Science in China Series A: Mathematics

, Volume 46, Issue 3, pp 312–325 | Cite as

On the regularity of trust region-cg algorithm: With application to deconvolution problem

  • Yanfei Wang


Deconvolution problem is a main topic in signal processing. Many practical applications are required to solve deconvolution problems. An important example is image reconstruction. Usually, researchers like to use regularization method to deal with this problem. But the cost of computation is high due to the fact that direct methods are used. This paper develops a trust region-cg method, a kind of iterative methods to solve this kind of problem. The regularity of the method is proved. Based on the special structure of the discrete matrix, FFT can be used for calculation. Hence combining trust region-cg method with FFT is suitable for solving large scale problems in signal processing.


ill- posed problems efficient implementation trust region- cg method regularity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Guan Zhaozhi, Xu Wenyuan, Jia Peizhang et al., Signal Processing and Analysis (in Chinese), Beijing: Science Press, 1986.Google Scholar
  2. 2.
    Roggemann, M., Welsh, B., Roggemann, M. C. et al., Imaging through turbulence, Boca Raton, Florida: CRC Press, 1996.Google Scholar
  3. 3.
    Vogel, C. R., Computational methods for inverse problems, SIAM Frontiers in Applied Mathematics Series, No. 23, Philadelphia: SIAM, 2002.Google Scholar
  4. 4.
    Michael, K. Ng., Plemmons, R. J., Qiao, S. Z. et al., Regularized blind deconvolution using recursive inverse filtering, in Proc. HK97 Conference on Scientific Computation (eds. Golub, G., Liu, S., Luk, F. et al.), Berlin: Springer-Verlag, 1997.Google Scholar
  5. 5.
    Wang Yanfei, Yuan Yaxiang, Zhang Hongchao et al., A trust region-CG algorithm for deblurring problem in atmospheric image reconstruction, Scicence in China, Ser. A, 2002, 45: 731–740.MATHMathSciNetGoogle Scholar
  6. 6.
    Yuan Yaxiang, On the truncated conjugate gradient method, Math. Prog., 2000, 87: 561–571.MATHCrossRefGoogle Scholar
  7. 7.
    Yuan Yaxiang, Problems on convergence of unconstrained optimization algorithms, in Numerical Linear Algebra and Optimization (ed. Yuan, Y. X.), Beijing/New York: Science Press, 1999, 95–107.Google Scholar
  8. 8.
    Kaltenbacher, B., Some Newton-type methods for the regularization of nonlinear ill-posed problems, Inverse Problems, 1997, 13: 729–753.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kaltenbacher, B., On Broyden’s method for nonlinear ill-posed problems, Numer. Funct. Anal. Opt., 1998, 19.Google Scholar
  10. 10.
    Tikhonov, A. N., Arsenin, V. Y., Tikhonov, A. N. et al., Solutions of ill-posed problems, New York: Wiley, 1977.MATHGoogle Scholar
  11. 11.
    Engl, H. W., Hanke, M., Neubauer, A. et al., Regularization of inverse problems, Dordrecht: Kluwer Academic, 1996.MATHGoogle Scholar
  12. 12.
    Powell, M. J. D., Convergence properties of a class of minimization algorithms, in Nonlinear Programming (eds. Mangasarian, O. L., Meyer, R. R., Robinson, S. M.), New York: Academic Press, 1975, 2: 1–27.Google Scholar
  13. 13.
    Toint, Ph. L., Towards an efficient sparsity exploiting Newton method for minimization, in Sparse Matrices and Their Uses (ed. Duff, I.), Berlin: Academic Press, 1981, 57–88.Google Scholar
  14. 14.
    Steihaug, T., The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal., 1983, 20: 626–637.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hanke, M., Iterative regularization techniques in image reconstruction, Proceedings of the Conference Mathematical Methods in Inverse Problems for Partial Differential Equations, Mt. Holyoke: Springer-Verlag, 1998.Google Scholar
  16. 16.
    Vogel, C. R., Oman, M. E. et al., Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Transactions on Image Processing, 1998, 7: 813–824.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science in China Press 2003

Authors and Affiliations

  1. 1.Laboratory of Remote Sensing Information Sciences,Institute of Remote Sensing ApplicationsChinese Academy of SciencesBeijingChina
  2. 2.State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingChina

Personalised recommendations