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 iterationdependent, 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.