Journal of Mathematical Imaging and Vision

, Volume 54, Issue 1, pp 106–116 | Cite as

On the Application of the Spectral Projected Gradient Method in Image Segmentation

  • Laura Antonelli
  • Valentina De Simone
  • Daniela di Serafino


We investigate the application of the nonmonotone spectral projected gradient (SPG) method to a region-based variational model for image segmentation. We consider a “discretize-then-optimize” approach and solve the resulting nonlinear optimization problem by an alternating minimization procedure that exploits the SPG2 algorithm by Birgin et al. (SIAM J Optim 10(4):1196–1211, 2000). We provide a convergence analysis and perform numerical experiments on several images, showing the effectiveness of this procedure.


Image segmentation Region-based variational model  Spectral projected gradient 

Mathematics Subject Classification

68U10 65K05 90C30 



We wish to thank Giovanni Pisante for useful discussions concerning variational models for image segmentation. We are also grateful to the anonymous referees for their useful comments, which helped us to improve the quality of this work. This work was partially supported by INdAM-GNCS (2014 Project First-order optimization methods for image restoration and analysis; 2015 Project Numerical Methods for Non-convex/Nonsmooth Optimization and Applications) and by MIUR (FIRB 2010 Project No. RBFR106S1Z002).


  1. 1.
    Aujol, J.F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63(1), 85–104 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barzilai, J., Borwein, J.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bertsekas, D.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)MATHGoogle Scholar
  4. 4.
    Birgin, E., Martínez, J., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10(4), 1196–1211 (2000)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Birgin, E., Martínez, J., Raydan, M.: Algorithm 813: SPG—software for convex-constrained optimization. ACM Trans. Math. Softw. 27(3), 340–349 (2001)MATHCrossRefGoogle Scholar
  6. 6.
    Birgin, E., Martínez, J., Raydan, M.: Spectral projected gradient methods: review and perspectives. J. Stat. Softw. 60(3) (2014)Google Scholar
  7. 7.
    Bonettini, S.: Inexact block coordinate descent methods with application to non-negative matrix factorization. IMA J. Numer. Anal. 31(4), 1431–1452 (2011)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bonettini, S., Zanella, R., Zanni, L.: A scaled gradient projection method for constrained image deblurring. Inverse Probl. 25(1), 015002 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bresson, X., Esedo\(\bar{\rm {g}}\)lu, S., Vandergheynst, P., Thiran, J.P., Osher, S.: Fast global minimization of the active contour/snake model. J. Math. Imaging Vis. 28(2), 151–167 (2007)Google Scholar
  10. 10.
    Brown, E., Chan, T., Bresson, X.: Completely convex formulation of the Chan-Vese image segmentation model. Int. J. Comput. Vis. 98(1), 103–121 (2012)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004)MathSciNetGoogle Scholar
  12. 12.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)MATHCrossRefGoogle Scholar
  14. 14.
    Chan, T., Esedo\(\bar{\rm {g}}\)lu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)Google Scholar
  15. 15.
    Chan, T., Sandberg, B., Moelich, M.: Some recent developments in variational image segmentation. In: Tai, X.C., Lie, K.A., Chan, T., Osher, F. (eds.) Image Processing Based on Partial Differential Equations, Mathematics and Visualization Series, pp. 175–201. Springer, Heidelberg (2005)Google Scholar
  16. 16.
    Dai, Y.H., Fletcher, R.: Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming. Numerische Mathematik 100(1), 21–47 (2005)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Dai, Y.H., Yuan, Y.: Analysis of monotone gradient methods. J. Ind Manag. Optim. 1(2), 181–192 (2005)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    De Asmundis, R., di Serafino, D., Riccio, F., Toraldo, G.: On spectral properties of steepest descent methods. IMA J. Numer. Anal. 33(4), 1416–1435 (2013)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    De Asmundis, R., di Serafino, D., Hager, W., Toraldo, G., Zhang, H.: An efficient gradient method using the Yuan steplength. Comput. Optim. Appl. 59(3), 541–563 (2014)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Esser, E.: Applications of Lagrangian–based alternating direction methods and connections to split Bregman. CAM Technical Report 09-31, UCLA, Los Angeles. (2009)
  21. 21.
    Figueiredo, M., Nowak, R., Wright, S.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1(4), 586–598 (2007)CrossRefGoogle Scholar
  22. 22.
    Fletcher, R.: A limited memory steepest descent method. Math. Program. Ser. A 135(1–2), 413–436 (2012)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Goldstein, T., Osher, S.: The split Bregman method for l1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Goldstein, T., Bresson, X., Osher, S.: Geometric applications of the split Bregman method: segmentation and surface reconstruction. J. Sci. Comput. 45(1–3), 272–293 (2010)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Num. Anal. 23(4), 707–716 (1986)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Jung, M., Kang, M., Kang, M.: Variational image segmentation models involving non-smooth data-fidelity terms. J. Sci. Comput. 59(2), 277–308 (2014)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 45(2), 577–685 (1989)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Papadakis, N., Yildizoǧlu, R., Aujol, J.F., Caselles, V.: High-dimension multilabel problems: convex or nonconvex relaxation? SIAM J. Imaging Sci. 6(4), 2603–2639 (2013)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Setzer, S.: Operator splittings, Bregman methods and frame shrinkage in image processing. Int. J. Comput. Vis. 92(3), 265–280 (2011)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1(3), 248–272 (2008)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Yildizoǧlu, R., Aujol, J.F., Papadakis, N.: Active contours without level sets. In: ICIP 2012—IEEE International Conference on Image Processing (Orlando, FL, Sept. 30–Oct. 3, 2012), pp. 2549–2552. IEEE (2012)Google Scholar
  32. 32.
    Yu, G., Qi, L., Dai, Y.H.: On nonmonotone Chambolle gradient projection algorithms for total variation image restoration. J. Math. Imaging Vis. 35(2), 143–154 (2005)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhu, M., Wright, S., Chan, T.: Duality-based algorithms for total-variation-regularized image restoration. Comput. Optim. Appl. 47(3), 377–400 (2010)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Laura Antonelli
    • 1
  • Valentina De Simone
    • 2
  • Daniela di Serafino
    • 2
  1. 1.Institute for High-Performance Computing and Networking (ICAR), CNRNaplesItaly
  2. 2.Department of Mathematics and PhysicsSecond University of NaplesCasertaItaly

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