Parallel projection methods for set theoretic signal reconstruction and restoration

An attempt is made to lay a theoretical and computational foundation for the use of parallel projection methods in set-theoretic signal restoration and reconstruction. A general method of parallel projections (MOPP) for solving feasibility problems in Hilbert spaces is proposed. In the proposed scheme, an elementary iteration consists in projecting the current estimate simultaneously onto selected sets and forming a relaxed convex combination of the projections. MOPP is seen to have substantial advantages over the method of successive projections, the widely used serial projection method which it generalizes: straightforward implementation on a parallel computing architecture, an explicit condition on the relaxation coefficients for faster convergence, and convergence to weighted least-squares solutions for inconsistent set-theoretic formulations. Convergence results for this algorithm in the convex and nonconvex cases are presented. Both consistent and inconsistent set-theoretic formulations are considered. Practical issues pertaining to the utilization of the method are discussed.<>