Radio Astronomical Image Formation using Constrained Least Squares and Krylov Subspaces

Abstract

Image formation for radio astronomy can be defined as estimating the spatial power distribution of celestial sources over the sky, given an array of antennas. One of the challenges with image formation is that the problem becomes ill-posed as the number of pixels becomes large. The introduction of constraints that incorporate a-priori knowledge is crucial. In this paper we show that in addition to non-negativity, the magnitude of each pixel in an image is also bounded from above. Indeed, the classical "dirty image" is an upper bound, but a much tighter upper bound can be formed from the data using array processing techniques. This formulates image formation as a least squares optimization problem with inequality constraints. We propose to solve this constrained least squares problem using active set techniques, and the steps needed to implement it are described. It is shown that the least squares part of the problem can be efficiently implemented with Krylov subspace based techniques, where the structure of the problem allows massive parallelism and reduced storage needs. The performance of the algorithm is evaluated using simulations.