Image recovery by convex combinations of projections

Abstract

The problem of image recovery in a Hilbert space setting by using convex projections (by projection we always mean minimum distance projection onto a closed convex set) may be stated as follows:. the original (unknown) image f is known a priori to belong to the intersection Co of r well-defined dosed convex sets C 1, .. C r in a Hilbert space H; hence f E Co = n;= 1 Ci; given only the projection operators Pi onto the individual sets Ci (i = 1, ..., r), recover f by an iterative scheme. Practically, the image f is a function of two real variables, and the Hilbert space H is the (real or complex) space L2(Q), where Q c R2. Since the sets Ci are only assumed to be convex, the projections Pi are in general nonlinear. Iterative methods of finding a common point of sets by means of contraction-like operators can be found in [1, 2, 5J; these methods have been applied to image processing first by Youla and Webb [11]. A procedure with essentially the same ideas has been used in [7J for the extrapolation of bandlimited functions. In the methods of image recovery used until now [9-12, 7J, from the projections Pi (i = 1, ..., r) r operators Ti are constructed, and from these the composition operator T = Tr Tr -1 ... T2 T1; starting from an arbitrary element x of H the sequence {Tnx}:= 0 is shown to converge, at least weakly, to an element of Co. Two cases are important: the case where each Ti equals Pi (hence T= PrPr-1 ... P2P1), and the case where each Ti is given by Ti=l+)oi(Pi-l), with 1 the identity operator on H and Ai a relaxation parameter with 0 < Ai < 2 (when )oi = 1 for all i this reduces to the foregoing case). So, at each iteration step the computation has to be done sequentially. When a parallel computer is available the computing .process could be speeded up a lot if at each iteration step the contribution of the different 413