A note on antireflective boundary conditions and fast deblurring models

In a recent work Ng, Chan, and Tang introduced reflecting (Neumann) boundary conditions (BCs) for blurring models and proved that the resulting choice leads to fast algorithms for both deblurring and detecting the regularization parameters in the presence of noise. The key point is that Neumann BC matrices can be simultaneously diagonalized by the (fast) cosine transform DCT III. Here we propose antireflective BCs that can be related to τ structures, i.e., to the algebra of the matrices that can be simultaneously diagonalized by the (fast) sine transform DST I. We show that, in the generic case, this is a more natural modeling whose features are (a) a reduced analytical error since the zero (Dirichlet) BCs lead to discontinuity at the boundaries, the reflecting (Neumann) BCs lead to C0 continuity at the boundaries, while our proposal leads to C1 continuity at the boundaries; (b) fast numerical algorithms in real arithmetic for both deblurring and estimating regularization parameters. Finally, simple yet significant 1D and 2D numerical evidence is presented and discussed.