Derivative-free superiorization with component-wise perturbations

Original Paper
  • 24 Downloads

Abstract

Superiorization reduces, not necessarily minimizes, the value of a target function while seeking constraints compatibility. This is done by taking a solely feasibility-seeking algorithm, analyzing its perturbation resilience, and proactively perturbing its iterates accordingly to steer them toward a feasible point with reduced value of the target function. When the perturbation steps are computationally efficient, this enables generation of a superior result with essentially the same computational cost as that of the original feasibility-seeking algorithm. In this work, we refine previous formulations of the superiorization method to create a more general framework, enabling target function reduction steps that do not require partial derivatives of the target function. In perturbations that use partial derivatives, the step-sizes in the perturbation phase of the superiorization method are chosen independently from the choice of the nonascent directions. This is no longer true when component-wise perturbations are employed. In that case, the step-sizes must be linked to the choice of the nonascent direction in every step. Besides presenting and validating these notions, we give a computational demonstration of superiorization with component-wise perturbations for a problem of computerized tomography image reconstruction.

Keywords

Superiorization Derivative-free Component-wise perturbations Image reconstruction Feasibility-seeking Perturbation resilience 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We greatly appreciate the constructive comments of two anonymous reviewers which helped us improve the paper.

References

  1. 1.
    Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18, 2419–2434 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bian, J., Siewerdsen, J., Han, X., Sidky, E., Prince, J., Pelizzari, C., Pan, X.: Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam CT. Phys. Med. Biol. 55, 6575–6599 (2010)CrossRefGoogle Scholar
  3. 3.
    Butnariu, D., Davidi, R., Herman, G.T., Kazantsev, I.G.: Stable convergence behavior under summable perturbations of a class of projection methods for convex feasibility and optimization problems. IEEE J. Sel. Top. Sign. Proces. 1, 540–547 (2007)CrossRefGoogle Scholar
  4. 4.
    Censor, Y.: Superiorization and perturbation resilience of algorithms: a bibliography compiled and continuously updated. http://math.haifa.ac.il/yair/bib-superiorization-censor.html see also: arXiv:1506.04219
  5. 5.
    Censor, Y.: Weak and strong superiorization: Between feasibility-seeking and minimization. An. Stiint. ale Univ. Ovidius Constanta-Ser. Mat. 23, 41–54 (2015)MathSciNetMATHGoogle Scholar
  6. 6.
    Censor, Y.: Can linear superiorization be useful for linear optimization problems?. Inverse Prob. 33, 044006 (2017)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Censor, Y., Davidi, R., Herman, G.T.: Perturbation resilience and superiorization of iterative algorithms. Inverse Prob. 26, 065008 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Censor, Y., Davidi, R., Herman, G.T., Schulte, R.W., Tetruashvili, L.: Projected subgradient minimization versus superiorization. J. Optim. Theory Appl. 160, 730–747 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Censor, Y., Herman, G.T., Jiang, M. (eds.): Superiorization: theory and applications. Inverse Problems 33(4). Special Issue (2017)Google Scholar
  10. 10.
    Censor, Y., Zaslavski, A.: Strict Fejér monotonicity by superiorization of feasibility-seeking projection methods. J. Optim. Theory Appl. 165, 172–187 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chan, T., Esedoglu, S., Park, F., Yip, A.: Total variation image restoration: overview and recent developments. In: Handbook of Mathematical Models in Computer Vision, pp. 17–31. Springer Science+Business Media, Inc (2006)Google Scholar
  12. 12.
    Combettes, P., Luo, J.: An adaptive level set method for nondifferentiable constrained image recovery. IEEE Trans. Image Process. 11, 1295–1304 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Combettes, P., Pesquet, J.C.: Image restoration subject to a total variation constraint. IEEE Trans. Image Process. 13, 1213–1222 (2004)CrossRefGoogle Scholar
  14. 14.
