Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Constrained TV\(_p\)-\(\ell _2\) Model for Image Restoration

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The popular total variation (TV) model for image restoration (Rudin et al. in Phys D 60(1–4):259-268, 1992) can be formulated as a Maximum A Posteriori estimator which uses a half-Laplacian image-independent prior favoring sparse image gradients. We propose a generalization of the TV prior, referred to as TV\(_p\), based on a half-generalized Gaussian distribution with shape parameter p. An automatic estimation of p is introduced so that the prior better fits the real images’ gradient distribution; we will show that, in general, the estimated p value does not necessarily require to be close to zero. The restored image is computed by using an alternating directions methods of multipliers procedure. In this context, a novel result in multivariate proximal calculus is presented which allows for the efficient solution of the proposed model. Numerical examples show that the proposed approach is particularly efficient and well suited for images characterized by a wide range of gradient distributions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Buades, A., Coll, B., Morel, J.M.: The staircasing effect in neighborhood filters and its solution. IEEE Trans. Image Process. 15, 1499–1505 (2006)

    Article  Google Scholar 

  2. 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), 1–22 (2011)

    Article  MATH  Google Scholar 

  3. Chan, R.H., Tao, M., Yuan, X.M.: Constrained total variational deblurring models and fast algorithms based on alternating direction method of multipliers. SIAM J. Imaging Sci. 6, 680–697 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chan, R.H., Lanza, A., Morigi, S., Sgallari, F.: An adaptive strategy for the restoration of textured images using fractional order regularization. Numer. Math. Theory Methods Appl. (NMTMA) 6(1), 276–296 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Chan, T., Esedoglu, S., Park, F., Yip, A.: Total variation image restoration. Overview and recent developments. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Handbook of Mathematical Models in Computer Vision, pp. 17–31. Springer, New York (2006)

    Chapter  Google Scholar 

  6. Cho, T.S., Zitnick, C.L., Joshi, N., Kang, S.B., Szeliski, R., Freeman, W.T.: Image restoration by matching gradient distributions. IEEE Trans. Pattern Anal. Mach. Intell. 34/4, 683–694 (2012)

    Google Scholar 

  7. Christiansen, M., Hanke, M.: Deblurring methods using antireflective boundary conditions. SIAM J. Sci. Comput. 30, 855–872 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)

    Book  MATH  Google Scholar 

  9. Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984)

    Article  MATH  Google Scholar 

  10. Gray, R.M., Davisson, L.D.: An Introduction to Statistical Signal Processing. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  11. Hong, M., Luo, Z., Razaviyayn, M.: Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems (2014). Preprint, arXiv:1410.1390

  12. He, B., Yuan, X.: On the O(1/n) convergence rate of the Douglas–Rachford alternating direction method. SIAM J. Numer. Anal. 50(2), 700–709 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Keren, D., Werman, M.: Probabilistic analysis of regularization. IEEE Trans. Pattern Anal. Mach. Intell. 15(10), 982–995 (1993)

    Article  Google Scholar 

  14. Krishnan, D., Fergus, R.: Fast image deconvolution using hyper-Laplacian priors. In: Bengio, Y., Schuurmans, D., Lafferty, J.D., Williams, C.K.I., Culotta, A. (eds.) Advances in Neural Information Processing Systems 22, pp. 1033–1041 (2009)

  15. Kunisch, K., Pock, T.: A bilevel optimization approach for parameter learning in variational models. SIAM J. Imaging Sci. 6(2), 938–983 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, R.W., Xu, T.: A robust alternating direction method for constrained hybrid variational deblurring model. arXiv:1309.0123v2 (2013)

  17. Nikolova, M., Ng, M.K., Zhang, S., Ching, W.: Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization. SIAM J. Imaging Sci. 1(1), 2–25 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nikolova, M., Ng, M., Tam, C.: Software is available at http://www.math.hkbu.edu.hk/~mng/imaging-software.html

