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    <completedYear/>
    <publishedYear>2013</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>doctoralthesis</type>
    <publisherName>Technische Universität München</publisherName>
    <publisherPlace>München</publisherPlace>
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    <contributingCorporation/>
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    <title language="eng">Reconstructions in limited angle x-ray tomography: Characterization of classical reconstructions and adapted curvelet sparse regularization</title>
    <abstract language="eng">This thesis is devoted to the problem of tomographic reconstruction at limited angular range. In the first part, we prove a characterization of filtered backprojection reconstructions from limited angle data. Moreover, we develop a strategy for artifact reduction and stabilization. In the second part, we introduce a new edge-preserving reconstruction algorithm for limited angle tomography and analyze this algorithm mathematically. Some numerical experiments are also presented.</abstract>
    <abstract language="deu">Diese Arbeit beschäftigt sich mit der tomographischen Rekonstruktion bei eingeschränktem Winkelbereich. Im ersten Teil wird eine Charakterisierung der Rekonstruktionen mittels gefilterter Rückprojektion bewiesen. Darüberhinaus wird eine Strategie zur Artefaktreduktion und Stabilisierung entwickelt. Der zweite Teil stellt einen neuen kantenerhaltenden Rekonstruktionsalgorithmus für die Computertomographie bei eingeschränktem Winkelbereich vor and analysiert diesen mathematisch. Numerische Experimente runden die Arbeit ab.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:91-diss-20130328-1115037-0-3</identifier>
    <identifier type="url">https://mediatum.ub.tum.de/?id=1115037</identifier>
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    <licence>Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG</licence>
    <author>Jürgen Frikel</author>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>Computertomographie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>Optimale Rekonstruktion</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>Regularisierungsverfahren</value>
    </subject>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othpublikationsherkunft" number="">Externe Publikationen</collection>
    <collection role="othforschungsschwerpunkt" number="16311">Digitalisierung</collection>
  </doc>
  <doc>
    <id>6044</id>
    <completedYear/>
    <publishedYear>2023</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
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    <belongsToBibliography>0</belongsToBibliography>
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    <title language="eng">A complementary ℓ1 -TV reconstruction algorithm for limited data CT</title>
    <abstract language="eng">In a variety of tomographic applications, data cannot be fully acquired, leading to severely underdetermined image reconstruction. Conventional methods result in reconstructions with significant artifacts. In order to remove these artifacts, regularization methods have to be applied that incorporate additional information. An important example is TV reconstruction which is well known to efficiently compensate for missing data and well reduces reconstruction artifacts. At the same time, however, tomographic data is also contaminated by noise, which poses an additional challenge. The use of a single regularizer within a variational regularization framework must therefore account for both the missing data and the noise. However, a single regularizer may not be ideal for both tasks. For example, the TV regularizer is a poor choice for noise reduction over different scales, in which case ℓ1 curvelet regularization methods work well. To address this issue, in this paper we introduce a novel variational regularization framework that combines the advantages of two different regularizers. The basic idea of our framework is to perform reconstruction in two stages, where the first stage mainly aims at accurate reconstruction in the presence of noise, and the second stage aims at artifact reduction. Both reconstruction stages are connected by a data proximity condition. The proposed method is implemented and tested for limited-view CT using a combined curvelet-TV approach. We define and implement a curvelet transform adapted to the limited view problem and demonstrate the advantages of our approach in a series of numerical experiments in this context.</abstract>
    <identifier type="arxiv">arXiv:2304.11599</identifier>
    <licence>Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG</licence>
    <author>Simon Göppel</author>
    <author>Jürgen Frikel</author>
