Matrix Structures and Image Restoration: Boundary Conditions, Re-Blurring, and Regularizing Multigrid-Type Algorithms

We consider the de-blurring problem of noisy and blurred images in the case of space invariant point spread functions (PSFs). The use of appropriate boundary conditions (see [2,10,12]) leads to linear systems with structured coefficient matrices related to space invariant operators like Toeplitz, circulants, trigonometric matrix algebras etc. We can obtain an effective and fast solver by combining the optimally convergent algebraic multigrid described in [11,1] with the Tikhonov regularization (see [3]). A completely alternative proposal is to apply the latter algebraic multigrid (which is designed ad hoc for structured matrices) with the low-pass projectors typical of the classical geometrical multigrid employed in a PDEs context. Thus, using an appropriate smoother, we obtain an iterative regularizing method (see [9]). Unfortunately, the normal equations approach used in connection with popular regularization processes (Tikhonov, CGNE, Landweber etc.) spoils the structure of matrix algebra and the modelistic features of the most precise boundary conditions i.e. reflective [10] and anti-reflective [12,5]. A remedy both for the computational and modelistic problems is to replace the transposition operation (A → A ) by the correlation operation (A → A, see [4]): we called this idea “re-blurring” (for a comprehensive discussion on this subject see [8,7,6,4]).