Iterative relaxation methods for image reconstruction

Abstract

The problem of recovering an image (a function of two variables) from experimentally available integrals of its grayness over thin strips is of great importance in a large number of scientific areas.An important version of the problem in medicine is that of obtaining the exact density distribution within the human body from X-ray projections.One approach that has been taken to solve this problem consists of translating the available information into a system of linear inequalities.The size and the sparsity of the resulting system of inequalities (typically, 25,000 inequalities with less than 1% of the coefficients nonzero) makes methods using successive relaxations computationally attractive.A variety of such methods have been proposed with differing relaxation parameters.In this paper we show that for a consistent system of linear inequalities, any sequence of relaxation parameters lying strictly between zero and two generates a sequence of vectors which converges to a solution.Under the same assumptions, for a system of linear equations, the relaxation method converges to the minimum norm solution.Previously proposed techniques are shown to be special cases of our procedure with different choices of relaxation parameters.Thus, our results provide proofs of convergence for previously used algorithms where no such proofs existed before.We discuss the practical consequences for image reconstruction of the choice of the relaxation parameters.

Solution of underdetermined nonlinear equations by stationary iteration methods

“…Other convergence results have been obtained by Herman and his coworkers [11], McCormick [18], and Elfving [6] for a block-iterative version. The popularity of the method in problems like image reconstruction (Gordon et al [9]) and image restoration ( ) is mainly due to its computational simplicity and to the fact that sparsity of the corresponding matrices can easily be exploited.…”