Block-iterative interior point optimization methods for image reconstruction from limited data

Iterative algorithms for image reconstruction often involve minimizing some cost function h(x) that measures the degree of agreement between the measu red data and a theoretical parametrized model. In addition, one may wish to have x satisfy certain constraints. It is usually the case that the cost f unction is the sum of simpler functions: [GRAPHICS] Partitioning the set (i = 1,..., 1) as the union of the disjoint sets B-n, n = 1,..., N, we let [GRAPHICS] The method presented here is block iterative, in the sense that at each ste p only the gradient of a single h(n)(x) is employed. Convergence can be sig nificantly accelerated, compared to that of the single-block (N = 1) method , through the use of appropriately chosen scaling factors. The algorithm is an interior point method, in the sense that the images x(k+1) obtained at each step of the iteration satisfy the desired constraints. Here the constr aints are imposed by having the next iterate x(k+1) satisfy the gradient eq uation delF(x(k+1)) = delF (x(k)) - t(n) delh(n) (x(k)), for appropriate scalars t(n), where the convex function F is defined and di fferentiable only on vectors satisfying the constraints. Special cases of the algorithm that apply to tomographic image reconstructi on, and permit inclusion of upper and lower bounds on individual pixels, ar e presented. The focus here is on the development of the underlying converg ence theory of the algorithm. Behaviour of special cases has been considere d elsewhere.