    <author>Markus Haltmeier</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Image reconstruction</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>limited data</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>artifact reduction</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>sparse regularization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>complementary image reconstruction</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>wedge-adapted curvelets</value>
    </subject>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othforschungsschwerpunkt" number="16311">Digitalisierung</collection>
  </doc>
  <doc>
    <id>5303</id>
    <completedYear/>
    <publishedYear>2021</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>conferencepresentation</type>
    <publisherName/>
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    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Combining reconstruction and edge detection in CT</title>
    <abstract language="eng">We present two methods that combine image reconstruction and edge detection in computed tomography (CT) scans. Our first method is as an extension of the prominent filtered backprojection algorithm. In our second method we employ ℓ1-regularization for stable calculation of the gradient. As opposed to the first method, we show that this approach is able to compensate for undersampled CT data.</abstract>
    <parentTitle language="deu">BVM Workshop 2023, 02.-04. Juli 2023, Braunschweig</parentTitle>
    <identifier type="url">https://www.bvm-workshop.org/wp-content/uploads/2021/02/3091_poster.pdf</identifier>
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    <licence>Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG</licence>
    <author>Jürgen Frikel</author>
    <author>Simon Göppel</author>
    <author>Markus Haltmeier</author>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othforschungsschwerpunkt" number="16311">Digitalisierung</collection>
  </doc>
  <doc>
    <id>5366</id>
    <completedYear/>
    <publishedYear>2012</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>conferencepresentation</type>
    <publisherName/>
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    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
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    <title language="eng">Adapted curvelet sparse regularization in limited angle tomography</title>
    <parentTitle language="eng">Summer School: New Trends and Directions in Harmonic Analysis, Fractional Operator Theory, and Image Analysis, Inzell, Germany, September 17 - 21, 2012</parentTitle>
    <identifier type="url">https://www.fim.uni-passau.de/fileadmin/dokumente/fakultaeten/fim/lehrstuhl/forster-heinlein/Inzell2012/Inzell_Frikel_2012_small.pdf</identifier>
    <enrichment key="opus.import.date">2022-08-24T10:21:08+00:00</enrichment>
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    <licence>Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG</licence>
    <author>Jürgen Frikel</author>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othpublikationsherkunft" number="">Externe Publikationen</collection>
    <collection role="othforschungsschwerpunkt" number="16311">Digitalisierung</collection>
  </doc>
  <doc>
    <id>2384</id>
    <completedYear/>
    <publishedYear>2020</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>163</pageFirst>
    <pageLast>178</pageLast>
    <pageNumber/>
    <edition>1st ed.</edition>
    <issue/>
    <volume/>
    <type>bookpart</type>
    <publisherName>Birkhäuser; Springer International Publishing</publisherName>
    <publisherPlace>Cham</publisherPlace>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
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    <title language="eng">Sparse Regularization of Inverse Problems by Operator-Adapted Frame Thresholding</title>
    <abstract language="eng">We analyze sparse frame based regularization of inverse problems by means of a diagonal frame decomposition (DFD) for the forward operator, which generalizes the SVD. The DFD allows to define a non-iterative (direct) operator-adapted frame thresholding approach which we show to provide a convergent regularization method with linear convergence rates. These results will be compared to the well-known analysis and synthesis variants of sparse ℓ1-regularization which are usually implemented thorough iterative schemes. If the frame is a basis (non-redundant case), the three versions of sparse regularization, namely synthesis and analysis variants of ℓ1-regularization as well as the DFD thresholding are equivalent. However, in the redundant case, those three approaches are pairwise different.</abstract>
    <parentTitle language="eng">Mathematics of Wave Phenomena</parentTitle>
    <identifier type="isbn">978-3-030-47173-6</identifier>
    <identifier type="doi">10.1007/978-3-030-47174-3_10</identifier>
    <identifier type="isbn">978-3-030-47174-3</identifier>
    <enrichment key="opus.import.date">2022-01-20T13:06:22+00:00</enrichment>
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    <enrichment key="BegutachtungStatus">peer-reviewed</enrichment>
