A Paradigm for Reconstruction

In [192] Youla suggests that problems in signal processing and image restoration might be viewed geometrically and the method of projection onto convex sets (POCS) employed to solve such inverse problems. In the survey paper [193] he examines the POCS method as a particular case of iterative algorithms for finding fixed points of nonexpansive mappings. This point of view is increasingly important in applications such as medical imaging and a number of recent papers have addressed the theoretical and practical issues involved [10, 11, 9, 36, 40, 43, 71, 72, 74].

A subset C of R ^N is convex if the line segment joining any two of its members lies entirely within C. In the plane R ² the set C of all points whose distance to the origin is less than one is convex; if we include the boundary of C, that is, the circumference of the circle, the set is also closed. But the circumference alone is not a convex set. If C is a closed convex set and x is not in C, then there is a unique point in C closer to x than any other member of C; that point is called the metric projection of x onto C, written P _C x. If the set is not convex, there need not be a unique nearest point; the circle of radius one (not...