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Deblurring Methods Using Antireflective Boundary Conditions

In this paper we consider the numerical solution of self-adjoint deblurring problems on bounded intervals. For these problems it has recently been shown that appropriate modeling of the solution near the boundary of the interval may significantly improve the numerical reconstructions. Among the alternatives the so-called antireflective boundary condition appears to be the best known choice. Here we develop an appropriate, i.e., stable and efficient implementation of this model in two steps, namely by (i) transforming the problem to homogeneous boundary values, and (ii) using the fast sine transform to solve the transformed problem. This approach allows us to incorporate regularization in a very straightforward way. Numerical reconstructions are superior qualitatively and quantitatively to those obtained with the reblurring method of Donatelli and Serra-Capizzano.