Statistical image reconstruction in X-Ray computed tomography yields
large-scale regularized linear least-squares problems with nonnegativity
bounds, where the memory footprint of the operator is a concern. Discretizing
images in cylindrical coordinates results in significant memory savings, and
allows parallel operator-vector products without on-the-fly computation of the
operator, without necessarily decreasing image quality. However, it
deteriorates the conditioning of the operator. We improve the Hessian
conditioning by way of a block-circulant scaling operator and we propose a
strategy to handle nondiagonal scaling in the context of projected-directions
methods for bound-constrained problems. We describe our implementation of the
scaling strategy using two algorithms: TRON, a trust-region method with exact
second derivatives, and L-BFGS-B, a linesearch method with a limited-memory
quasi-Newton Hessian approximation. We compare our approach with one where a
change of variable is made in the problem. On two reconstruction problems, our
approach converges faster than the change of variable approach, and achieves
much tighter accuracy in terms of optimality residual than a first-order
method.Comment: 19 pages, 7 figure