Covariance-Preconditioned Iterative Methods for Nonnegatively Constrained Astronomical Imaging

Abstract

We consider the problem of solving ill-conditioned linear systems $A\bfx=\bfb$ subject to the nonnegativity constraint $\bfx\geq\bfzero$, and in which the vector $\bfb$ is a realization of a random vector $\hat{\bfb}$, i.e., $\bfb$ is noisy. We explore what the statistical literature tells us about solving noisy linear systems; we discuss the effect that a substantial black background in the astronomical object being viewed has on the underlying mathematical and statistical models; and, finally, we present several covariance-based preconditioned iterative methods that incorporate this information. Each of the methods presented can be viewed as an implementation of a preconditioned modified residual-norm steepest descent algorithm with a specific preconditioner, and we show that, in fact, the well-known and often used Richardson-Lucy algorithm is one such method. Ill-conditioning can inhibit the ability to take advantage of a priori statistical knowledge, in which case a more traditional preconditioning approach may be appropriate. We briefly discuss this traditional approach as well. Examples from astronomical imaging are used to illustrate concepts and to test and compare algorithms.