    <licence>Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG</licence>
    <author>Jürgen Frikel</author>
    <author>Markus Haltmeier</author>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othforschungsschwerpunkt" number="16311">Digitalisierung</collection>
  </doc>
  <doc>
    <id>3146</id>
    <completedYear/>
    <publishedYear>2018</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>1</pageFirst>
    <pageLast>28</pageLast>
    <pageNumber/>
    <edition/>
    <issue>2</issue>
    <volume>34</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
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    <title language="eng">Efficient regularization with wavelet sparsity constraints in photoacoustic tomography</title>
    <abstract language="eng">In this paper, we consider the reconstruction problem of photoacoustic tomography (PAT) with a flat observation surface. We develop a direct reconstruction method that employs regularization with wavelet sparsity constraints. To that end, we derive a wavelet-vaguelette decomposition (WVD) for the PAT forward operator and a corresponding explicit reconstruction formula in the case of exact data. In the case of noisy data, we combine the WVD reconstruction formula with soft-thresholding, which yields a spatially adaptive estimation method. We demonstrate that our method is statistically optimal for white random noise if the unknown function is assumed to lie in any Besov-ball. We present generalizations of this approach and, in particular, we discuss the combination of PAT-vaguelette soft-thresholding with a total variation (TV) prior. We also provide an efficient implementation of the PAT-vaguelette transform that leads to fast image reconstruction algorithms supported by numerical results.</abstract>
    <parentTitle language="eng">Inverse Problems</parentTitle>
    <identifier type="doi">10.1088/1361-6420/aaa0ac</identifier>
    <note>Corresponding authors: Jürgen Frikel and Markus Haltmeier</note>
    <enrichment key="opus.import.date">2022-03-17T06:33:09+00:00</enrichment>
    <enrichment key="opus.source">sword</enrichment>
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    <enrichment key="BegutachtungStatus">peer-reviewed</enrichment>
    <author>Jürgen Frikel</author>
    <author>Markus Haltmeier</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Image reconstruction</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>photoacoustic tomography</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>sparsity constraints</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>variational regularization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>wavelet-vaguelette decomposition</value>
    </subject>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othforschungsschwerpunkt" number="16315">Information und Kommunikation</collection>
    <collection role="oaweg" number="">Hybrid Open Access - OA-Veröffentlichung in einer Subskriptionszeitschrift/-medium</collection>
    <collection role="oaweg" number="">Corresponding author der OTH Regensburg</collection>
  </doc>
  <doc>
    <id>7332</id>
    <completedYear/>
    <publishedYear>2024</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue>10</issue>
    <volume>12</volume>
    <type>article</type>
    <publisherName>MDPI</publisherName>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Data-Proximal Complementary ℓ1-TV Reconstruction for Limited Data Computed Tomography</title>
    <abstract language="eng">In a number of tomographic applications, data cannot be fully acquired, resulting in severely underdetermined image reconstruction. Conventional methods in such cases lead to reconstructions with significant artifacts. To overcome these artifacts, regularization methods are applied that incorporate additional information. An important example is TV reconstruction, which is known to be efficient in compensating for missing data and reducing reconstruction artifacts. On the other hand, tomographic data are also contaminated by noise, which poses an additional challenge. The use of a single regularizer must therefore account for both the missing data and the noise. A particular regularizer may not be ideal for both tasks. For example, the TV regularizer is a poor choice for noise reduction over multiple scales, in which case ℓ1 curvelet regularization methods are well suited. To address this issue, in this paper, we present a novel variational regularization framework that combines the advantages of different regularizers. The basic idea of our framework is to perform reconstruction in two stages. The first stage is mainly aimed at accurate reconstruction in the presence of noise, and the second stage is aimed at artifact reduction. Both reconstruction stages are connected by a data proximity condition. The proposed method is implemented and tested for limited-view CT using a combined curvelet–TV approach. We define and implement a curvelet transform adapted to the limited-view problem and illustrate the advantages of our approach in numerical experiments.</abstract>
