A Wavelet-based multilevel approach for blind deconvolution problems

We develop a multilevel approach to solving a blind deconvolution problem, with the ultimate intent of recovering signals which are known to have edges. First, we discuss how to generate a hierarchy of blind deconvolution problems by means of the Haar wavelet transform, and we give a modified regularized total least norm approach for solving the resulting coarse-grid problems. Use of the Haar wavelet transform for intergrid manipulation is motivated by the fact that they can preserve desirable properties of the blurring matrix when restricted to the coarse grid, and because their orthonormality helps with the interpretability of the noise and subproblems in the hierarchy. Recognizing that in blind deconvolution problems the blurring matrices are often assumed to be structured, we subsequently discuss treatment of the case when both the known linear operator and the unknown perturbation to the operator are banded Toeplitz matrices. For this case, since the matrix structure is inherited at coarser levels, a modified regularized structured total least norm approach is introduced, and a quasi-Newton method is employed to solve the coarse-grid and residual correction problems. Numerical examples show the potential of our multilevel method to recover both signals with edges and blurring operators.