Distributed Computation of Linear Inverse Problems with Application to Computed Tomography

Abstract

The inversion of linear systems is a fundamental step in many inverse problems. Computational challenges exist when trying to invert large linear systems, where limited computing resources mean that only part of the system can be kept in computer memory at any one time. We are here motivated by tomographic inversion problems that often lead to linear inverse problems. In state of the art x-ray systems, even a standard scan can produce 4 million individual measurements and the reconstruction of x-ray attenuation profiles typically requires the estimation of a million attenuation coefficients. To deal with the large data sets encountered in real applications and to utilise modern graphics processing unit (GPU) based computing architectures, combinations of iterative reconstruction algorithms and parallel computing schemes are increasingly applied. Although both row and column action methods have been proposed to utilise parallel computing architectures, individual computations in current methods need to know either the entire set of observations or the entire set of estimated x-ray absorptions, which can be prohibitive in many realistic big data applications. We present a fully parallelizable computed tomography (CT) image reconstruction algorithm that works with arbitrary partial subsets of the data and the reconstructed volume. We further develop a non-homogeneously randomised selection criteria which guarantees that sub-matrices of the system matrix are selected more frequently if they are dense, thus maximising information flow through the algorithm. A grouped version of the algorithm is also proposed to further improve convergence speed and performance. Algorithm performance is verified experimentally.