    <parentTitle language="eng">Mathematics</parentTitle>
    <identifier type="issn">2227-7390</identifier>
    <identifier type="doi">10.3390/math12101606</identifier>
    <note>Corresponding author der OTH Regensburg: Jürgen Frikel (und Markus Haltmeier)</note>
    <enrichment key="BegutachtungStatus">peer-reviewed</enrichment>
    <enrichment key="CorrespondingAuthor">Jürgen Frikel und Markus Haltmeier</enrichment>
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    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Simon Göppel</author>
    <author>Jürgen Frikel</author>
    <author>Markus Haltmeier</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>image reconstruction</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>limited data</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>artifact reduction</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>sparse regularization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>total variation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>wedge-adapted curvelets</value>
    </subject>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="oaweg" number="">Gold Open Access- Erstveröffentlichung in einem/als Open-Access-Medium</collection>
    <collection role="othforschungsschwerpunkt" number="16311">Digitalisierung</collection>
    <collection role="oaweg" number="">Corresponding author der OTH Regensburg</collection>
  </doc>
  <doc>
    <id>2285</id>
    <completedYear/>
    <publishedYear>2020</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue>12</issue>
    <volume>36</volume>
    <type>article</type>
    <publisherName>IOP Publishing</publisherName>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
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    <title language="eng">A new 3D model for Magnetic Particle Imaging using realistic magnetic field topologies for algebraic reconstruction</title>
    <abstract language="eng">We derive a new 3D model for magnetic particle imaging (MPI) that is able to incorporate realistic magnetic fields in the reconstruction process. In real MPI scanners, the generated magnetic fields have distortions that lead to deformed magnetic low-field volumes with the shapes of ellipsoids or bananas instead of ideal field-free points (FFP) or lines (FFL), respectively. Most of the common model-based reconstruction schemes in MPI use however the idealized assumption of an ideal FFP or FFL topology and, thus, generate artifacts in the reconstruction. Our model-based approach is able to deal with these distortions and can generally be applied to dynamic magnetic fields that are approximately parallel to their velocity field. We show how this new 3D model can be discretized and inverted algebraically in order to recover the magnetic particle concentration. To model and describe the magnetic fields, we use decompositions of the fields in spherical harmonics. We complement the description of the new model with several simulations and experiments, exploring the effects of magnetic fields distortion and reconstruction parameters on the reconstruction.</abstract>
    <parentTitle language="eng">Inverse Problems</parentTitle>
    <identifier type="doi">10.1088/1361-6420/abb446</identifier>
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    <licence>Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG</licence>
    <author>Gaël Bringout</author>
    <author>Wolfgang Erb</author>
    <author>Jürgen Frikel</author>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othforschungsschwerpunkt" number="16312">Energie und Mobilität</collection>
    <collection role="othforschungsschwerpunkt" number="16316">Produktion und Systeme</collection>
  </doc>
  <doc>
    <id>2290</id>
    <completedYear/>
    <publishedYear>2020</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
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    <title language="eng">Regularization of inverse problems by filtered diagonal frame decomposition</title>
    <abstract language="eng">The characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this work we introduce and analyze the concept of filtered diagonal frame decomposition which extends the standard filtered singular value decomposition to the frame case. Frames as generalized singular system allows to better adapt to a given class of potential solutions. In this paper, we show that filtered diagonal frame decomposition yield a convergent regularization method. Moreover, we derive convergence rates under source type conditions and prove order optimality under the assumption that the considered frame is a Riesz-basis.</abstract>
    <identifier type="url">https://arxiv.org/pdf/2008.06219.pdf</identifier>
    <identifier type="url">https://applied-math.uibk.ac.at/images/Preprints/2021/pp_76_2021.pdf</identifier>
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    <author>Andrea Ebner</author>
    <author>Jürgen Frikel</author>
