Development of a Solvability Map

Main Article Content

Gengsheng L. Zeng, Fellow, IEEE


From time to time, it is necessary to determine whether there are sufficient measurements for the image reconstruction task especially when a non-standard scanning geometry is used. When the imaging system can be approximately modeled as a system of linear equations, the condition number of the system matrix indicates whether the entire system can be stably solved as a whole. When the system as a whole cannot be stably solved, the Moore-Penrose pseudo inverse matrix can be evaluated through the singular value decomposition (SVD) and then a generalized solution can be obtained. However, these methods are not practical because they require the computer memory to store the whole system matrix, which is often too large to store. Also, we do not know if the generalized solution is good enough for the application in mind. This paper proposes a practical image solvability map, which can be obtained for any practical image reconstruction algorithm. This image solvability map measures the reconstruction errors for each location using a large number of computer-simulated random phantoms. In other words, the map is generated by a Monte Carlo approach.

Index Terms: Internal problem, Inverse problem, Image reconstruction, Biomedical imaging, Computed Tomography, Computer simulations, Monte Carlo

Keywords: Internal problem, Inverse problem, Image reconstruction, Biomedical imaging, Computed Tomography, Computer simulations, Monte Carlo

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How to Cite
ZENG, Gengsheng L.. Development of a Solvability Map. Medical Research Archives, [S.l.], v. 10, n. 11, nov. 2022. ISSN 2375-1924. Available at: <>. Date accessed: 29 mar. 2023. doi:
Research Articles


1. Orlov SS. Theory of three-dimensional reconstruction: II, The recovery operator. Sov. Physics-Crystallography. 1975; 20:701-709.
2. Tuy HK. An inversion formula for cone-beam reconstruction algorithm. SIAM J. Appl. Math. 1983;43:546-52.
3. Kirillov AA. On a problem of I M Gel'fand. Sov. Math. Dokl. 1961;2: 268-269.
4. Zeng GL, Gullberg GT, Foresti SA. Eigen analysis of cone-beam scanning geometries. Proceedings of the 1995 International meeting on fully three-dimensional image reconstruction in radiology and nuclear medicine. 1995; 261-265, Aix-les-Bains, Savoie, France, July 4-6.
5. Lanczos C. An iterative method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Stand. 1950;45:255-282.
6. Zhang B, Zeng GL. Two-dimensional iterative region-of-interest (ROI) reconstruction from truncated projection data. Med. Phys. 2007; 34: 935-944.
7. Golub GH, Van Loan CF. Matrix computations (3rd ed.). 1996; Baltimore: Johns Hopkins. pp. 257–258. ISBN 978-0-8018-5414-9.
8. Wirgin A. The inverse Crime. Mathematical Physics, arXiv:math-ph/0401050v1 (math-phy) 2004.
9. Zeng GL, Medical Imaging Reconstruction, A Tutorial. 2009; ISBN: 978-3-642-05367-2, 978-7-04-020437-7, Higher Education Press, Springer, Beijing.
10. Natterer F. The Mathematics of Computerized Tomography. 2001; SIAM eBook,
11. Zeng GL, Gullberg GT. An SVD study of truncated transmission data in SPECT. IEEE Trans. Nucl. Sci. 1997; 44(1):107-111.
12. Mao Y, Zeng GL. Tailored ML-EM algorithm for reconstruction of truncated projection data using few view angles. Phys. Med. Biol. 2013;58: N157-N169.
13 Hatamikia S, Biguri A, Kronreif G, Figl M, Russ T, Kettenbach J, et al. Toward on-the-fly trajectory optimization for C-arm CBCT under strong kinematic constraints. PLoS ONE., 2021;16(2): e0245508.
14. Ritschl L, Kuntz J, Fleischmann CH, Kachelrieß M. The rotate-plus-shift C-arm trajectory. Part I. Complete data with less than 180° rotation. Med. Phys. 2016; 43(5): 2295-2302.
15. Chityala R, Hoffmann KR, Bednarek DR, Rudin S. Region of interest (ROI) computed tomography. Proc SPIE Int Soc Opt Eng., 2004; 5368(2): 534-541.
16. Clackdoyle R, Defrise M. Tomographic reconstruction in the 21st century. Region-of-interest reconstruction from incomplete data. IEEE Signal Processing. 2010;60:60–80.