Abstract
Tomography is concerned with the reconstruction of images from their projections. In this paper, we consider the reconstruction problem for a class of tomography problems, where the images are restricted to binary grey levels. For any given set of projections, we derive an upper bound on the difference between any two binary images having these projections, and a bound on the difference between a particular binary image and any binary image having the given projections. Both bounds are evaluated experimentally for different geometrical settings, based on simulated projection data for a range of images.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Alpers, A.: Instability and Stability in Discrete Tomography. Ph.D. thesis, Technische Universität München. Shaker Verlag, Aachen (2003) ISBN 3-8322-2355-X
Alpers, A., Brunetti, S.: Stability results for the reconstruction of binary pictures from two projections. Image and Vision Computing 25(10), 1599–1608 (2007)
Alpers, A., Gritzmann, P.: On stability, error correction, and noise compensation in discrete tomography. SIAM Journal on Discrete Mathematics 20(1), 227–239 (2006)
Ben-israel, A., Greville, T.N.E.: Generalized inverses: Theory and applications. Canadian Math. Soc. (2002)
Hajdu, L., Tijdeman, R.: Algebraic aspects of discrete tomography. J. Reine Angew. Math. 534, 119–128 (2001)
Herman, G.T.: Fundamentals of Computerized Tomography: Image reconstruction from projections. Springer, Heidelberg (2009)
Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms and Applications. Birkhäuser, Boston (1999)
Herman, G.T., Kuba, A. (eds.): Advances in Discrete Tomography and its Applications. Birkhäuser, Boston (2007)
Jinschek, J.R., Batenburg, K.J., Calderon, H.A., Kilaas, R., Radmilovic, V., Kisielowski, C.: 3-D reconstruction of the atomic positions in a simulated gold nanocrystal based on discrete tomography. Ultramicroscopy 108(6), 589–604 (2007)
Midgley, P.A., Dunin-Borkowski, R.E.: Electron tomography and holography in materials science. Nature Materials 8(4), 271–280 (2009)
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)
Van der Sluis, A., Van der Vorst, H.A.: SIRT and CG-type methods for the iterative solution of sparse linear least-squares problems. Linear Algebra Appl. 130, 257–302 (1990)
Van Dalen, B.: On the difference between solutions of discrete tomography problems. Journal of Combinatorics and Number Theory 1, 15–29 (2009)
Van Dalen, B.: On the difference between solutions of discrete tomography problems II. Pure Mathematics and Applications 20, 103–112 (2009)
Van Dalen, B.: Stability results for uniquely determined sets from two directions in discrete tomography. Discrete Mathematics 309, 3905–3916 (2009)
Zhua, J., Li, X., Ye, Y., Wang, G.: Analysis on the strip-based projection model for discrete tomography. Discrete Appl. Math. 156(12), 2359–2367 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Batenburg, K.J., Fortes, W., Hajdu, L., Tijdeman, R. (2011). Bounds on the Difference between Reconstructions in Binary Tomography. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds) Discrete Geometry for Computer Imagery. DGCI 2011. Lecture Notes in Computer Science, vol 6607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19867-0_31
Download citation
DOI: https://doi.org/10.1007/978-3-642-19867-0_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-19866-3
Online ISBN: 978-3-642-19867-0
eBook Packages: Computer ScienceComputer Science (R0)