NWO-TOP: The synthesis of signal processing and radio astronomical calibration and imaging techniques

 

Outline of research

Scientific context

With the discovery of sources of celestial radio emission by Karl Jansky and Grote Reber, before and during the second world-war, radio astronomy was born. After the war engineers and astronomers in the UK, Australia, the Netherlands and the USA teamed up to build ever more sophisticated receivers and bigger telescopes. The quest for improved angular resolution has dominated the early technical developments in radio astronomy: radio interferometry was born. Today, many of the world's finest and most productive radio telescopes are arrays of dishes based on the principle of aperture synthesis: using interferometry they synthesize an aperture of much larger dimensions. The Westerbork Synthesis Radio telescope (WSRT, 3 km size) and the VLA (35 km, in New Mexico, USA) are premier examples of telescopes build on this principle.

In the period from 1960-1970 major radio astronomical discoveries were made that had a lasting influence on astronomy: quasars, CMB-radiation (Cosmic Microwave Background, 2.73 K) and pulsars, discoveries which led to Nobel prizes and paradigm shifts, and which transformed our understanding of the Universe. Using separate telescopes on intercontinental baselines angular resolutions of one-thousandth of an arcsecond (the size of a dinnerplate on the Moon) had been achieved by 1970. In the last 30 years the technical developments in radio astronomy have shifted to greater flexibility and versatility, improved sensitivity and higher image fidelity. And all of those over an increasingly broader radio frequency spectrum, from sub-mm to several meters wavelength.

ASTRON, and the Dutch radio astronomical community with it, have an outstanding international reputation in the construction, operation and exploitation of radio telescopes. They have held this position for 60 years, almost from the creation of the Netherlands Foundation for Radio Astronomy (NFRA, previously SRZM, now ASTRON) in 1949. This year the WSRT telescope will turn 40 years old and another decade-long period of front-line research is approaching, following the installation of focal-planearrays (FPA) in the telescope in 2012.

In June 2010, the LOFAR (Low Frequency Array) telescope was officially opened. This is a stepping stone for the SKA, the Square Kilometre Array , which will be the largest project ever in (radio-)astronomy with global support and corresponding governance structure.

New, larger and more complex radio telescopes bring new challenges. Foremost among these is the calibration of the data in order to remove the atmospheric and instrumental effects which corrupt the exceedingly faint signals from cosmic sources. Indeed, the scientific success of the new generation of radio telescopes will depend critically on the ability to calibrate the data, and to deliver 'thermal-noise-limited' performance. That is to say, the quality of the products of synthesis array telescopes - like images, spectra and temporal lightcurves of the objects that are visible in the wide fields of view - must justify the expenses made to build the enormous collecting area and infrastructure.

Aims of this project

At the end of the grant period, in 2013-2014, we hope to have demonstrated how to calibrate LOFAR to fundamental and signal processing limited levels. We also want to contribute to the training of the next generation of leading engineers/scientists that will be crucial for getting the most out of the next generation radio arrays.

Among the first astronomical applications to come out of the collaboration between the involved groups we hope will be the detection of the extremely faint signals from the Epoch of Reionization (the LOFAR EoR project).

Interferometry and calibration: a brief introduction and jargon

Interferometers measure the spatial coherence of the electric field from (cosmic) radio sources. The quantity measured by an interferometer is called the visibility. The Van Cittert - Zernike theorem relates these visibilities to the spatial brightness distribution on the sky through a Fourier transform. A distributed array of telescopes, or antennas, produces a large number of visibilities: the more the better. The aperture plane in which we measure the visibilities is called the u-v plane. Using a technique known as earth rotation aperture synthesis the WSRT samples the u-v plane with great regularity. The images that can be made from these data is distorted (corrupted) by atmospheric and instrumental effects, leading to imperfect images with artifacts or 'errors'. These errors are usually proportional to the intensity of the sources that are investigated.