    <author>Dirk Lorenz</author>
    <author>Johannes Schwab</author>
    <author>Markus Haltmeier</author>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othforschungsschwerpunkt" number="16311">Digitalisierung</collection>
    <collection role="oaweg" number="">Diamond Open Access - OA-Veröffentlichung ohne Publikationskosten (Sponsoring)</collection>
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  <doc>
    <id>2462</id>
    <completedYear/>
    <publishedYear>2021</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>153</pageFirst>
    <pageLast>157</pageLast>
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    <edition/>
    <issue/>
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    <type>conferenceobject</type>
    <publisherName>Springer Vieweg</publisherName>
    <publisherPlace>Wiesbaden, Germany</publisherPlace>
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    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
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    <title language="eng">Combining Reconstruction and Edge Detection in Computed Tomography</title>
    <abstract language="eng">We present two methods that combine image reconstruction and edge detection in computed tomography (CT) scans. Our first method is as an extension of the prominent filtered backprojection algorithm. In our second method we employ l1-regularization for stable calculation of the gradient. As opposed to the first method, we show that this approach is able to compensate for undersampled CT data.</abstract>
    <parentTitle language="eng">Bildverarbeitung für die Medizin 2021 : Proceedings, German Workshop on Medical Image Computing: Regensburg, March 7-9, 2021</parentTitle>
    <identifier type="isbn">978-3-658-33197-9</identifier>
    <identifier type="doi">10.1007/978-3-658-33198-6_37</identifier>
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    <licence>Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG</licence>
    <author>Jürgen Frikel</author>
    <author>Simon Göppel</author>
    <author>Markus Haltmeier</author>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othforschungsschwerpunkt" number="16314">Lebenswissenschaften und Ethik</collection>
  </doc>
  <doc>
    <id>3592</id>
    <completedYear/>
    <publishedYear>2022</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue>8</issue>
    <volume>10</volume>
    <type>article</type>
    <publisherName>MDPI</publisherName>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2022-04-15</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Feature Reconstruction from Incomplete Tomographic Data without Detour</title>
    <abstract language="eng">In this paper, we consider the problem of feature reconstruction from incomplete X-ray CT data. Such incomplete data problems occur when the number of measured X-rays is restricted either due to limit radiation exposure or due to practical constraints, making the detection of certain rays challenging. Since image reconstruction from incomplete data is a severely ill-posed (unstable) problem, the reconstructed images may suffer from characteristic artefacts or missing features, thus significantly complicating subsequent image processing tasks (e.g., edge detection or segmentation).&#13;
In this paper, we introduce a framework for the robust reconstruction of convolutional image features directly from CT data without the need of computing a reconstructed image first. Within our framework, we use non-linear variational regularization methods that can be adapted to a variety of feature reconstruction tasks and to several limited data situations. The proposed variational regularization method minimizes an energy functional being the sum of a feature dependent datafitting term and an additional penalty accounting for specific properties of the features. In our numerical experiments, we consider instances of edge reconstructions from angular under-sampled data and show that our approach is able to reliably reconstruct feature maps in this case.</abstract>
    <parentTitle language="eng">Mathematics</parentTitle>
    <identifier type="doi">10.3390/math10081318</identifier>
    <enrichment key="BegutachtungStatus">peer-reviewed</enrichment>
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    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Simon Göppel</author>
    <author>Jürgen Frikel</author>
    <author>Markus Haltmeier</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Computed tomography</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Radon transform</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>reconstruction</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>limited data</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>sparse data</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>feature reconstruction,</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>edge detection</value>
    </subject>
