The University of Montana
Department of Mathematical Sciences

Technical report #1/2006


Covariance-Preconditioned Iterative Methods For Nonnegatively Constrained Astronomical Imaging

John Bardsley
Department of Mathematical Sciences
University of Montana

and

James Nagy
Department of Mathematics and Computer Science
Emory University

Abstract

We consider the problem of solving ill-conditioned linear systems Ax=b subject to the nonnegativity constraint x0, and in which the vector b is a realization of a random vector , i.e. b 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.

Keywords:image restoration, linear models, preconditioning, statistical methods, weighted least squares

AMS Subject Classification: 65F20, 65F30

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