We argue that some inverse problems arising in imaging can be efficiently treated using only single-precision (or other reduced-precision) arithmetic, using a combination of old ideas (first-order methods, polynomial preconditioners), and new ones (bilateral filtering, total variation). Using single precision, and having structures which parallelize in the ways needed to take advantage of low-cost/high-performance multi-core/SIMD architectures, this framework is especially suited to embedded image reconstruction applications like medical imaging. We show with a simulated magnetic resonance imaging problem that this method can be numerically effective. Since the convergence/error analysis is particularly simple for pure quadratic objectives, this approach can also be used in embedded environments with fixed computation budgets, or certification requirements. Simple analysis for the quadratic case also serves as a basis for the analysis of nonlinear problems solved via a sequence of quadratic approximations. We include one example of a nonlinear, nonquadratic penalty function.
AdvOL2007-001, McMaster University, January/2006