Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections

Solving a convex set theoretic image recovery problem amounts to finding a point in the intersection of closed and convex sets in a Hilbert space, The projection onto convex sets (POCS) algorithm, in which an initial estimate is sequentially projected onto the individual sets according to a periodic schedule, has been the most prevalent tool to solve such problems, Nonetheless, POCS has several shortcomings: It converges slowly, it is ill suited for implementation on parallel processors, and it requires the computation of exact projections at each iteration, In this paper, we propose a general parallel projection method (EMOPSP) that overcomes these shortcomings, At each iteration of EMOPSP, a convex combination of subgradient projections onto some of the sets is formed and the update is obtained via relaxation, The relaxation parameter may vary over an iteration-dependent, extrapolated range that extends beyond the interval ]0,2] used in conventional projection methods, EMOPSP not only generalizes existing projection-based schemes, but it also converges very efficiently thanks to its extrapolated relaxations, Theoretical convergence results are presented as well as numerical simulations.