## Abstract

We consider the problem of solving ill-conditioned linear systems Ax = b subject to the nonnegativity constraint x ≥ 0, and in which the vector b is a realization of a random vector b, 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.

Original language | English |
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Pages (from-to) | 1184-1197 |

Number of pages | 14 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 27 |

Issue number | 4 |

DOIs | |

State | Published - 2006 |

## Keywords

- Image restoration
- Linear models
- Preconditioning
- Statistical methods
- Weighted least squares