BICAV: a block-iterative parallel algorithm for sparse systems with pixel-related weighting

Abstract

Abstract-Component averaging (CAV) was recently introduced by Censor, Gordon, and Gordon as a new iterative parallel technique suitable for large and sparse unstructured systems of linear equations. Based on earlier work of Byrne and Censor, it uses diagonal weighting matrices, with pixel-related weights determined by the sparsity of the system matrix. CAV is inherently parallel (similar to the very slowly converging Cimmino method) but its practical convergence on problems of image reconstruction from projections is similar to that of the algebraic reconstruction technique (ART). Parallel techniques are becoming more important for practical image reconstruction since they are relevant not only for supercomputers but also for the increasingly prevalent multiprocessor workstations. This paper reports on experimental results with a block-iterative version of component averaging (BICAV). When BICAV is optimized for block size and relaxation parameters, its very first iterates are far superior to those of CAV, and more or less on a par with ART. Similar to CAV, BICAV is also inherently parallel. The fast convergence is demonstrated on problems of image reconstruction from projections, using the SNARK93 image reconstruction software package. Detailed plots of various measures of convergence, and reconstructed images are presented.

Bounds on the largest singular value of a matrix and the convergence of simultaneous and block‐iterative algorithms for sparse linear systems

“…For w i = 1, this is the CARP1 algorithm of [13] (see also [11,8,9]). The simultaneous DROP algorithm of [7] requires only that the weights w i be positive, but dividing each w i by their maximum, max i {w i }, while multiplying each λ k by the same maximum,…”

“…Convergence of CAV then follows, as does convergence of several other methods, including the ART, Landweber's method, the SART [1], the block-iterative CAV (BICAV) [9], the CARP1 method of Gordon and Gordon [13], and a block-iterative variant of CARP1 obtained from the DROP method of Censor et al [7]. Convergence of most of these methods was also established in [15], using a unifying framework of a block-iterative Landweber algorithm, but without deriving upper bounds for the largest eigenvalue of a general A † A.…”

Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. II. Singular weights

“…As follows from estimates (22), (23), (29), (33), (36), (38), (41) The paper [1] reviews the literature on direct and iterative methods of computing weighted pseudoinverses and weighted normal pseudosolutions. Noteworthy are the papers [31][32][33][34], which contain some algorithmic results that use weighted pseudoinversion with singular weights and consider problems of parallel computations for the algorithms proposed.…”

Iterative algorithms with seminorm‐induced oblique projections

“…The motivation of this paper comes from the recent work of Censor et al [14,15] who used generalized oblique projections for the solution of large and sparse systems of linear equations arising in the fully discretized approach to the problem of image reconstruction from projections, see, for example, Herman [22] and Censor [11]. In order to achieve significant acceleration in the algorithm's behavior, they had to use generalized oblique projections (as they called them) which allow zeros on the diagonals of the weighting matrices for the fully simultaneous projections algorithm that they used.…”

“…Since the work of [14,15] is limited to linear equations, it is natural to inquire whether it can be extended, and if so, in what directions and to what extent? Is it possible to extend the definition of the generalized oblique projections of [14,15] to cover convex sets which are not linear (i.e., not hyperplanes or halfspaces)?…”