    Davidi, R., Herman, G.T., Censor, Y.: Perturbation-resilient block-iterative projection methods with application to image reconstruction from projections. Int. Trans. Oper. Res. 16, 505–524 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Defrise, M., Vanhove, C., Liu, X.: An algorithm for total variation regularization in high-dimensional linear problems. Inverse Prob. 27, 065002 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Garduño, E., Herman, G.T.: Superiorization of the ML-EM algorithm. IEEE Trans. Nucl. Sci. 61, 162–172 (2014)CrossRefGoogle Scholar
  17. 17.
    Garduño, E., Herman, G.T.: Computerized tomography with total variation and with shearlets. Inverse Prob. 33, 044011 (2017)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Garduño, E., Herman, G.T., Davidi, R.: Reconstruction from a few projections by 1-minimization of the Haar transform. Inverse Prob. 27, 055006 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gibali, A., Petra, S.: DC-programming versus 0-superiorization for discrete tomography. Analele Stiintifice ale Universitatii Ovidius Constanta-Seria Matematica. Accepted for publication. Available on ResearchGate (2017)Google Scholar
  20. 20.
    Hansen, P.C., Saxild-Hansen, M.: AIR Tools–A MATLAB package of algebraic iterative reconstruction methods. J. Comput. Appl. Math. 236, 2167–2178 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Helou Neto, E., De Pierro, Á.: Incremental subgradients for constrained convex optimization: a unified framework and new methods. SIAM J. Optim. 20, 1547–1572 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Helou Neto, E., De Pierro, Á.: On perturbed steepest descent methods with inexact line search for bilevel convex optimization. Optimization 60, 991–1008 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Herman, G.T.: Fundamentals of Computerized Tomography, 2nd edn. Springer-Verlag, London (2009)Google Scholar
  24. 24.
    Herman, G.T., Garduño, E., Davidi, R., Censor, Y.: Superiorization: an optimization heuristic for medical physics. Med. Phys. 39, 5532–5546 (2012)CrossRefGoogle Scholar
  25. 25.
    Marquina, A., Osher, S.: Image super-resolution by TV-regularization and Bregman iteration. J. Sci. Comput. 37, 367–382 (2008)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    MATLAB: A high-level language and interactive environment system by Mathworks. http://www.mathworks.com/products/matlab
  27. 27.
    Needell, D., Ward, R.: Stable image reconstruction using total variation minimization. SIAM J. Imag. Sci. 6, 1035–1058 (2013)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Nikazad, T., Davidi, R., Herman, G.T.: Accelerated perturbation-resilient block-iterative projection methods with application to image reconstruction. Inverse Prob 28, 035005 (2012)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Nurminski, E.: Envelope stepsize control for iterative algorithms based on Fejer processes with attractants. Optim. Methods Soft. 25, 97–108 (2010)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rios, L., Sahinidis, N.: Derivative-free optimization: a review of algorithms and comparison of software implementations. J. Glob. Optim. 56, 1247–1293 (2013)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena 60, 259–268 (1992)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Shen, J., Chan, T.: Mathematical models for local nontexture inpaintings. SIAM J. Appl. Math. 62, 1019–1043 (2002)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Sidky, E., Duchin, Y., Pan, X., Ullberg, C.: A constrained, total-variation minimization algorithm for low-intensity x-ray CT. Med. Phys. 38, S117—S125 (2011)CrossRefGoogle Scholar
  34. 34.
    Sidky, E., Pan, X.: Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization. Phys. Med. Biol. 53, 4777–4807 (2008)CrossRefGoogle Scholar
  35. 35.
    Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imag. Sci. 1, 248–272 (2008)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Zhang, H.M., Wang, L.Y., Yan, B., Li, L., Xi, X.Q., Lu, L.Z.: Image reconstruction based on total-variation minimization and alternating direction method in linear scan computed tomography. Chin. Phys. B 22, 078701 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HaifaHaifaIsrael
  2. 2.Department of MathematicsUniversity of California Los AngelesLos AngelesUSA
  3. 3.Division of Biomedical Engineering Sciences, Department of Basic Sciences, School of MedicineLoma Linda UniversityLoma LindaUSA

Personalised recommendations