke, Using the Navier-Cauchy equation for motion estimation in dynamic imaging. 2020.
  - BibTeX
  - Link
  BibTeX
  @misc{hahn2020using, archiveprefix = {arXiv}, author = {Hahn, B. N. and Kienle-Garrido, M. L. and Klingenberg, C. and Warnecke, S.}, eprint = {2009.04212}, primaryclass = {math.AP}, title = {Using the Navier-Cauchy equation for motion estimation in dynamic imaging}, url = {https://arxiv.org/abs/2009.04212}, year = 2020 }
  Link
  https://arxiv.org/abs/2009.04212
4. 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.
  BibTeX
  @article{Blanke_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.}, author = {Blanke, Stephanie E and Hahn, Bernadette N and Wald, Anne}, doi = {10.1088/1361-6420/abb5e1}, journal = {Inverse Problems}, number = 12, pages = 124001, publisher = {{IOP} Publishing}, title = {Inverse problems with inexact forward operator: iterative regularization and application in dynamic imaging}, url = {https://doi.org/10.1088/1361-6420/abb5e1}, volume = 36, year = 2020 }
  Link
  https://doi.org/10.1088/1361-6420/abb5e1
2019
1. Mathematisches Forschungsinstitut Oberwolfach, Tomographic Inverse Problems: Theory and Applications, Workshop Reports, 2019.
  - BibTeX
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  BibTeX
  @article{https://doi.org/10.14760/owr-2019-4, author = {{Mathematisches Forschungsinstitut Oberwolfach}}, doi = {10.14760/OWR-2019-4}, journal = {Workshop Reports}, language = {en}, publisher = {Mathematisches Forschungsinstitut Oberwolfach}, title = {Tomographic Inverse Problems: Theory and Applications}, url = {http://publications.mfo.de/handle/mfo/3720}, year = 2019 }
  Link
  http://publications.mfo.de/handle/mfo/3720
2. 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
  - BibTeX
  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.
  BibTeX
  @inproceedings{kluth2019spatiotemporal, 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.}, author = {Kluth, T. and Hahn, B. N. and Brandt, C.}, booktitle = {Proc. Int. Workshop Magnetic Particle Imaging}, title = {Spatio-temporal concentration reconstruction using motion priors in magnetic particle imaging}, year = 2019 }
3. 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.
  BibTeX
  @article{hahn2019efficient, 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.}, author = {Hahn, B. N. and Kienle Garrido, M.-L.}, doi = {10.1088/1361-6420/ab178b}, journal = {Inverse Problems}, number = 9, pages = 094005, title = {An efficient reconstruction approach for a class of dynamic imaging operators}, url = {https://doi.org/10.1088/1361-6420/ab178b}, volume = 35, year = 2019 }
  Link
  https://doi.org/10.1088/1361-6420/ab178b
2018
1. T. Schuster, B. Hahn, and M. Burger, Dynamic inverse problems: modelling—regularization—numerics, Inverse Problems, vol. 34, no. 4, p. 040301, Mar. 2018.
  - BibTeX
  - Link
  BibTeX
  @article{Schuster_2018, author = {Schuster, Thomas and Hahn, Bernadette and Burger, Martin}, doi = {10.1088/1361-6420/aab0f5}, issn = {1361-6420}, journal = {Inverse Problems}, month = {03}, number = 4, pages = 040301, publisher = {IOP Publishing}, title = {Dynamic inverse problems: modelling—regularization—numerics}, url = {http://dx.doi.org/10.1088/1361-6420/aab0f5}, volume = 34, year = 2018 }
  Link
  http://dx.doi.org/10.1088/1361-6420/aab0f5
2. 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.
  BibTeX
  @article{rigaud2018compton, 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.}, author = {Rigaud, G. and Hahn, B. N.}, doi = {10.1088/1361-6420/aabf0b}, journal = {Inverse Problems}, number = 7, pages = 075004, title = {3D Compton scattering imaging and contour reconstruction for a class of Radon transforms}, url = {https://doi.org/10.1088/1361-6420/aabf0b}, volume = 34, year = 2018 }
  Link
  https://doi.org/10.1088/1361-6420/aabf0b
2017
1. 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.
  BibTeX
  @article{hahn2017motion, 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.}, author = {Hahn, B. N.}, doi = {10.1007/s11220-017-0159-6}, journal = {Sensing and Imaging}, number = 10, pages = {1-20}, title = {Motion Estimation and Compensation Strategies in Dynamic Computerized Tomography}, url = {https://doi.org/10.1007/s11220-017-0159-6}, volume = 18, year = 2017 }
  Link
  https://doi.org/10.1007/s11220-017-0159-6
2. 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.
  BibTeX
  @article{hahn2017motion, 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.}, author = {Hahn, B. N.}, doi = {10.1088/1361-6420/aa8d7b}, journal = {Inverse Problems}, number = 11, pages = 114001, title = {A motion artefact study and locally deforming objects in computerized tomography}, url = {https://doi.org/10.1088/1361-6420/aa8d7b}, volume = 33, year = 2017 }
