The goal of this paper is to lay a theoretical and computational foundation for the use of parallel projection methods in set theoretic signal restoration and reconstruction. A new projection algorithm is proposed in which the projections of the current estimate onto selected sets are computed simultaneously and the update is a relaxed convex combination of these projections. 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.
IN TRO D U C T I 0 NThe set theoretic signal recovery (restoration or reconstruction) problem is to produce an estimate of an original signal which is consistent with all prior knowledge as well as with the data. By recasting this concept in a signal space, the class of signals consistent with a particular piece of information is associated with a property set. The problem is then to find a feasible signal, i.e., a point in the intersection of all the property sets.l Since this feasibility problem can usually not be solved in one step, it can be approached via iterative methods by building a sequence of signals converging in some sense t o a signal in the intersection of the sets.Over the past decade, the set theoretic approach has been applied to a broad spectrum of signal recovery problems, e.g., tomography, image deblurring, signal extrapolation, phase recovery, and holography [2]. In almost all of these applications, the method of suctheoretic solutions [2], [9].2 MOSP consists in projecting sequentially an initial estimate onto the sets in a cyclic manner. Although easy to implement, this scheme displays three major shortcomings. First, because of its serial nature, it is not well suited for implementation on parallel processors. Indeed, at each iteration, only one of the sets can be acted upon. Second, although conceptually MOSP can be accelerated by relaxing the iterations, there exists no practical, general rule to determine the relaxation coefficients accordingly. Third, MOSP provides poor solutions if the feasibility set (the intersection of the property sets) is empty, a common instance in some fields. In this case, it will yield (at best) a point which can be guaranteed to belong to only one of the sets.In this paper, we present a general method of parallel projections (MOPP) to compute set theoretic solutions. In the proposed scheme, which generalizes MOSP, an elementary iteration consists in projecting the current estimate simultaneously onto selected sets and forming a relaxed convex combination of the projections. The main thrust of this approach is to alleviate the afore mentioned shortcomings of MOSP and provide a theoretically sound and computationally efficient framework for solving set theoretic problems, in particular in multi-dimensional signal reconstruction and restoration. The paper is organized as follows. First, MOSP is reviewed and its limitations are discussed. We then proceed...
