Deconvolution: a wavelet frame approach

Abstract

This paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches, which has already appeared in Chan et al. (SIAM J. Sci. ) 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 ). 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

Blind motion deblurring using multiple images

“…1 (a) and the blurred image Fig. 1 (b) with a geometrical perturbation h defined in (7). In other words, the estimated blur kernel p is based on the following perturbed version of the original equation:…”

“…Step 2 is a non-blind image deblurring problem, which has been studied extensively in the literature; see, for instances, [1,22,8,7,18,5]. However, there is one more error source in Step 2 than the traditional non-blind deblurring problem has, that is, the error in the intermediate blur kernel p (k+1) i used for deblurring.…”

“…However, there is one more error source in Step 2 than the traditional non-blind deblurring problem has, that is, the error in the intermediate blur kernel p (k+1) i used for deblurring. Inspired by recent non-blind deblurring techniques which are based on sparse approximation to the image under certain tight frame systems ( [7,4]), we also use the sparsity constraint on the clear image g under tight frame systems to regularize the non-blind deblurring. And we use a modified version of so-called linearized Bregman iteration (See [24,32,31,16,20,11,25,4,5,6,15]) to achieve impressive robustness to image noises, alignment errors, and, more importantly, perturbations on the given intermediate blur kernels.…”

“…In our implementation, we use a multilevel tight framelet decomposition without down-sampling under the Neumann (symmetric) boundary condition. A detailed description can be found in [7].…”

“…In the past, there have been extensive studies on robust non-blind deconvolution algorithms (e.g. [1,22,8,7,18]). In the case of blind deconvolution, both the blur kernel p and the clear image g are unknown.…”