    <collection role="ddc" number="5">Naturwissenschaften und Mathematik</collection>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="oaweg" number="">Gold Open Access- Erstveröffentlichung in einem/als Open-Access-Medium</collection>
    <collection role="othforschungsschwerpunkt" number="16315">Information und Kommunikation</collection>
  </doc>
  <doc>
    <id>3165</id>
    <completedYear/>
    <publishedYear>2016</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>2301</pageFirst>
    <pageLast>2318</pageLast>
    <pageNumber/>
    <edition/>
    <issue>4</issue>
    <volume>48</volume>
    <type>article</type>
    <publisherName>SIAM</publisherName>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Limited Data Problems for the Generalized Radon Transform in Rⁿ</title>
    <abstract language="eng">We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in R-n. We analyze general filtered backprojection type reconstruction methods for limited data with filters given by general pseudodifferential operators. We provide microlocal characterizations of visible and added singularities in R-n and define modified versions of reconstruction operators that do not generate added artifacts. We calculate the symbol of our general reconstruction operators as pseudodifferential operators and provide conditions for the filters under which the reconstruction operators are elliptic for the visible singularities. If the filters are chosen according to those conditions, we show that almost all visible singularities can be recovered reliably. Our work generalizes the results for the classical line transforms in R-2 and the classical reconstruction operators (that use specific filters). In our proofs, we employ a general paradigm that is based on the calculus of Fourier integral operators. Since this technique does not rely on explicit expressions of the reconstruction operators, it enables us to analyze more general imaging situations.</abstract>
    <parentTitle language="eng">SIAM JOURNAL ON MATHEMATICAL ANALYSIS</parentTitle>
    <identifier type="doi">10.1137/15M1045405</identifier>
    <enrichment key="opus.import.date">2022-03-17T06:33:09+00:00</enrichment>
    <enrichment key="opus.source">sword</enrichment>
    <enrichment key="opus.import.user">importuser</enrichment>
    <enrichment key="BegutachtungStatus">peer-reviewed</enrichment>
    <licence>Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG</licence>
    <author>Jürgen Frikel</author>
    <author>Eric Todd Quinto</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>ARTIFACTS</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>computed tomography</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Fourier integral operators</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Lambda tomography</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>limited angle tomography</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>LOCAL TOMOGRAPHY</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>microlocal analysis</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Radon transforms</value>
    </subject>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othforschungsschwerpunkt" number="16316">Produktion und Systeme</collection>
  </doc>
  <doc>
    <id>3171</id>
    <completedYear/>
    <publishedYear>2018</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue>5</issue>
    <volume>34</volume>
    <type>article</type>
    <publisherName>IOP Publishing</publisherName>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Mathematical analysis of the 1D model and reconstruction schemes for magnetic particle imaging</title>
    <abstract language="eng">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.</abstract>
    <parentTitle language="eng">Inverse Problems</parentTitle>
    <identifier type="doi">10.1088/1361-6420/aab8d1</identifier>
    <enrichment key="opus.import.date">2022-03-17T06:53:51+00:00</enrichment>
    <enrichment key="opus.source">sword</enrichment>
    <enrichment key="opus.import.user">importuser</enrichment>
    <enrichment key="BegutachtungStatus">peer-reviewed</enrichment>
    <licence>Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG</licence>
    <author>Wolfgang Erb</author>
    <author>Andreas Weinmann</author>
    <author>Mandy Ahlborg</author>
    <author>Christina Brandt</author>
    <author>Gael Bringout</author>
    <author>Thorsten M. Buzug</author>
    <author>Jürgen Frikel</author>
    <author>C. Kaethner</author>
    <author>Tobias Knopp</author>
    <author>T. März</author>
    <author>Martin Moddel</author>
    <author>Martin Storath</author>
