Journal of Mathematical Imaging and Vision

, Volume 29, Issue 1, pp 79–91 | Cite as

Minimization of a Detail-Preserving Regularization Functional for Impulse Noise Removal

  • Jian-Feng Cai
  • Raymond H. Chan
  • Carmine Di Fiore
Article

Abstract

Recently, a powerful two-phase method for restoring images corrupted with high level impulse noise has been developed. The main drawback of the method is the computational efficiency of the second phase which requires the minimization of a non-smooth objective functional. However, it was pointed out in (Chan et al. in Proc. ICIP 2005, pp. 125–128) that the non-smooth data-fitting term in the functional can be deleted since the restoration in the second phase is applied to noisy pixels only. In this paper, we study the analytic properties of the resulting new functional ℱ. We show that ℱ, which is defined in terms of edge-preserving potential functions φα, inherits many nice properties from φα, including the first and second order Lipschitz continuity, strong convexity, and positive definiteness of its Hessian. Moreover, we use these results to establish the convergence of optimization methods applied to ℱ. In particular, we prove the global convergence of some conjugate gradient-type methods and of a recently proposed low complexity quasi-Newton algorithm. Numerical experiments are given to illustrate the convergence and efficiency of the two methods.

Keywords

Image Processing Variational Method Optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Astola, J., Kuosmanen, P.: Fundamentals of Nonlinear Digital Filtering. CRC, Boca Raton (1997) Google Scholar
  2. 2.
    Black, M., Rangarajan, A.: On the unification of line processes, outlier rejection, and robust statistics with applications to early vision. Int. J. Comput. Vis. 19, 57–91 (1996) CrossRefGoogle Scholar
  3. 3.
    Bortoletti, A., Di Fiore, C., Fanelli, S., Zellini, P.: A new class of quasi-Newtonian methods for optimal learning in MLP-networks. IEEE Trans. Neural Netw. 14, 263–273 (2003) CrossRefGoogle Scholar
  4. 4.
    Bouman, C., Sauer, K.: On discontinuity-adaptive smoothness priors in computer vision. IEEE Trans. Pattern Anal. Mach. Intell. 17, 576–586 (1995) CrossRefGoogle Scholar
  5. 5.
    Bovik, A.: Handbook of Image and Video Processing. Academic Press, San Diego (2000) MATHGoogle Scholar
  6. 6.
    Cai, J.F., Chan, R.H., Morini, B.: Minimization of an edge-preserving regularization functional by conjugate gradient type methods. In: Image Processing Based on Partial Differential Equations: Proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8–12, 2005, pp. 109–122. Springer, Berlin (2007) CrossRefGoogle Scholar
  7. 7.
    Chan, R.H., Ho, C.-W., Nikolova, M.: Convergence of Newton’s method for a minimization problem in impulse noise removal. J. Comput. Math. 22, 168–177 (2004) MATHMathSciNetGoogle Scholar
  8. 8.
    Chan, R.H., Ho, C.W., Nikolova, M.: Salt-and-pepper noise removal by median-type noise detector and edge-preserving regularization. IEEE Trans. Image Process. 14, 1479–1485 (2005) CrossRefGoogle Scholar
  9. 9.
    Chan, R.H., Ho, C.W., Leung, C.Y., Nikolova, M.: Minimization of detail-preserving regularization functional by Newton’s method with continuation. In: Proceedings of IEEE International Conference on Image Processing, pp. 125–128. Genova, Italy (2005) Google Scholar
  10. 10.
    Chan, R.H., Hu, C., Nikolova, M.: An iterative procedure for removing random-valued impulse noise. IEEE Signal Process. Lett. 11, 921–924 (2004) CrossRefGoogle Scholar
  11. 11.
    Charbonnier, P., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Deterministic edge-preserving regularization in computed imaging. IEEE Trans. Image Process. 6, 298–311 (1997) CrossRefGoogle Scholar
  12. 12.
    Chen, T., Wu, H.R.: Space variant median filters for the restoration of impulse noise corrupted images. IEEE Trans. Circuits Syst. II 48, 784–789 (2001) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs (1983) MATHGoogle Scholar
  14. 14.
    Dong, Y., Chan, R., Xu, S.: A detection statistic for random-valued impulse noise. IEEE Trans. Image Process. 16, 1112–1120 (2007) CrossRefGoogle Scholar
  15. 15.
    Di Fiore, C., Fanelli, S., Lepore, F., Zellini, P.: Matrix algebras in quasi-Newton methods for unconstrained minimization. Numer. Math. 94, 479–500 (2003) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Di Fiore, C., Lepore, F., Zellini, P.: Hartley-type algebras in displacement and optimization strategies. Linear Algebra Appl. 366, 215–232 (2003) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Green, P.J.: Bayesian reconstructions from emission tomography data using a modified EM algorithm. IEEE Trans. Med. Imag. MI-9, 84–93 (1990) CrossRefGoogle Scholar
  18. 18.
    Huang, T.S., Yang, G.J., Tang, G.Y.: Fast two-dimensional median filtering algorithm. IEEE Trans. Acoust. Speech Signal Process. 1, 13–18 (1979) CrossRefGoogle Scholar
  19. 19.
    Hwang, H., Haddad, R.A.: Adaptive median filters: new algorithms and results. IEEE Trans. Image Process. 4, 499–502 (1995) CrossRefGoogle Scholar
  20. 20.
    Leung, S., Osher, S.: Global minimization of the active contour model with TV-inpainting and two-phase denoising. In: Proceeding of the 3rd IEEE workshop on Variational, Geometric and Level Set Methods in Computer Vision, pp. 149–160, 2005 Google Scholar
  21. 21.
    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 (2000) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20, 99–120 (2004) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (1999) MATHGoogle Scholar
  24. 24.
    Nodes, T.A., Gallagher, N.C. Jr.: The output distribution of median type filters. IEEE Trans. Commun. 32, 532–541 (1984) CrossRefGoogle Scholar
  25. 25.
    Pok, G., Liu, J.-C., Nair, A.S.: Selective removal of impulse noise based on homogeneity level information. IEEE Trans. Image Process. 12, 85–92 (2003) CrossRefGoogle Scholar
  26. 26.
    Sun, J., Zhang, J.: Global convergence of conjugate gradient methods without line search. Ann. Oper. Res. 103, 161–173 (2001) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Vogel, C.R., Oman, M.E.: Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE Trans. Image Process. 7, 813–824 (1998) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Jian-Feng Cai
    • 1
  • Raymond H. Chan
    • 1
  • Carmine Di Fiore
    • 2
  1. 1.Department of MathematicsThe Chinese University of Hong KongShatinHong Kong
  2. 2.Department of MathematicsUniversity of Roma “Tor Vergata”RomaItaly
  3. 3.Temasek Laboratories and Department of MathematicsNational University of SingaporeSingaporeSingapore

Personalised recommendations