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 measured 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 function is the sum of simpler functions: h(x) = ∑i = 1Ihi(x). Partitioning the set {i = 1,...,I} as the union of the disjoint sets Bn,n = 1,...,N, we let hn(x) = ∑iBnhi(x). The method presented here is block iterative, in the sense that at each step only the gradient of a single hn(x) is employed. Convergence can be significantly 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 xk + 1 obtained at each step of the iteration satisfy the desired constraints. Here the constraints are imposed by having the next iterate xk + 1 satisfy the gradient equation ∇F(xk + 1) = ∇F(xk)-tn∇hn(xk), for appropriate scalars tn, where the convex function F is defined and differentiable only on vectors satisfying the constraints. Special cases of the algorithm that apply to tomographic image reconstruction, and permit inclusion of upper and lower bounds on individual pixels, are presented. The focus here is on the development of the underlying convergence theory of the algorithm. Behaviour of special cases has been considered elsewhere.