Choosing parameters in block-iterative or ordered-subset reconstruction algorithms

Viewed abstractly, all the algorithms considered here are designed to provide a nonnegative solution x to the system of linear equations y=Px, where y is a vector with positive entries and P a matrix whose entries are nonnegative and with no purely zero columns. The expectation maximization maximum likelihood (EMML) method as it occurs in emission tomography and the simultaneous multiplicative algebraic reconstruction technique (SMART) are slow to converge on large data sets; accelerating convergence through the use of block-iterative or ordered subset versions of these algorithms is a topic of considerable interest. These block-iterative versions involve relaxation and normalization parameters the correct selection of which may not be obvious to all users. The algorithms are not faster merely by virtue of being block-iterative; the correct choice of the parameters is crucial. Through a detailed discussion of the theoretical foundations of these methods we come to a better understanding of the precise roles these parameters play.