Science in China Series A: Mathematics

, Volume 48, Issue 2, pp 169–184 | Cite as

A new trust region algorithm for image restoration

  • Zaiwen Wen
  • Yanfei Wang
Article
  • 42 Downloads

Abstract

The image restoration problems play an important role in remote sensing and astronomical image analysis. One common method for the recovery of a true image from corrupted or blurred image is the least squares error (LSE) method. But the LSE method is unstable in practical applications. A popular way to overcome instability is the Tikhonov regularization. However, difficulties will encounter when adjusting the so-called regularization parameter α. Moreover, how to truncate the iteration at appropriate steps is also challenging. In this paper we use the trust region method to deal with the image restoration problem, meanwhile, the trust region subproblem is solved by the truncated Lanczos method and the preconditioned truncated Lanczos method. We also develop a fast algorithm for evaluating the Kronecker matrix-vector product when the matrix is banded. The trust region method is very stable and robust, and it has the nice property of updating the trust region automatically. This releases us from tedious finding the regularization parameters and truncation levels. Some numerical tests on remotely sensed images are given to show that the trust region method is promising.

Keywords

trust region algorithm image restoration Lanczos method Kronecker matrix-vector product preconditioning 

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References

  1. 1.
    Roggemann, M., Welsh, B., Imaging Through Turbulence, Boca Raton: CRC Press, 1996.Google Scholar
  2. 2.
    Hanke, M., Iterative Regularization Techniques in Image Reconstruction, in Proceedings of the Conference Mathematical Methods in Inverse Problems for Partial Differential Equations, Mt. Holyoke: Springer-Verlag, 1998.Google Scholar
  3. 3.
    Vogel, C. R., Oman, M. E., Fast Robust Total Variation-based Reconstruction of Noisy, Blurred Images, IEEE Transactions on Image Processing, 1998, 7: 813–824.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Tikhonov, A. N., Arsenin, V. Y., Solutions of Ill-Posed Problems, New York: Wiley, 1977.MATHGoogle Scholar
  5. 5.
    Wang, Y. F., Yuan, Y. X., Zhang, H. C. et al., A Trust Region-CG Algorithm for Deblurring Problem in Atmospheric Image Reconstruction, Science in China, Ser. A, 2002, 45: 731–740.MATHMathSciNetGoogle Scholar
  6. 6.
    Wang, Y. F., On the Regularity of Trust Region-CG Algorithm: with Application to Image Deconvlution Problem, Science in China, Ser. A, 2003, 46: 312–325.Google Scholar
  7. 7.
    Conn, A. R., Gould, N., Toint, Ph. L. et al., Trust Region Methods, CGT publications, 1999.Google Scholar
  8. 8.
    Yuan, Y. X., Sun, W. Y., On the Theory and Methods for Optimization, Beijing: Science Press, 2001.Google Scholar
  9. 9.
    Steihaug, T., The Conjugate Gradient Method and Trust Regions in Large Scale Optimization, SIAM J. Numer. Anal., 1983, 20: 626–637.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    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
  11. 11.
    Gould, N., Lucidi, S., Roma, M. et al., Solving the trust region subproblem using the Lanczos method, SIAM Journal on Optimization, 1999, 9: 504–525.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chan, R. H., Michael, K. N. et al., Conjugate gradient methods for Toeplitz systems, SIAM Review, 1996, 38(3): 427–482.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science in China Press 2005

Authors and Affiliations

  • Zaiwen Wen
    • 1
  • Yanfei Wang
    • 2
  1. 1.State Key Laboratory of Scientific Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingChina
  2. 2.National Key Laboratory on Remote Sensing Science, Institute of Remote Sensing ApplicationsChinese Academy of SciencesBeijingChina

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