Improved image deblurring with anti-reflective boundary conditions and re-blurring

Anti-reflective boundary conditions (BCs) have been introduced recently in connection with fast deblurring algorithms. In the noise free case, it has been shown that they reduce substantially artifacts called ringing effects with respect to other classical choices (zero Dirichlet, periodic, reflective BCs) and lead to $O(n^2log(n))$ arithmetic operations, where $n^2$ is the size of the image. In the one-dimensional case, for noisy data, we proposed a successful approach called re-blurring: more specifically, when the PSF is symmetric, the normal equations product $A^TA$ is replaced by $A^2$, where $A$ is the blurring operator (see Donatelli et al. Inverse Problems, 21, pp. 169--182). Our present goal is to consider higher dimensions and extend to nonsymmetric PSFs the computational and theoretical analysis. In this more general framework, suitable for real applications, the new proposal is to replace $A^T$ by $A'$ in the normal equations, where $A'$ is the blurring matrix related to the current BCs with PSF rotated by $180$ degrees. We notice that, although with zero Dirichlet and periodic BCs the re-blurring approach is equivalent to the normal equations scheme, since there $A' = A^T$, the novelty concerns both reflective BCs and anti-reflective BCs, where in general $A' eq A^T$. We show that the re-blurring with anti-reflective BCs leads to a large reduction of the ringing effects arising in classical deblurring schemes. A wide set of numerical experiments concerning 2D images and nonsymmetric PSFs confirms the effectiveness of our proposal.