  Link
  https://doi.org/10.1088/1361-6420/aa8d7b
2016
1. 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.
  BibTeX
  @article{hahn2016detectable, 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.}, author = {Hahn, B. N. and Quinto, E. T.}, doi = {10.1137/16M1057917}, journal = {SIAM J. Imaging Sciences}, number = 3, pages = {1195-1225}, title = {Detectable singularities from dynamic Radon data}, url = {http://epubs.siam.org/doi/10.1137/16M1057917}, volume = 9, year = 2016 }
  Link
  http://epubs.siam.org/doi/10.1137/16M1057917
2. 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.
  BibTeX
  @article{hahn2016space, 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.}, author = {Hahn, B. N.}, doi = {10.1088/0266-5611/32/2/025006}, journal = {Inverse Problems}, number = 2, pages = 025006, title = {Null space and resolution in dynamic computerized tomography}, url = {http://dx.doi.org/10.1088/0266-5611/32/2/025006}, volume = 32, year = 2016 }
  Link
  http://dx.doi.org/10.1088/0266-5611/32/2/025006
2015
1. 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.
  BibTeX
  @article{hahn2015dynamic, 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.}, author = {Hahn, B. N.}, doi = {10.3934/ipi.2015.9.395}, journal = {Inverse Problems & Imaging}, number = 2, pages = {395-413}, title = {Dynamic linear inverse problems with moderate movements of the object: Ill-posedness and regularization}, url = {http://dx.doi.org/10.3934/ipi.2015.9.395}, volume = 9, year = 2015 }
  Link
  http://dx.doi.org/10.3934/ipi.2015.9.395
2. 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.
  BibTeX
  @article{gerth2015method, 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.}, author = {Gerth, D. and Hahn, B. N. and Ramlau, R.}, doi = {10.1088/0266-5611/31/6/065002}, journal = {Inverse Problems}, number = 6, pages = { 065002}, title = {The method of the approximate inverse for atmospheric tomography}, url = {http://dx.doi.org/10.1088/0266-5611/31/6/065002}, volume = 31, year = 2015 }
  Link
  http://dx.doi.org/10.1088/0266-5611/31/6/065002
2014
1. 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.
  BibTeX
  @article{hahn2014reconstruction, 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.}, author = {Hahn, B. N.}, doi = {10.1515/jip-2012-0094}, journal = {Journal of Inverse and Ill-posed Problems}, number = 3, pages = {323-339}, title = {Reconstruction of dynamic objects with affine deformations in computerized tomography}, url = {https://doi.org/10.1515/jip-2012-0094}, volume = 22, year = 2014 }
  Link
  https://doi.org/10.1515/jip-2012-0094
2. 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.
  BibTeX
  @article{hahn2014efficient, 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.}, author = {Hahn, B. N.}, doi = {10.1088/0266-5611/30/3/035008}, journal = {Inverse Problems}, number = 3, pages = 035008, title = {Efficient algorithms for linear dynamic inverse problems with known motion}, url = {http://dx.doi.org/10.1088/0266-5611/30/3/035008}, volume = 30, year = 2014 }
  Link
  http://dx.doi.org/10.1088/0266-5611/30/3/035008
2013
1. 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.
  BibTeX
  @article{hahn2013combined, 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.}, author = {Hahn, B. N. and Louis, A. K. and Maisl, M. and Schorr, C.}, doi = {10.1088/0957-0233/24/12/125601}, journal = {Meas. Sci. Technol}, number = 12, pages = 125601, title = {Combined reconstruction and edge detection in dimensioning}, url = {http://dx.doi.org/10.1088/0957-0233/24/12/125601}, volume = 24, year = 2013 }
  Link
  http://dx.doi.org/10.1088/0957-0233/24/12/125601
2012
1. 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.
  BibTeX
  @article{hahn2012reconstruction, 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.}, author = {Hahn, B. N. and Louis, A. K.}, doi = {10.1088/0266-5611/28/4/045013}, journal = {Inverse Problems}, number = 4, pages = 045013, title = {Reconstruction in the three-dimensional parallel scanning geometry with application in synchrotronbased x-ray tomography}, url = {https://doi.org/10.1088/0266-5611/28/4/045013}, volume = 28, year = 2012 }
  Link
  https://doi.org/10.1088/0266-5611/28/4/045013
2. B. N. Hahn, Reconstruction of dynamic objects in computerized tomography, Oberwolfach Reports, vol. 9, pp. 3069-3071B, 2012.
  - BibTeX
  BibTeX
  @article{hahn2012reconstruction, author = {Hahn, B. N.}, journal = {Oberwolfach Reports}, pages = {3069-3071B}, title = {Reconstruction of dynamic objects in computerized tomography}, volume = 9, year = 2012 }

