Deconvolution: a wavelet frame approach
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This paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches, which has already appeared in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b) as wavelet frame based high resolution image reconstruction methods. We first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation, which completes the formulation given in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b). This formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames which satisfy certain minimization properties. These missing coefficients are recovered iteratively together with a built-in denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating. This approach has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the simulation results given in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b). However, an analysis of convergence as well as the stability of algorithms and the minimization properties of solutions were absent in those papers. This paper is to establish the theoretical foundation of this wavelet frame approach. In particular, a proof of convergence, an analysis of the stability of algorithms and a study of the minimization property of solutions are given.
Mathematics Subject Classification (2000)42C40 65T60 68U99
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- 3.Borup, L., Grivonbal, R., Nielsen, M.: Tight wavelet frames in Lebesgue and Sobolev spaces. J. Funct. Spaces Appl. 2(3) (2004)Google Scholar
- 5.Chan, R., Chan, T., Shen, L., Shen, Z.: A wavelet method for high-resolution image reconstruction with displacement errors. In: IEEE Signal Processing Society. Proceedings of the 2001 International Symposium of Intelligent Multimedia, Video and Speech Processing, Hong Kong, pp. 24–27. IEEE, USA (2001)Google Scholar
- 10.Chan, R., Shen, Z., Xia, T.: Resolution enhancement for video clips: tight frame approach. In: Proceedings of IEEE International Conference on Advanced Video and Signal-Based Surveillance, Italy, pp. 406–410 (2005)Google Scholar
- 11.Chan, R., Shen, Z., Xia, T.: A framelet algorithm for enchancing video stills. Appl. Comput. Harmon. Anal (2007, in press)Google Scholar
- 17.Daubechies I.: Ten lectures on wavelets. CBMS Conference Series in Applied Mathematics 61, SIAM, Philadelphia (1992)Google Scholar
- 20.De Mol, C., Defrise, M.: A note on wavelet-based inversion algorithms. In: Nashed, M., Scherzer, O. (eds.) Inverse problems, image analysis, and medical imaging, New Orleans, LA, 2001, pp. 85–96. Contemp. Math. 313, Amer. Math. Soc., Providence, RI (2002)Google Scholar
- 24.Engl, H., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht, Boston (1996)Google Scholar
- 27.Jia, R., Micchelli, C.: Using the refinement equations for the construction of pre-wavelets. II. Powers of two. In: Laurent, P., Mehaute, A., Schumaker, L. (ed.) First International Conference on Curves and Surfaces, Chamonix-Mont-Blanc, 1990. Curves and Surfaces, pp. 209–246. Academic, Boston (1991)Google Scholar
- 33.Tikhonov A.N. (1963). On the solution of incorrectly put problems and the regularization method. Soviet Math. Doklady 4: 1035–1038 Google Scholar