  19. Nikolova, M., Ng, M.K., Tam, C.-P.: Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. Trans. Imaging Proc. 19(12), 3073–3088 (2010)

    Article  MathSciNet  Google Scholar 

  20. Ng, M.K., Chan, R.H., Tang, W.C.: A fast algorithm for deblurring models with Neumann boundary conditions. SIAM J. Sci. Comput. 21, 851–866 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rodriguez, P., Wohlberg, B.: Efficient minimization method for a generalized total variation functional. IEEE Trans. Image Process. 18, 2(322-332) (2009)

    Article  MathSciNet  Google Scholar 

  22. Rodrguez, P.: Multiplicative updates algorithm to minimize the generalized total variation functional with a non-negativity constraint. In: Proceedings of the IEEE international conference on image processing (ICIP), (Hong Kong), pp. 2509–2512 (2010)

  23. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Saquib, S.S., Bouman, C.A., Sauer, K.: ML parameter estimation for Markov random fields with applications to Bayesian tomography. IEEE Trans. Image Process. 7(7), 1029–1044 (1998)

    Article  Google Scholar 

  25. Sha, F., Lin, Y., Saul, L., Lee, D.: Multiplicative updates for nonnegative quadratic programming. Neural Comput. 19(8), 2004–2031 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sidky, E.Y., Chartrand, R., Boone, J.M., Pan, X.: Constrained \(T_p\) V minimization for enhanced exploitation of gradient sparsity: application to CT image reconstruction. IEEE J. Transl. Eng. Health Med. 2, 1–18 (2014). doi:10.1109/JTEHM.2014.2300862

    Article  Google Scholar 

  27. Song, K.-S.: A globally convergent and consistent method for estimating the shape parameter of a generalized gaussian distribution. IEEE Trans. Inf. Theory 52(2), 510–527 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Strong, D., Chan, T.: Edge-preserving and scale-dependent properties of total variation regularization. Inverse Probl. 19, 165–187 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tao, M., Yang, J.: Alternating direction algorithm for total variation deconvolution in image reconstruction. Department of Mathematics, Nanjing University, Tech. Rep. TR0918 (2009)

  30. Varanasi, M., Aazhang, B.: Parametric generalized Gaussian density estimation. J. Acoust. Soc. Am. 86(4), 1404–1414 (1989)

    Article  Google Scholar 

  31. Vogel, C., Oman, M.: Iterative methods for total variation denoising. SIAM J. Sci. Comput. 17(1–4), 227–238 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wen, Y., Chan, R.H.: Parameter selection for total variation based image restoration using discrepancy principle. IEEE Trans. Image Process. 21(4), 1770–1781 (2012)

    Article  MathSciNet  Google Scholar 

  33. Yan, J., Lu, W.-S.: Image denoising by generalized total variation regularization and least squares fidelity. J. Multidimens. Syst. Signal Process. 26(1), 243–266 (2015)

    Article  MathSciNet  Google Scholar 

  34. Yu, S., Zhang, A., Li, H.: A review of estimating the shape parameter of generalized Gaussian distribution. J. Comput. Inf. Syst. 8(21), 9055–9064 (2012)

    Google Scholar 

  35. Zhu, M., Chan, T.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration. UCLA CAM Report 08-34, (2007)

  36. Zuo, W., Meng, D., Zhang, L., Feng, X., Zhang, D.: A generalized iterated shrinkage algorithm for non-convex sparse coding. In: IEEE international conference on computer vision (ICCV), pp. 217–224 (2013)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bologna, Bologna, Italy

    Alessandro Lanza, Serena Morigi & Fiorella Sgallari

Authors
  1. Alessandro Lanza
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Serena Morigi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fiorella Sgallari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Serena Morigi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lanza, A., Morigi, S. & Sgallari, F. Constrained TV\(_p\)-\(\ell _2\) Model for Image Restoration. J Sci Comput 68, 64–91 (2016). https://doi.org/10.1007/s10915-015-0129-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0129-x

Keywords

Search

Navigation