Unified Hilbert space approach to iterative least-squares linear signal restoration

Abstract

We deal with iterative least-squares solutions of the linear signal-restoration problem g = Af. First, several existing techniques for solving this problem with different underlying models are unified. Specifically, the following are shown to be special cases of a general iterative procedure [ BialyH., Arch. Ration. Mech. Anal.4, 166 ( 1959)] for solving linear operator equations in Hilbert spaces: (1) a Van Cittert-type algorithm for deconvolution of discrete and continuous signals; (2) an iterative procedure for regularization when g is contaminated with noise; (3) a Papoulis–Gerchberg algorithm for extrapolation of continuous signals [ PapoulisA., IEEE Trans. Circuits Syst.CAS-22, 735 ( 1975); GerchbergR. W., Opt. Acta21, 709 ( 1974)]; (4) an iterative algorithm for discrete extrapolation of band-limited infinite-extent discrete signals {and the minimum-norm property of the extrapolation obtained by the iteration [ JainA.RanganathS., IEEE Trans. Acoust. Speech Signal Process. ASSP-29, ( 1981)]}; and (5) a certain iterative procedure for extrapolation of band-limited periodic discrete signals [ TomV., IEEE Trans. Acoust. Speech Signal Process.ASSP-29, 1052 ( 1981)]. The Bialy algorithm also generalizes the Papoulis–Gerchberg iteration to cases in which the ideal low-pass operator is replaced by some other operators. In addition a suitable modification of this general iteration is shown. This technique leads us to new iterative algorithms for band-limited signal extrapolation. In numerical simulations some of these algorithms provide a fast reconstruction of the sought signal.