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 (typically, 25,000 inequalities with fewer than 1 percent of the coefficients nonzero) makes methods using successive relaxations computationally attractive, as compared to other ways of solving systems of inequalities. In this paper, it is shown that, for a consistent system of linear inequalities, any sequence of relaxation parameters lying strictly between 0 and 2 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. The practical consequences for image reconstruction of the choice of the relaxation parameters are discussed.

Efficient Statistical Equilibria in Markets

“…Moreover, the advances of modern computer science now admits applications with fairly large offer sets. Balancing algorithms of Bregman type have long since been used in situations more than 25 000 different equations/inequalities, see, e.g., Herman et al (1978). Hence we suggest that offer sets with a similar or even higher order of magnitude may be within reach.…”

Component-Averaged Row Projections: A Robust, Block-Parallel Scheme for Sparse Linear Systems

“…KACZ has been studied very extensively, both theoretically and experimentally. Its convergence with relaxation parameters, for consistent systems, has been shown by Herman, Lent, and Lutz [22] and by Trummer [33]. Tanabe [32] proved that when the system is inconsistent, it converges cyclically; i.e., for each hyperplane, the sequence of projections on that hyperplane converges to a limit.…”

A Study of Two Dimensional Tomography Reconstruction of Temperature and Gas Concentration in a Combustion Field Using TDLAS

“…where k is the iteration index in the ART procedure, and λ (0 < λ < 2) is the relaxation coefficient that plays an important role in accuracy performance and determining convergence rate [26]. It is evident that the λ represents the contribution of the absorption at j-grid to the integral i-th beam.…”