Abstract
We present an approach to binary tomography by variational reconstruction and difference-of-convex-functions (DC) programming. Because we use a standard functional comprising a reconstruction error and a smoothness prior, the integer conditions are relaxed to convex box constraints. Complementing the functional with a concave penalty term allows a gradual enforcement of binary solutions. A DC-programming approach leads to an iterative reconstruction algorithm that is also applicable to large-scale problems. We show that hidden parameters, which model uncertainties of the imaging process, can be estimated as part of the variational reconstruction. Besides presenting a concise overview over recent results, we also include novel results concerning the optimization performance of our approach.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Baumeister, J.: Stable Solution of Inverse Problems. F. Vieweg & Sohn, Braunschweig/Wiesbaden, Germany (1987).
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SI AM Review, 38, 367–426 (1996).
Beck, A., Teboulle, M.: Global optimality conditions for quadratic optimization problems with binary constraints. SIAM J. Optimiz 11, 179–188 (2000).
Birgin, E.G., Martinez, J.M., Raydan, M.: Algorithm 813: SPG-software for convex-constrained optimization. ACM Trans. Math. Softw., 27, 340–349 (2001).
Censor, Y., Zenios, S.A.: Parallel Optimization: Theory, Algorithms, and Applications. Oxford Univ. Press, New York, NY (1998).
Demoment, G.: Image reconstruction and restoration: Overview of common estimation structures and problems. IEEE Trans. Acoustics, Speech, Signal Proc., 37, 2024–2036 (1989).
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht, The Netherlands (1996).
Floudas, C.A., Visweswaran, V.: Quadratic optimization. In: Horst, R., Pardalos, R.M. (eds.): Handbook of Global Optimization, Kluwer Acad. Publ., Dordrecht, The Netherlands, pp. 217–269 (1995).
Giannessi, F., Niccolucci, F.: Connections between nonlinear and integer programming problems. In Symposia Mathematica, Vol. 19, Academic Press, Orlando, FL, pp. 161–176 (1976).
Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd ed., Johns Hopkins Univ. Press, Baltimore, MD (1997).
Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography. J. Theor. Biol. 29, 471–481 (1970).
Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. Springer, Berlin, Germany (1993).
Herman, G.T.: Mathematical Methods in Tomography. Springer, Berlin, Germany (1992).
Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms, and Applications. Birkhauser, Boston, MA (1999).
Herman, G.T., Kuba, A.: Discrete tomography in medical imaging. Proc. IEEE , 91, 1612–1626 (2003).
Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd ed., Springer, Berlin, Germany (1996).
Kaczmarz, S.: Angenaherte Auflosung von Systemen linearer Gleichungen. Bull. Acad. Polon. Sci. et Let. A, pp. 355–357 (1937).
McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions. Wiley, New York, NY (1996).
Pham Dinh, T., Elbernoussi, S.: Duality in d.c. (difference of convex functions) optimization subgradient methods. In: Hoffmann, K.-H., Hiriart-Urruty, J.B., Lemarechal, C., Zowe, J. (eds.), Trends in Mathematical Optimization, Birkauser, Basel, Switzerland, pp. 277–293 (1988).
Pham Dinh, T., Hoai An, L.T.: A D.C. optimization algorithm for solving the trust-region subproblem. SIAM J. Optim., 8, 476–505 (1998).
Popa, C., Zdunek, R.: Kaczmarz extended algorithm for tomographic image reconstruction from limited data. Math. Comput. Simul., 65, 579–598 (2004).
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin, Germany (1998).
Schiile, T., Schnorr, C., Weber, S., Hornegger, J.: Discrete tomography by convex-concave regularization and D.C. programming. Discr. Appl. Math., 151, 229–243 (2005).
Schiile, T., Weber, S., Schnorr, C.: Adaptive reconstruction of discrete-valued objects from few projections. Electr. Notes Discr. Math., 20, 365–384 (2005).
Vogel, C.R.: Computational Methods for Inverse Problems. SIAM, Philadelphia, PA (2002).
S. Weber, Schiile, T., Schnorr, C., Kuba, A.: Binary tomography with deblurring.In: Reulke, R., Eckardt, U., Flach, B., Knauer, U., Polthier, K. (eds.),Combinatorial Image Analysis. Springer, Berlin, Germany, pp. 375–388 (2006).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Birkhäuser Boston
About this chapter
Cite this chapter
Schnörr, C., Schüle, T., Weber, S. (2007). Variational Reconstruction with DC-Programming. In: Herman, G.T., Kuba, A. (eds) Advances in Discrete Tomography and Its Applications. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4543-4_11
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4543-4_11
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-3614-2
Online ISBN: 978-0-8176-4543-4
eBook Packages: EngineeringEngineering (R0)