Deconvolution: a wavelet frame approach

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.