Let T be a (possibly nonlinear) continuous operator on Hilbert space H. If, for some starting vector x, the orbit sequence {T k x, k = 0, 1, . . .} converges, then the limit z is a fixed point of T ; that is, T z = z. An operator N on a Hilbert space H is nonexpansive (ne) if, for each x and y in H, N x − N y x − y . Even when N has fixed points the orbit sequence {N k x} need not converge; consider the example N = −I , where I denotes the identity operator. However, for any α ∈ (0, 1) the iterative procedure defined by x k+1 = (1 − α)x k + α N x k 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.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. These include the Gerchberg-Papoulis method for bandlimited extrapolation, the SART algorithm of Anderson and Kak, the Landweber and projected Landweber algorithms, simultaneous and sequential methods for solving the convex feasibility problem, the ART and Cimmino methods for solving linear systems of equations, the CQ algorithm for solving the split feasibility problem and Dolidze's procedure for the variational inequality problem for monotone operators.

Let C and Q be nonempty closed convex sets in R N and R M , respectively, and A an M by N real matrix. The split feasibility problem (SFP) is to find x ∈ C with Ax ∈ Q, if such x exist. An iterative method for solving the SFP, called the CQ algorithm, has the following iterative step: x k+1 = P C (x k + γ A T (P Q − I)Ax k), where γ ∈ (0, 2/L) with L the largest eigenvalue of the matrix A T A and P C and P Q denote the orthogonal projections onto C and Q, respectively; that is, P C x minimizes c − x , over all c ∈ C. The CQ algorithm converges to a solution of the SFP, or, more generally, to a minimizer of P Q Ac − Ac over c in C, whenever such exist. The CQ algorithm involves only the orthogonal projections onto C and Q, which we shall assume are easily calculated, and involves no matrix inverses. If A is normalized so that each row has length one, then L does not exceed the maximum number of nonzero entries in any column of A, which provides a helpful estimate of L for sparse matrices. Particular cases of the CQ algorithm are the Landweber and projected Landweber methods for obtaining exact or approximate solutions of the linear equations Ax = b; the algebraic reconstruction technique of Gordon, Bender and Herman is a particular case of a block-iterative version of the CQ algorithm. One application of the CQ algorithm that is the subject of ongoing work is dynamic emission tomographic image reconstruction, in which the vector x is the concatenation of several images corresponding to successive discrete times. The matrix A and the set Q can then be selected to impose constraints on the behaviour over time of the intensities at fixed voxels, as well as to require consistency (or near consistency) with measured data.

The related problems of minimizing the functionals F(x)=alphaKL(y,Px)+(1-alpha)KL(p,x) and G(x)=alphaKL(Px,y)+(1-alpha)KL(x,p), respectively, over the set of vectors x=/>0 are considered. KL(a, b) is the cross-entropy (or Kullback-Leibler) distance between two nonnegative vectors a and b. Iterative algorithms for minimizing both functionals using the method of alternating projections are derived. A simultaneous version of the multiplicative algebraic reconstruction technique (MART) algorithm, called SMART, is introduced, and its convergence is proved.

Analysis of convergence of the algebraic reconstruction technique (ART) shows it to be predisposed to converge to a solution faster than simultaneous methods, such as those of the Cimmino-Landweber type, the expectation maximization maximum likelihood method for the Poisson model (EMML), and the simultaneous multiplicative ART (SMART), which use all the data at each step. Although the choice of ordering of the data and of relaxation parameters are important, as Herman and Meyer have shown, they are not the full story. The analogous multiplicative ART (MART), which applies only to systems y=Px in which y>0, P= or >0 and a nonnegative solution is sought, is also sequential (or "row-action"), rather than simultaneous, but does not generally exhibit the same accelerated convergence relative to its simultaneous version, SMART. By dividing each equation by the maximum of the corresponding row of P, we find that this rescaled MART (RMART) does converge faster, when solutions exist, significantly so in cases in which the row maxima are substantially less than one. Such cases arise frequently in tomography and when the columns of P have been normalized to have sum one. Between simultaneous methods, which use all the data at each step, and sequential (or row-action) methods, which use only a single data value at each step, there are the block-iterative (or ordered subset) methods, in which a single block or subset of the data is processed at each step. The ordered subset EM (OSEM) of Hudson et al. is significantly faster than the EMML, but often fails to converge. The "rescaled block-iterative" EMML (RBI-EMML) is an accelerated block-iterative version of EMML that converges, in the consistent case, to a solution, for any choice of subsets; it reduces to OSEM when the restrictive "subset balanced" condition holds. Rescaled block-iterative versions of SMART and MART also exhibit accelerated convergence.

The simultaneous MART algorithm (SMART) and the expectation maximization method for likelihood maximization (EMML) are extended to block-iterative versions, BI-SMART and BI-EMML, that converge to a solution in the feasible case, for any choice of subsets. The BI-EMML reduces to the "ordered subset" EMML of Hudson et al. (1992, 1994) when their "subset balanced" property holds.