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.
Similar content being viewed by others
References
Roggemann, M., Welsh, B., Imaging Through Turbulence, Boca Raton: CRC Press, 1996.
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.
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.
Tikhonov, A. N., Arsenin, V. Y., Solutions of Ill-Posed Problems, New York: Wiley, 1977.
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.
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.
Conn, A. R., Gould, N., Toint, Ph. L. et al., Trust Region Methods, CGT publications, 1999.
Yuan, Y. X., Sun, W. Y., On the Theory and Methods for Optimization, Beijing: Science Press, 2001.
Steihaug, T., The Conjugate Gradient Method and Trust Regions in Large Scale Optimization, SIAM J. Numer. Anal., 1983, 20: 626–637.
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.
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.
Chan, R. H., Michael, K. N. et al., Conjugate gradient methods for Toeplitz systems, SIAM Review, 1996, 38(3): 427–482.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wen, Z., Wang, Y. A new trust region algorithm for image restoration. Sci. China Ser. A-Math. 48, 169–184 (2005). https://doi.org/10.1360/03ys0178
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1360/03ys0178