@unpublished{FrikelQuinto, author = {Frikel, J{\"u}rgen and Quinto, Eric Todd}, title = {A paradigm for the characterization of artifacts in tomography}, doi = {10.48550/arXiv.1409.4103}, abstract = {We present a paradigm for characterization of artifacts in limited data tomography problems. In particular, we use this paradigm to characterize artifacts that are generated in reconstructions from limited angle data with generalized Radon transforms and general filtered backprojection type operators. In order to find when visible singularities are imaged, we calculate the symbol of our reconstruction operator as a pseudodifferential operator.}, language = {en} } @inproceedings{Frikel, author = {Frikel, J{\"u}rgen}, title = {A new framework for sparse regularization in limited angle x-ray tomography}, series = {2010 7th IEEE International Symposium on Biomedical Imaging: From Nano to Macro ; Rotterdam 14.04.2010 - 17.04.2010}, booktitle = {2010 7th IEEE International Symposium on Biomedical Imaging: From Nano to Macro ; Rotterdam 14.04.2010 - 17.04.2010}, publisher = {IEEE}, address = {Piscataway, NJ}, isbn = {978-1-4244-4125-9}, issn = {1945-7928}, doi = {10.1109/ISBI.2010.5490113}, pages = {824 -- 827}, abstract = {We propose a new framework for limited angle tomographic reconstruction. Our approach is based on the observation that for a given acquisition geometry only a few (visible) structures of the object can be reconstructed reliably using a limited angle data set. By formulating this problem in the curvelet domain, we can characterize those curvelet coefficients which correspond to visible structures in the image domain. The integration of this information into the formulation of the reconstruction problem leads to a considerable dimensionality reduction and yields a speedup of the corresponding reconstruction algorithms.}, language = {en} } @article{Frikel, author = {Frikel, J{\"u}rgen}, title = {Sparse regularization in limited angle tomography}, series = {Applied and Computational Harmonic Analysis}, volume = {34}, journal = {Applied and Computational Harmonic Analysis}, number = {1}, publisher = {Elsevier}, doi = {10.1016/j.acha.2012.03.005}, pages = {117 -- 141}, abstract = {We investigate the reconstruction problem of limited angle tomography. Such problems arise naturally in applications like digital breast tomosynthesis, dental tomography, electron microscopy, etc. Since the acquired tomographic data is highly incomplete, the reconstruction problem is severely ill-posed and the traditional reconstruction methods, e.g. filtered backprojection (FBP), do not perform well in such situations. To stabilize the reconstruction procedure additional prior knowledge about the unknown object has to be integrated into the reconstruction process. In this work, we propose the use of the sparse regularization technique in combination with curvelets. We argue that this technique gives rise to an edge-preserving reconstruction. Moreover, we show that the dimension of the problem can be significantly reduced in the curvelet domain. To this end, we give a characterization of the kernel of the limited angle Radon transform in terms of curvelets and derive a characterization of solutions obtained through curvelet sparse regularization. In numerical experiments, we will show that the theoretical results directly translate into practice and that the proposed method outperforms classical reconstructions.}, language = {en} } @article{FrikelQuinto, author = {Frikel, J{\"u}rgen and Quinto, Eric Todd}, title = {Artifacts in Incomplete Data Tomography with Applications to Photoacoustic Tomography and Sonar}, series = {SIAM Journal on Applied Mathematics}, volume = {75}, journal = {SIAM Journal on Applied Mathematics}, number = {2}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia, Pa.}, issn = {1095-712X}, doi = {10.1137/140977709}, pages = {703 -- 725}, abstract = {We develop a paradigm using microlocal analysis that allows one to characterize the visible and added singularities in a broad range of incomplete data tomography problems. We give precise characterizations for photoacoustic and thermoacoustic tomography and sonar, and provide artifact reduction strategies. In particular, our theorems show that it is better to arrange sonar detectors so that the boundary of the set of detectors does not have corners and is smooth. To illustrate our results, we provide reconstructions from synthetic spherical mean data as well as from experimental photoacoustic data.}, language = {en} } @article{FrikelQuinto, author = {Frikel, J{\"u}rgen and Quinto, Eric Todd}, title = {Limited Data Problems for the Generalized Radon Transform in Rⁿ}, series = {SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, volume = {48}, journal = {SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, number = {4}, publisher = {SIAM}, doi = {10.1137/15M1045405}, pages = {2301 -- 2318}, abstract = {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.}, language = {en} }