The deconvolution problem: Fast algorithms including the preconditioned conjugate-gradient to compute a MAP estimator

Different aspects of the deconvolution problem are discussed and a solution by the MAP estimator using the a priori knowledge of smoothness and nonnegativity of the unknown signal is proposed. Several fast algorithms are examined and the conjugate-gradient (CG) method is show to have several advantages: it allows computation of a constrained solution and when used with a preconditioning technique it can be faster than other algorithms for general near-to-Toeplitz systems, in particular, for Toeplitz-block-Toeplitz systems.