Advances in Computational Mathematics

, Volume 45, Issue 2, pp 589–610 | Cite as

On the discontinuity of images recovered by noncovex nonsmooth regularized isotropic models with box constraints

  • Chao Zeng
  • Chunlin WuEmail author


Nonconvex nonsmooth regularizations have exhibited the ability of restoring images with neat edges in many applications, which has been provided a mathematical explanation by analyzing the discontinuity of the local minimizers of the variational models. Since in many applications the pixel intensity values in digital images are restricted in a certain given range, box constraints are adopted to improve the restorations. A similar property of nonconvex nonsmooth regularization for box-constrained models has been proved in the literature. While many theoretical results are available for anisotropic models, we investigate the isotropic case. We establish similar theoretical results for isotropic nonconvex nonsmooth models with box constraints. Numerical experiments are presented to validate our theoretical results.


Box constraint Image restoration Nonconvex nonsmooth regularization 

Mathematics Subject Classification (2010)

49K30 49N45 49N60 90C26 94A08 94A12 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to thank the anonymous referees for the careful reading of the manuscript and providing valuable suggestions that helped improve this paper. This work is supported by Postdoctoral Science Foundation of China (2016M601248), National Natural Science Foundation of China (Grants 11301289, 11531013 and 11871035), Recruitment Program of Global Young Expert, and the Fundamental Research Funds for the Central Universities.


  1. 1.
    Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18 (11), 2419–2434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bergmann, R., Chan, R.H., Hielscher, R., Persch, J., Steidl, G.: Restoration of manifold-valued images by half-quadratic minimization. Inverse Probl. Imaging 10(2), 281–304 (2017)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bian, W., Chen, X.: Linearly constrained non-lipschitz optimization for image restoration. SIAM J. Imag. Sci. 8(4), 2294–2322 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bian, W., Chen, X.: Optimality and Complexity for Constrained Optimization Problems with Nonconvex Regularization. Mathematics of Operations Research (2017)Google Scholar
  5. 5.
    Bondy, J., Murty, U.: Graph theory (graduate texts in mathematics) (2008)Google Scholar
  6. 6.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    Chan, R.H., Tao, M., Yuan, X.: Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers. SIAM J. Imag. Sci. 6(1), 680–697 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, P.Y., Selesnick, I.W.: Group-sparse signal denoising: non-convex regularization, convex optimization. IEEE Trans. Signal Process. 62(13), 3464–3478 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, X., Ng, M.K., Zhang, C.: Non-Lipschitz-regularization and box constrained model for image restoration. IEEE Trans. Image Process. 21(12), 4709–4721 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, X., Zhou, W.: Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization. SIAM J. Imag. Sci. 3 (4), 765–790 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chouzenoux, E., Jezierska, A., Pesquet, J.C., Talbot, H.: A majorize-minimize subspace approach for 2 0 image regularization. SIAM J. Imag. Sci. 6(1), 563–591 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Han, Y., Wang, W.W., Feng, X.C.: A new fast multiphase image segmentation algorithm based on nonconvex regularizer. Pattern Recogn. 45(1), 363–372 (2012)CrossRefzbMATHGoogle Scholar
  13. 13.
    Hanke, M., Nagy, J.G., Vogel, C.: Quasi-Newton approach to nonnegative image restorations. Linear Algebra Appl. 316(1–3), 223–236 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hintermüller, M., Wu, T.: Nonconvex TVq -models in image restoration: analysis and a trust-region regularization based superlinearly convergent solver. SIAM J. Imag. Sci. 6(3), 1385–1415 (2013)CrossRefzbMATHGoogle Scholar
  15. 15.
    Jung, M., Kang, M.: Efficient nonsmooth nonconvex optimization for image restoration and segmentation. J. Sci. Comput. 62(2), 336–370 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Krishnan, D., Lin, P., Yip, A.M.: A primal-dual active-set method for non-negativity constrained total variation deblurring problems. IEEE Trans. Image Process. 16(11), 2766–2777 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Nagy, J.G.: Enforcing nonnegativity in image reconstruction algorithms. Proc Spie 4121, 182–190 (2000)CrossRefGoogle Scholar
  18. 18.
    Nikolova, M.: Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares. Multiscale Model. Simul. 4(3), 960–991 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nikolova, M., Ng, M.K., Tam, C.-P.: Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. IEEE Trans. Image Process. 19 (12), 3073–3088 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nikolova, M., Ng, M.K., Tam, C.P.: On 1 data fitting and concave regularization for image recovery. SIAM J. Scientific Comput. 35(1), A397–A430 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nikolova, M., Ng, M.K., Zhang, S., Ching, W.K.: Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization. SIAM J. Imag. Sci. 1(1), 2–25 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ochs, P., Chen, Y., Brox, T., Pock, T.: ipiano: Inertial proximal algorithm for non-convex optimization. SIAM J. Imag. Sci. 7(2), 1388–1419 (2014)CrossRefzbMATHGoogle Scholar
  23. 23.
    Ochs, P., Dosovitskiy, A., Brox, T., Pock, T.: On iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision. SIAM J. Imag. Sci. 8(1), 331–372 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Robini, M.C., Yang, F., Zhu, Y.: Inexact half-quadratic optimization for linear inverse problems. SIAM J. Imag. Sci. 11(2), 1078–1133 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Robini, M.C., Zhu, Y.: Generic half-quadratic optimization for image reconstruction. SIAM J. Imag. Sci. 8(3), 1752–1797 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena 60(1), 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Weinmann, A., Demaret, L., Storath, M.: Total variation regularization for manifold-valued data. SIAM J. Imag. Sci. 7(4), 2226–2257 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wu, C., Liu, Z.-F., Wen, S.: A general truncated regularization framework for contrast-preserving variational signal and image restoration: motivation and implementation. arXiv:1611.08817 (2016)
  29. 29.
    Wu, C., Tai, X.-C.: Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imag. Sci. 3(3), 300–339 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Yang, L., Pong, T.K., Chen, X.: Alternating direction method of multipliers for a class of nonconvex and nonsmooth problems with applications to background/foreground extraction. SIAM J. Imag. Sci. 10(1), 74–110 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zeng, C., Wu, C.: On the edge recovery property of noncovex nonsmooth regularization in image restoration. SIAM J. Numer. Anal. 56(2), 1168–1182 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematical SciencesNankai UniversityTianjinChina

Personalised recommendations