Summed squared distance error reduction by simultaneous multiprojections and applications

A parallel projection scheme in which projections are performed simultaneously on all constraints at each iteration is presented. It is a multiprojections scheme because different distance functions can be minimized for each projection as long as they are based on weighted L² norms. The use of multiple distance functions enables simplifications of the various projection operations, leading to an efficient implementation of this algorithm. This is true especially for constrained deconvolution type problems. Due to the inherent parallelism of the algorithm, projections onto more than two sets can be performed, even if one (or more) of the sets are nonconvex, maintaining a monotone decrease of a cost functional (with no further restrictions). This is in contrast to the serial projections onto convex sets (POCS) algorithm where no more than two sets can be treated if one of the sets is nonconvex. Our method is a special case of a multiprojections method proposed by Censor and Elfving where generalized distance functions of the Bregman type are used. The restriction to weighted L² norms leads to a simple and explicit form of this algorithm and allows relaxation and the use of nonconvex sets.