Convergent block-iterative algorithms for image reconstruction from inconsistent data

It has been shown that convergence to a solution can be significantly accelerated for a number of iterative image reconstruction algorithms, including simultaneous Cimmino-type algorithms, the "expectation maximization" method for maximizing likelihood (EMML) and the simultaneous multiplicative algebraic reconstruction technique (SMART), through the use of rescaled block-iterative (BI) methods. These BI methods involve partitioning the data into disjoint subsets and using only one subset at each step of the iteration. One drawback of these methods is their failure to converge to an approximate solution in the inconsistent case, in which no image consistent with the data exists; they are always observed to produce limit cycles (LCs) of distinct images, through which the algorithm cycles. No one of these images provides a suitable solution, in general. The question that arises then is whether or not these LC vectors retain sufficient information to construct from them a suitable approximate solution; we show that they do. To demonstrate that, we employ a "feedback" technique in which the LC vectors are used to produce a new "data" vector, and the algorithm restarted. Convergence of this nested iterative scheme to an approximate solution is then proven. Preliminary work also suggests that this feedback method may be incorporated in a practical reconstruction method.