A unified treatment of some iterative algorithms in signal processing and image reconstruction

A wide variety of iterative procedures used in signal processing and image reconstruction and elsewhere are special cases of the KM iterative procedure, for particular choices of the ne operator N.

Abstract

Let T be a (possibly nonlinear) continuous operator on Hilbert space . If, for some starting vector x, the orbit sequence {Tkx,k = 0,1,...} converges, then the limit z is a fixed point of T; that is, Tz = z. An operator N on a Hilbert space is nonexpansive (ne) if, for each x and y in , Even when N has fixed points the orbit sequence {Nkx} need not converge; consider the example N = −I, where I denotes the identity operator. However, for any the iterative procedure defined by converges (weakly) to a fixed point of N whenever such points exist. This is the Krasnoselskii–Mann (KM) approach to finding fixed points of ne operators.