Fast transforms for high order boundary conditions in deconvolution problems

We study strategies to increase the precision in deconvolution models, while maintaining the complexity of the related numerical linear algebra procedures (matrix-vector product, linear system solution, computation of eigenvalues, etc.) of the same order of the celebrated fast Fourier transform. The key idea is the choice of a suitable functional basis to represent signals and images. Starting from an analysis of the spectral decomposition of blurring matrices associated to the antireflective boundary conditions introduced in Serra Capizzano (SIAM J. Sci. Comput. 25(3):1307–1325, 2003), we extend the model for preserving polynomials of higher degree and fast computations also in the nonsymmetric case. We apply the proposed model to Tikhonov regularization with smoothing norms and the generalized cross validation in order to choose the regularization parameter. A selection of numerical experiments shows the effectiveness of the proposed techniques.