Variational Reconstruction with DC-Programming

  • C. Schnörr
  • T. Schüle
  • S. Weber
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


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.


Iterative Reconstruction Algorithm Quadratic Optimization Problem Binary Constraint Discrete Tomography Vessel System 
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.


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  1. 1.
    Baumeister, J.: Stable Solution of Inverse Problems. F. Vieweg & Sohn, Braunschweig/Wiesbaden, Germany (1987).MATHGoogle Scholar
  2. 2.
    Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SI AM Review, 38, 367–426 (1996).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beck, A., Teboulle, M.: Global optimality conditions for quadratic optimization problems with binary constraints. SIAM J. Optimiz 11, 179–188 (2000).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Birgin, E.G., Martinez, J.M., Raydan, M.: Algorithm 813: SPG-software for convex-constrained optimization. ACM Trans. Math. Softw., 27, 340–349 (2001).MATHCrossRefGoogle Scholar
  5. 5.
    Censor, Y., Zenios, S.A.: Parallel Optimization: Theory, Algorithms, and Applications. Oxford Univ. Press, New York, NY (1998).Google Scholar
  6. 6.
    Demoment, G.: Image reconstruction and restoration: Overview of common estimation structures and problems. IEEE Trans. Acoustics, Speech, Signal Proc., 37, 2024–2036 (1989).CrossRefGoogle Scholar
  7. 7.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht, The Netherlands (1996).MATHGoogle Scholar
  8. 8.
    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).Google Scholar
  9. 9.
    Giannessi, F., Niccolucci, F.: Connections between nonlinear and integer programming problems. In Symposia Mathematica, Vol. 19, Academic Press, Orlando, FL, pp. 161–176 (1976).Google Scholar
  10. 10.
    Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd ed., Johns Hopkins Univ. Press, Baltimore, MD (1997).Google Scholar
  11. 11.
    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).CrossRefGoogle Scholar
  12. 12.
    Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. Springer, Berlin, Germany (1993).Google Scholar
  13. 13.
    Herman, G.T.: Mathematical Methods in Tomography. Springer, Berlin, Germany (1992).Google Scholar
  14. 14.
    Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms, and Applications. Birkhauser, Boston, MA (1999).MATHGoogle Scholar
  15. 15.
    Herman, G.T., Kuba, A.: Discrete tomography in medical imaging. Proc. IEEE , 91, 1612–1626 (2003).CrossRefGoogle Scholar
  16. 16.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd ed., Springer, Berlin, Germany (1996).MATHGoogle Scholar
  17. 17.
    Kaczmarz, S.: Angenaherte Auflosung von Systemen linearer Gleichungen. Bull. Acad. Polon. Sci. et Let. A, pp. 355–357 (1937).Google Scholar
  18. 18.
    McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions. Wiley, New York, NY (1996).Google Scholar
  19. 19.
    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).Google Scholar
  20. 20.
    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).MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Popa, C., Zdunek, R.: Kaczmarz extended algorithm for tomographic image reconstruction from limited data. Math. Comput. Simul., 65, 579–598 (2004).CrossRefMathSciNetGoogle Scholar
  22. 22.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin, Germany (1998).MATHGoogle Scholar
  23. 23.
    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).CrossRefGoogle Scholar
  24. 24.
    Schiile, T., Weber, S., Schnorr, C.: Adaptive reconstruction of discrete-valued objects from few projections. Electr. Notes Discr. Math., 20, 365–384 (2005).CrossRefGoogle Scholar
  25. 25.
    Vogel, C.R.: Computational Methods for Inverse Problems. SIAM, Philadelphia, PA (2002).Google Scholar
  26. 26.
    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).CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • C. Schnörr
    • 1
  • T. Schüle
    • 1
  • S. Weber
    • 1
  1. 1.Dept. M&CS, CVGPR-GroupUniversity of MannheimMannheimGermany

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