    <author>A. Weber</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>FORMULATION</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>ill-posedness</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>INTERPOLATION</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>inverse problems</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>magnetic particle imaging</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>model-based reconstruction</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>MPI</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>NODES</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>reconstruction operator</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>RELAXATION</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>RESOLUTION</value>
    </subject>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othforschungsschwerpunkt" number="16314">Lebenswissenschaften und Ethik</collection>
  </doc>
  <doc>
    <id>4814</id>
    <completedYear/>
    <publishedYear>2022</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>613</pageFirst>
    <pageLast>617</pageLast>
    <pageNumber/>
    <edition/>
    <issue>4</issue>
    <volume>30</volume>
    <type>article</type>
    <publisherName>De Gruyter</publisherName>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2022-06-25</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Research biography of a distinguished expert in the field of inverse problems: Professor Eric Todd Quinto</title>
    <abstract language="eng">This article gives a brief overview of the research in microlocal analysis, tomography, and integral geometry of Professor Eric Todd Quinto, Robinson Professor of Mathematics at Tufts University, along with the collaborators and colleagues who influenced his work.</abstract>
    <parentTitle language="eng">Journal of Inverse and Ill-posed Problems</parentTitle>
    <identifier type="doi">10.1515/jiip-2022-0031</identifier>
    <identifier type="issn">1569-3945</identifier>
    <enrichment key="BegutachtungStatus">peer-reviewed</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <licence>Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG</licence>
    <author>Mark Agranovsky</author>
    <author>Jan Boman</author>
    <author>Alemdar Hasanov</author>
    <author>Raluca Felea</author>
    <author>Jürgen Frikel</author>
    <author>Venky Krishnan</author>
    <author>Roman Novikov</author>
    <author>Ronny Ramlau</author>
    <author>Cristiana Sebu</author>
    <collection role="ddc" number="5">Naturwissenschaften und Mathematik</collection>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othforschungsschwerpunkt" number="16317">Sensorik</collection>
    <collection role="oaweg" number="">Bronze Open Access - frei zugänglich aber ohne Lizenzhinweis</collection>
  </doc>
  <doc>
    <id>5093</id>
    <completedYear/>
    <publishedYear>2016</publishedYear>
    <thesisYearAccepted/>
    <language>deu</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue>1</issue>
    <volume>32</volume>
    <type>article</type>
    <publisherName>Institute of Physics</publisherName>
    <publisherPlace>Bristol [u.a.]</publisherPlace>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="deu">On artifacts in limited data spherical Radon transform: Curved observation surface</title>
    <abstract language="eng">We study the limited data problem of the spherical Radon transform in two and three-dimensional spaces with general acquisition surfaces. In such situations, it is known that the application of filtered-backprojection reconstruction formulas might generate added artifacts and degrade the quality of reconstructions. In this article, we explicitly analyze a family of such inversion formulas, depending on a smoothing function that vanishes to order k on the boundary of the acquisition surfaces. We show that the artifacts are k orders smoother than their generating singularity. Moreover, in two-dimensional space, if the generating singularity is conormal satisfying a generic condition then the artifacts are even k+1/2 orders smoother than the generating singularity. Our analysis for three-dimensional space contains an important idea of lifting up space. We also explore the theoretical findings in a series of numerical experiments. Our experiments show that a good choice of the smoothing function leads to a significant improvement of reconstruction quality.</abstract>
    <parentTitle language="deu">Inverse Problems</parentTitle>
    <identifier type="issn">1361-6420</identifier>
    <identifier type="issn">0266-5611</identifier>
    <identifier type="doi">10.1088/0266-5611/32/1/015012</identifier>
    <enrichment key="opus.import.date">2022-08-05T07:16:41+00:00</enrichment>
    <enrichment key="opus.source">sword</enrichment>
    <enrichment key="opus.import.user">importuser</enrichment>
    <enrichment key="BegutachtungStatus">peer-reviewed</enrichment>
    <licence>Keine Lizenz - Es gilt das deutsche Urheberrecht: § 53 UrhG</licence>
    <author>Lyudmyla L. Barannyk</author>
    <author>Jürgen Frikel</author>
    <author>Linh V. Nguyen</author>
    <collection role="institutes" number="FakIM">Fakultät Informatik und Mathematik</collection>
    <collection role="othforschungsschwerpunkt" number="16316">Produktion und Systeme</collection>
  </doc>
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