Flipped structured matrix-sequences in image deblurring with reflective and anti-reflective boundary conditions

Motivated by the recent work (Donatelli et al. Electron. Trans. Numer. Anal. 59, 157-178 2023) on a preconditioned MINRES for flipped linear systems in imaging with Dirichlet boundary conditions (BCs), in this note we extend the scope of that research for including more precise BCs such as reflective and anti-reflective ones. We prove spectral results for the matrix-sequences associated to the original deblurring problem incorporating the considered BCs. More precisely, the resulting matrix-sequences are real quasi-symmetric i.e. real symmetric up to zero-distributed perturbations and this justifies the use of MINRES in the current setting. The theoretical spectral analysis is supported by a wide variety of numerical experiments, concerning the visualization of the spectra of the original matrices in various ways. We also report numerical tests regarding the convergence speed and regularization features of the associated GMRES and MINRES methods. Conclusions and open problems end the present study.