Parametric High Resolution Techniques for Radio Astronomical Imaging

Abstract

The increased sensitivity of future radio telescopes will result in requirements for higher dynamic range within the image as well as better resolution and immunity to interference. In this paper we propose a new matrix formulation of the imaging equation in the cases of non co-planar arrays and polarimetric measurements. Then we improve our parametric imaging techniques in terms of resolution and estimation accuracy. This is done by enhancing both the MVDR parametric imaging, introducing alternative dirty images and by introducing better power estimates based on least squares, with positive semi-definite constraints. We also discuss the use of robust Capon beamforming and semi-definite programming for solving the self-calibration problem. Additionally we provide statistical analysis of the bias of the MVDR beamformer for the case of moving array, which serves as a first step in analyzing iterative approaches such as CLEAN and the techniques proposed in this paper. Finally we demonstrate a full deconvolution process based on the parametric imaging techniques and show its improved resolution and sensitivity compared to the CLEAN method.

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

“…The dirty image formed by substituting (17) into (15) is called the MVDR dirty image [12]. Each pixel of the MVDR dirty image is given by…”

“…Similarly we can show that by defining a diagonal preconditioner and applying it to the WLS we can relate this new preconditioned WLS (PWLS) problem to the MVDR dirty image. This will also allow us to understand the relation between the WLS solution and the LS-MVI algorithm [12]. The diagonal preconditioner with this property given by (26).…”

“…Beyond these two techniques there are several extensions in various directions: extensions of the CLEAN algorithm to support multi-resolution and wavelets as well as non co-planar arrays and multiple wavelengths such as the W-projection [7] and the A-projection algorithm (see the overview paper [8]). MEM techniques have been also extended to take into account source structure through the use of multiresolution and wavelet based techniques [9], global non-negative least squares was proposed by Briggs [10], matrix based parametric imaging such as the Least Squares Minimum Variance Imaging (LS-MVI), maximum likelihood based techniques in [11], [12], [13] and sparse L1 reconstruction in [14] and [15]. Source modeling is an important issue and various techniques to improve modeling over simple point source models by using shapelets, wavelets and Gaussians [16] have been implemented.…”

“…In order to benefit from the vast literature related to solving least square problems on one hand and to gain from the non-linear processing offered by standard deconvolution techniques on the other hand, we propose to reformulate the imaging problem using an active set approach with two additional constraints as well as a low-dimensional fine fitting of the parameters. We generalize several of the sequential parametric techniques into an active-set weighted least squares algorithm [21] with weighting derived to allow the introduction of high resolution techniques such as the LS-MVI ( [12]). This reformulation will allow us to obtain computationally efficient imaging algorithms that are closely related to existing non-linear sequential source estimation techniques with the benefit of accelerated convergence due to tighter upper bounds on the power distribution over the complete image.…”

Synthetic aperture radio telescopes

“…MEM techniques have been also extended to take into account source structure through the use of multiresolution and wavelet based techniques [17]. Global non-negative least squares was proposed by Briggs [18], matrix based parametric imaging such as the Least Squares Minimum Variance Imaging (LS-MVI) and maximum likelihood based techniques in [19] and [20] and sparse L 1 reconstruction in [21] and [22]. Source modeling is an important issue and various techniques to improve modeling over simple point source models by using shapelets, wavelets and Gaussians [23] have been implemented.…”