Teaching

Current semester

Current courses can be found on the Teaching Page or directly in Campus.

Previous semesters

Summer term 2024:

research period

Winter term 2023/24:

Höhere Mathematik 3 für el, kyb, mecha, phys

Summer term 2023:

Master seminar: Dynamic inverse problems
Fortgeschrittene Analysis für SimTech 2

Winter term 2022/23:

Mathematical image processing
associated lecture (PD Dr. Gael Rigaud):
Masterseminar: Machine Learning meets inverse problems

Summer term 2022:

Introduction to Inverse Problems

Winter term 2021/22:

Introduction to Optimization
associated lecture (PD Dr. Gael Rigaud):
Masterseminar: Modern Challenges in Image processing and Imaging

Summer term 2021:

Mathematical Image Processing
Seminar zu Optimierung und inversen Problemen

Winter term 2020/21:

Introduction to Optimization
Seminar: Medizinische Bildgebung

Sommersemester 2020:

Regularization of Inverse problems
theorie and application
Seminar: Sparsity und Compressed Sensing

Vita

since 04 / 2020	Professor at University of Stuttgart
10 / 2018	Positive Evaluation of the Junior Prrofessorship
04 / 2016 - 03 / 2020	Junior Professor at University of Würzburg
01 / 2015 - 03 / 2016	Research and teaching assistant at the Institute of Applied Mathematics, Saarland University
09 / 2014 - 12 / 2014	Visiting Scholar and Lecturer at Tufts University, Medford, MA, USA
01 / 2011 - 08 / 2014	Research and teaching assistant at the Institute of Applied Mathematics, Saarland University
09 / 2009 - 12 / 2010	Graduate assistant within the DFG project "Combining Image Reconstruction and Image Evaluation", Saarland University
04 / 2008 - 10 / 2009	Student assistant at the Department of Mathematics, Saarland University

Activities in research and teaching

Member of the Editorial Board

Inverse Problems
Sensing and Imaging

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Awards

Lecture awared of the student council mathematics in the category Bachelor additional modules for lecture "Optimization" (summer term 2021)
Lecture award of the student council mathematics for digital teaching (summer term 2020)
EAIP Young Scientist Award for distinguished contributions to inverse problems (2018)
Highlights of IOP Inverse Problems (2012, 2014, 2015, 2016, 2017)

Bernadette Hahn-Rigaud

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Dynamic inverse problems

Imaging and tomography

Data analysis and image processing

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