Radio astronomical image formation using constrained least squares and Krylov subspaces

Abstract

Aims. Image formation for radio astronomy can be defined as estimating the spatial intensity distribution of celestial sources throughout 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. Methods. 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. We also propose a method for correcting for the possible mismatch between source positions and the pixel grid. This correction improves both the detection of sources and their estimated intensities. The performance of these algorithms is evaluated using simulations. Results. Based on parametric modeling of the astronomical data, a new imaging algorithm based on convex optimization, active sets, and Krylov-subspace-based solvers is presented. The relation between the proposed algorithm and sequential source removing techniques is explained, and it gives a better mathematical framework for analyzing existing algorithms. We show that by using the structure of the algorithm, an efficient implementation that allows massive parallelism and storage reduction is feasible. Simulations are used to compare the new algorithm to classical CLEAN. Results illustrate that for a discrete point model, the proposed algorithm is capable of detecting the correct number of sources and producing highly accurate intensity estimates.

A Fourier dimensionality reduction model for big data interferometric imaging

“…One would naturally want to estimate x 2 from the dirty image. This is supported by recent work showing that x can be bounded by the dirty image (Wijnholds & van der Veen 2011;Sardarabadi et al 2016). …”

Computationally efficient radio astronomical image formation using constrained least squares and and the MVDR beamformer

“…We have demonstrated its superiority over previously proposed CLS algorithms in a simulated experiments. The full version of this paper [23] will provide examples for the simulated 3C catalog as well as measured data. It will also provide the full implementation of the algorithm which is computationally simple but requires careful use of Krylov spaces to prevent the need to store very large matrices.…”

“…We end up with conclusions and extensions. Due to space limitations, the implementation details of the active set technique as well as comparison of the algorithm to other algorithms on real data will appear in the full version of this paper [23].…”

PRIFIRA: General regularization using prior-conditioning for fast radio interferometric imaging★

“…Alternatively, the problem can be regularized by posing a nonnegativity constraint on the solution (Briggs 1995). The resulting Non-negative LS (NNLS) optimization can be implemented using the active set method (Sardarabadi, Leshem & van der Veen 2016) and similarly consists of two levels of iterations: (i) an outer loop to iteratively find the sparse support of the image and (ii) an inner loop in which a dimension reduced version of the LS problem is solved.…”