Spectral analysis of the anti-reflective algebra

Anti-reflective boundary conditions have been studied in connection with fast deblurring algorithms, in the case of d-dimensional objects (signals for d=1, images for d=2). Here we study how, under the assumption of strong symmetry of the point spread functions and under mild degree conditions, the associated matrices depend on a symbol and define an algebra homomorphism. Furthermore, the eigenvalues can be exhaustively described in terms of samplings of the symbol and other related functions, and appropriate O(ndlog(n)) arithmetic operations algorithms can be derived for the related computations. These results, in connection with the use of the anti-reflective transform, are of interest when employing filtering type procedures for the reconstruction of noisy and blurred objects.