Abstract
The paper introduces a new tomography reconstruction approach for gray and binary image reconstruction. The proposed method intends to find a solution by searching for the best linear combination of the basis vectors of the null space of the projection matrix. One of the advantages of the proposed approach is that the projection error remains always extremely low, practically equal to zero, during the reconstruction process. The method applies a gradient based optimization algorithm. A short experimental evaluation, including three relevant and well-know algorithms for comparison, is presented.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Batenburg, K.J., Sijbers, J.: DART: a practical reconstruction algorithm for discrete tomography. IEEE Trans. Image Process. 20, 2542–2553 (2011)
Batenburg, K.J., Sijbers, J.: DART: a fast heuristic algebraic reconstruction algorithm for discrete tomography. In: Proceedings of International Conference on Image Processing (ICIP), pp. 133–136 (2007)
Birgin, E., Martínez, J.: Spectral conjugate gradient method for unconstrained optimization. Appl. Math. Optim. 43, 117–128 (2001)
Carmignato, S., Dewulf, W., Leach, R.: Industrial X-Ray Computed Tomography. Springer, Heidelberg (2018). https://doi.org/10.1007/978-3-319-59573-3
Gilbert, P.: Iterative methods for the three-dimensional reconstruction of an object from projections. J. Theor. Biol. 36(1), 105–117 (1972). https://doi.org/10.1016/0022-5193(72)90180-4, https://www.sciencedirect.com/science/article/pii/0022519372901804
Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography. J. Theor. Biol. 29(3), 471–481 (1970). https://doi.org/10.1016/0022-5193(70)90109-8, https://www.sciencedirect.com/science/article/pii/0022519370901098
Herman, G.T.: Image Reconstruction from Projections. Springer, Heidelberg (1980)
Herman, G.T., Kuba, A.: Discrete Tomography: Foundations, Algorithms and Applications. Birkhäuser (1999)
Herman, G.T., Kuba, A.: Advances in Discrete Tomography and Its Applications. Birkhäuser (2007)
Herman, G.T., Kuba, A.: Discrete tomography: Foundations, Algorithms, and Applications. Springer, Heidelberg (2012)
Kisner, S.J.: image reconstruction for X-ray computed tomography in security screening applications. Ph.D. thesis, USA (2013)
Lukić, T.: Discrete tomography reconstruction based on the multi-well potential. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds.) IWCIA 2011. LNCS, vol. 6636, pp. 335–345. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21073-0_30
Lukić, T., Balázs, P.: Binary tomography reconstruction based on shape orientation. Pattern Recogn. Lett. 79, 18–24 (2016)
Lukić, T., Balázs, P.: Limited-view binary tomography reconstruction assisted by shape centroid. Vis. Comput. (Springer) 38, 695–705 (2022)
Lukić, T., Lukity, A.: A spectral projected gradient optimization for binary tomography. In: Rudas, I.J., Fodor, J., Kacprzyk, J. (eds.) Computational Intelligence in Engineering. SCI, vol. 313, pp. 263–272. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15220-7_21
Lukić, T., Nagy, B.: Deterministic discrete tomography reconstruction method for images on triangular grid. Pattern Recogn. Lett. 49, 11–16 (2014)
Lukić, T., Nagy, B.: Regularized binary tomography on the hexagonal grid. Phys. Scripta 94, 025201(9pp) (2019)
Lukić, T., Balázs, P.: Shape circularity assisted tomography reconstruction. Phys. Scripta 95(10), 105211 (2020). https://doi.org/10.1088/1402-4896/abb633
Nocedal, J., Wright, S.J.: Numerical Optimization, 2e edn. Springer, New York (2006). https://doi.org/10.1007/978-0-387-40065-5
Palenstijn, W.J., Bédorf, J., Sijbers, J., Batenburg, K.J.: A distributed ASTRA toolbox. Adv. Struct. Chem. Imaging 2(1), 1–13 (2016). https://doi.org/10.1186/s40679-016-0032-z
Schüle, T., Schnörr, C., Weber, S., Hornegger, J.: Discrete tomography by convex-concave regularization and D.C. programming. Discrete Appl. Math. 151, 229–243 (2005)
Weber, S., Nagy, A., Schüle, T., Schnörr, C., Kuba, A.: A benchmark evaluation of large-scale optimization approaches to binary tomography. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 146–156. Springer, Heidelberg (2006). https://doi.org/10.1007/11907350_13
Acknowledgement
Authors acknowledge the financial support of Department of Fundamental Sciences, Faculty of Technical Sciences, University of Novi Sad, in the frame of the Project “Primena opštih disciplina u tehničkim i informatičkim naukama”. T. Lukić also acknowledges support received from the Hungarian Academy of Sciences through the DOMUS project.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Lukić, T., Kopanja, T. (2023). Tomography Reconstruction Based on Null Space Search. In: Barneva, R.P., Brimkov, V.E., Nordo, G. (eds) Combinatorial Image Analysis. IWCIA 2022. Lecture Notes in Computer Science, vol 13348. Springer, Cham. https://doi.org/10.1007/978-3-031-23612-9_15
Download citation
DOI: https://doi.org/10.1007/978-3-031-23612-9_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-23611-2
Online ISBN: 978-3-031-23612-9
eBook Packages: Computer ScienceComputer Science (R0)