Paola Boito : January 20, 2009

Optimal antireflective preconditioners for signal and image deblurring.

Abstract

We consider the problem of signal/image deblurring and denoising in the framework of a discrete, linear, space-invariant blurring model, which is associated with a structured linear system. Such a model is completely determined by a point-spread function (PSF) and a choice of boundary conditions (BC). We are interested in examining the behaviour of antireflective BC, proposed by Serra Capizzano et al., with a non (strongly) symmetric PSF. In particular, we wish to find optimal antireflective preconditioners for the associated Tikhonov regularization problem with reblurring, which is solved using an iterative method such as conjugate gradient or GMRES. A similar problem for the case of reflective BC was studied in [Chan et al. '99]; the authors showed that the optimal reflective preconditioner is given by the reflective blurring matrix associated with the symmetrised PSF. We conjecture that a similar result holds in the antireflective case and we prove that this is indeed the case for 1-dimensional signals. Numerical experiments show the effectiveness of the proposed method and allow to compare the results given by reflective and antireflective BC. This is joint work with S. Serra Capizzano and C. Tablino Possio.