Iterative methods for reconstructing images have been studied for decades. Because many of these methods, such as the EMML, are slow to converge, particularly for the large data sets typical of modern imaging, there has been growing interest in block-iterative (also called ordered-subset) methods for image reconstruction, due largely to the accelerated convergence some of these methods provide. A brief overview of the use of iterative reconstrction methods in medical imaging is given in [137]. The blockiterative methods of interest to us here can be derived as incremental optimization procedures, in which the cost function h( x) to be minimized can be decomposed as a sum of simpler functions, , and the iterative procedure involves the gradients of only a few of the h _i( x) at each step.

Our topic is the reconstruction of a discrete image from finite data pertaining to that image. Because realistic models relating the data to the image pixels (or voxels) typically preclude closed form solutions, we shall focus here on iterative algorithms. For reasons to be presented shortly, the algorithms we shall consider are optimization methods, in which we seek to maximize or minimize some function over the set of feasible images, that is, those satisfying whatever constraints, such as nonnegativity, we have imposed.

When the data is essentially noise-free, but insufficient to determine a unique image, one may choose that feasible image consistent with the data, for which some function, such as entropy, is maximized, or some measure...