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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