Journal of Mathematical Imaging and Vision

, Volume 27, Issue 3, pp 265–276 | Cite as

On Semismooth Newton’s Methods for Total Variation Minimization

  • Michael K. Ng
  • Liqun Qi
  • Yu-fei Yang
  • Yu-mei Huang


In [2], Chambolle proposed an algorithm for minimizing the total variation of an image. In this short note, based on the theory on semismooth operators, we study semismooth Newton’s methods for total variation minimization. The convergence and numerical results are also presented to show the effectiveness of the proposed algorithms.


semismooth Newton’s methods total variation denoising regularization 


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  1. 1.
    R. Acar and C.R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Problems, Vol. 10, pp. 1217–1229, 1994.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Chambolle, “An algorithm for total variation minimization and applications,” Journal of Mathematical Imaging and Vision, Vol. 20, pp. 89–97, 2004.CrossRefMathSciNetGoogle Scholar
  3. 3.
    T.F. Chan, G.H. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM J. Sci. Comput., Vol. 20, pp. 1964–1977, 1999.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley: New York, 1983.MATHGoogle Scholar
  5. 5.
    G.H. Golub and C. Van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, 1996.Google Scholar
  6. 6.
    H. Jiang and D. Ralph, “Global and local superlinear convergence analysis of Newton-type methods for semismooth equations with smooth least squares,” in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi (eds.), Kluwer Acad. Publ., Dordrecht, 1998, pp. 181–209.Google Scholar
  7. 7.
    G. Golub and C. Van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, 1996.Google Scholar
  8. 8.
    L. Qi, “Convergence analysis of some algorithms for solving nonsmooth equations,” Math. Oper. Res., Vol. 18, pp. 227–244, 1993.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    L. Qi and J. Sun, “A Nonsmooth version of Newton’s method,” Mathematical Programming, Vol. 58, pp. 353–367, 1993.CrossRefMathSciNetGoogle Scholar
  10. 10.
    L. Rudin, S. Osher and E. Fatemi, “Nonlinear total variation based noise removal algorithm,” Physica D, Vol. 60, pp. 259–268, 1992.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2007

Authors and Affiliations

  • Michael K. Ng
    • 1
  • Liqun Qi
    • 2
  • Yu-fei Yang
    • 3
  • Yu-mei Huang
    • 1
    • 4
  1. 1.Centre for Mathematical Imaging and Vision and Department of MathematicsHong Kong Baptist UniversityKowloon TongHong Kong
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHung HomHong Kong
  3. 3.College of Mathematics and EconometricsHunan UniversityChangshaChina
  4. 4.School of Information Science and EngineeringLanzhou UniversityLanzhouChina

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