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J Xray Sci Technol. 2017 Apr 5. doi: 10.3233/XST-16211. [Epub ahead of print]

Initial analysis of the middle problem in CT image reconstruction.

Yang J1,2,3, Yu H4, Wang G3.

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LMAM, School of Mathematical Sciences, Peking University, Beijing, China.
Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing, China.
Biomedical Imaging Center/Cluster, CBIS, Rensselaer Polytechnic Institute, Troy, NY, USA.
Department of Electrical and Computer Engineering, University of Massachusetts Lowell, Lowell, MA, USA.


The interior and exterior problems have been extensively studied in the field of reconstruction of computed tomography (CT) images, which lead to important theoretical and practical results. In this study, we formulate a middle problem of CT image reconstruction, which is more challenging than either the interior or exterior problems. In the middle problem of CT image reconstruction, projection data are measured through and only through the middle dough-like region, so that each projection profile misses data not only internally but also on both sides. For an object with a radially symmetric exterior, we proved that the middle problem could be uniquely solved if the middle ring-shaped zone is piecewise constant or there is a known sub-region inside this middle region. Then, we designed and evaluated a POCS-based algorithm for middle tomography, which is to reconstruct a middle image only from the available data. Finally, the remaining issues are also discussed for further research.


Middle problem; POCS; data truncation; image reconstruction; limited-angle tomography; middle tomography


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