Pros and Cons of Multiscale Procedures in Imaging and Inverse Problems
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I consider a multiscale procedure introduced in imaging by Tadmor, Nezzar and Vese and extended to inverse problems by Modin, Nachman and myself. I deal with the convergence properties of such a procedure, focusing on linear inverse problems such as deblurring. When the regularization term is a a suitable Hilbert space norm penalization, convergence holds and the multiscale procedure provides a more stable reconstruction. This is confirmed by numerical examples. On the other hand, for general regularization terms, related for instance to total variation penalization, convergence of the multiscale procedure may indeed fail.