A Method for Reducing the Complexity in the Reconstruction of a Blurred Signal

The regularized least squares methods for the solution of ill-posed inverse problems are summarized, and appropriate references are stated. Additionally, an adap-tive multi-scale algorithm is proposed to solve highly ill-posed inverse problems with fewer degrees of freedom and comparable performance. The algorithm controls the level of detail in the reconstruction by distributing the ¯ne scale information to the appropriate intervals in the overall estimation interval. The method is applied to a linear inverse problem, namely the reconstruction of a signal from its blurred and noisy version. The results are stated, compared with the regular ¯ne scale approach and the relevant properties are explained. This treatment is seen as a step for the application of the algorithm to the nonlinear inverse problems where it is forseen to provide a decrease in the complexity of the inversion as well as better convergence in the solution space than the regular approach. 1 Inroduction Least-squares estimation methods are commonly used in solving linear inverse problems. A linear inverse problem can be expressed as the problem of estimating the vector x based on the knowledge of a data vector b which is related to x as b = Ax + n. Here A is a known linear transformation matrix and n is a random noise vector. Inverse problems are typically ill-posed, meaning that small perturbations in the data can lead to large amplitude, non-physical artifacts in the reconstruction. Linear ill-posed problems arise in a variety of applications: astronomy [1], computerized tomography [2], electrocardiography [3], early vision [5] and meteorology [4] are just a few of these. Vast amount of literature on ill-posed problems exist in the setting of Hilbert spaces and other in¯nite dimensional spaces. See for example [6], [7],