Prior learning and convex-concave regularization of binary tomography. (English) Zbl 1179.68192

Herman, Gabor T. (ed.) et al., Proceedings of the workshop on discrete tomography and its applictions, New York, NY, USA, June 13–15, 2005. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 20, 313-327 (2005).
Summary: In our previous work, we introduced a convex-concave regularization approach to the reconstruction of binary objects from few projections within a limited range of angles. A convex reconstruction functional, comprising the projections equations and a smoothness prior, was complemented with a concave penalty term enforcing binary solutions. In the present work we investigate alternatives to the smoothness prior in terms of probabilistically learnt priors encoding local object structure. We show that the difference-of-convex-functions DC-programming framework is flexible enough to cope with this more general model class. Numerical results show that reconstruction becomes feasible under conditions where our previous approach fails.
For the entire collection see [Zbl 1109.65003].


68U10 Computing methodologies for image processing
15B36 Matrices of integers
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
Full Text: DOI


[1] Bertsekas, D., On the goldstein-levitin-Polyak gradient projection method, IEEE transactions on automatic control, 21, 174-184, (1976) · Zbl 0326.49025
[2] Besag, J., On the analysis of dirty pictures (with discussion), J.R. statist. soc., series B, 48, 259-302, (1986) · Zbl 0609.62150
[3] Borges, C., On the estimation of Markov random field parameters, IEEE trans. on pattern analysis and machine intelligence, 21, (1999)
[4] Chan, M.; Herman, G.; Levitan, E., Bayesian image reconstruction using image-modeling Gibbs priors, Int. J. imag. syst. technol., 9, 85-98, (1998)
[5] Dinh, T.P.; An, L.H., A d.c. optimization algorithm for solving the trust- region subproblem, SIAM J. optim., 8, 476-505, (1998) · Zbl 0913.65054
[6] Dinh, T.P.; Elbernoussi, S., Duality in d.c. (difference of convex functions) optimization subgradient methods, (), 277-293
[7] Gardner, R.; Gritzmann, P., Discrete tomography: determination of finite sets by x-rays, Trans. amer. math. soc., 349, 2271-2295, (1997) · Zbl 0873.52015
[8] Geman, D., Random fields and inverse problems in imaging, (), 113-193 · Zbl 0718.60119
[9] Geman, S.; Geman, D., Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE trans. patt. anal. Mach. intell., 6, 721-741, (1984) · Zbl 0573.62030
[10] Gritzmann, P.; Prangenberg, D.; de Vries, S.; Wiegelmann, M., Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography, Int. J. imag. syst. technol., 9, 101-109, (1998)
[11] ()
[12] Kuba, A.; Herman, G., Discrete tomography: A historical overview, (), 3-34 · Zbl 0959.92014
[13] Li, S., Markov random field modeling in image analysis, (2001), Springer Verlag Tokyo · Zbl 0978.68130
[14] Liao, H.Y.; Herman, G.T., Automated estimation of the parameters of Gibbs priors to be used in binary tomography, Discrete appl. math., 139, 149-170, (2004) · Zbl 1072.68634
[15] Matej, S.; Herman, G.; Vardi, A., Binary tomography on the hexagonal grid using Gibbs priors, Int. J. imag. syst. technol., 9, 126-131, (1998)
[16] Natterer, F.; Wübbeling, F., Mathematical methods in image reconstruction, (2001), SIAM Philadelphia · Zbl 0974.92016
[17] Rockafellar, R., Convex analysis, (1972), Princeton Univ. Press Princeton, NJ · Zbl 0224.49003
[18] Schüle, T., C. Schnörr, S. Weber and J. Hornegger, Discrete tomography by convex-concave regularization and d.c. programming, Technical report 15/2003, University of Mannheim (2003), to appear in Discrete Applied Mathematics
[19] Weber, S.; Schüle, T.; Schnörr, C.; Hornegger, J., A linear programming approach to limited angle 3d reconstruction from dsa projections, Special issue of methods of information in medicine, 4, 320-326, (2004)
[20] Winkler, G., Image analysis, random fields and dynamic Monte Carlo methods, Appl. of mathematics, 27, (1995), Springer-Verlag Heidelberg · Zbl 0821.68125
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.