Covariance-Preconditioned Iterative Methods for Nonnegatively Constrained Astronomical Imaging

• Published in: SIAM journal on matrix analysis and applications Vol. 27; no. 4; pp. 1184 - 1197
• Main Authors: Bardsley, Johnathan M., Nagy, James G.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2006
• Subjects: Algorithms; Applied mathematics; Cameras; Exact sciences and technology; Iterative methods; Mathematics; Noise; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Random variables; Sciences and techniques of general use; Sensors; Telescopes; weighted least squares; Iterative method; Random vector; Image restoration; linear models; Linear model; Statistical method; Linear system; Covariance; Least squares method; Imaging; 65F30; 65F20; Conditioning; Preconditioning; statistical methods
• ISSN: 0895-4798; 1095-7162
• DOI: 10.1137/040615043

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.