The specifics of method of consecutive projections in deconvolution problems

Abstract

Method of consecutive projections (MCP) is quite popular. The principal idea of the method is to present the convex set as intersection of simple elementary convex sets and project to sets, outer for the current point. To build projection onto elementary sets very easy, because elementary sets are semispaces. It was proved, that this algorithm is finitely convergent. Each iteration of MCP has 3 tasks: 1. choose elementary set for projection; 2. define direction; 3. calculate length of step. Typical features of the method for deconvolution problems are the large dimension and the presence of a large number of almost perpendicular directions of projection. We found that several sequential steps in perpendicular directions can be replaced by one step, indicating a simple way to build it. Although the theoretical speed of convergence of the method remained the same, for large-scale problems (of the order of hundreds of millions), characteristic, for example, for image processing, it was possible to reduce the counting time by hundreds of times.