The ratio between the peak brightness and the typical errors in an image is called the Dynamic Range. In 1980 the typical dynamic range in WSRT images, after a decade in which the WSRT had been the dominant synthesis array in the world, was between 100:1 and 1000:1. Calibration is supposed to remove the artifacts and lead to an error-free image which approaches the theoretical ('thermal') noise; the thermal noise is proportional to the ratio of the receiver noise and the total collecting area. It is also, under ideal conditions, going down as the inverse of the square root of the integration time. However, the stability of the atmosphere and the instrument are usually not good enough to achieve the thermal noise after a typical synthesis time of 12 hours. This all changed due to the 'invention' of selfcalibration around 1980. Selfcalibration uses the emission in the field itself to calibrate the data: hence the name 'selfcal'. It appeared to be an extraordinary successful technique. A special, WSRTspecific, variation on the theme of selfcalibration uses the redundancy in the array to reach extreme dynamic range images (the use of redundancy calibration is coming back again in the design of the LOFAR HBA-tiles station and could well become part of the SKA design).

In the 1990s, when telescopes had become more sensitive, it became clear that even selfcal was not capable of removing all artifacts from the data. Especially in those cases where very wide fields are imaged - with LOFAR at low frequencies, but also the WSRT already suffers from these effects - the effects of direction-dependent effects (DDEs) become obvious. DDEs violate the simple selfcal assumption that all corruptions can be assigned to a single telescope/antenna; this requires that all telescope beams are similar ('they should see the same sky'). Also the non-isoplanaticity of the ionospheric disturbances complicates matters. This has led to the concept of 'peeling' where effects towards bright sources are iteratively or jointly solved for and the sources 'peeled' from the data. A major mathematical development to describe these effects is the full polarization measurement Equation (ME) developed at ASTRON. It models the instrumental effects of a station as a product of 2x2 Jones matrices, in which the matrix elements are parametrized expressions. The implication of the presence of DDEs is that a large number of unknowns need to be solved for using a finite number of observables. This brings us into the area of parameter estimation and signal processing techniques. Approaches for joint calibration and deconvolution have been pioneered by Wijnholds for the LOFAR station calibration.

With improvements in calibration on the way, image deconvolution started to become a problem. Deconvolution is the process in which the instrumental response and the true source signals are disentangled. The traditional way of deconvolving images is using the CLEAN technique of Hogbom. Recently, several model based approaches using techniques from the field of signal processing have been proposed that nmay provide improved imaging results, especially in cases with which CLEAN is know to have difficulties. We propose a Ph.D. project to explore the application of several signal processing techniques to radio astronomical imaging problems to build the theory to make informed decisions on the choice of imaging and deconvolution routines.

Similar signal processing techniques can be used to calibrate the telescope. The development of self-calibration methods like the weighted alternating least squares method introduced by Wijnholds and Van der Veen, shows that the use of similar techniques for calibration and imaging may facilitation integration of these processes in a single self-calibration framework. Choices such as parameterization of the physical model and design of the array configuration can have tremendous impact on the availability of computationally efficient solutions.

PhD project: New approaches to radio astronomical image formation

Even modern radio interferometers subsample the spatial Fourier domain, so that unique image reconstruction is not possible without some further processing known as deconvolution. The deconvolution process uses some a-priori knowledge about the image to remove the effect of "dirty beam" side-lobes. One of the most frequently used methods for astronomical image deconvolution is CLEAN. This method is basically a sequential Least-Squares (LS) fitting procedure in which the brightest source location and power are estimated, and subsequently removed. This process continues until the residual image is noise-like. Over the years this method has been partially analyzed. However, full analysis is still lacking due to its iterative nature. Our initial work formulated the imaging problem as a parameter estimation problem. This allows the adoption of a range of techniques from array signal processing. The classical image formation then becomes a matched filter beamformer. However, it is known in array processing that for closely spaced sources this has a poor performance, and we proposed to insert the MVDR beamformer instead. This gave improved performance, but unfortunately MVDR does not preserve absolute power levels. Robust beamforming techniques take care of this. Along these lines, we have recently derived a better algorithm for parametric imaging, called the LS-MVI, which provides improved resolution over the classical CLEAN.

Methods like CLEAN and LS-MVI perform very well if the image consists of a number of discrete sources (but there may be some structure in these sources). The fact that these sources are discrete implies that not all points in the image contain actual astrophysical information, which suggests that there may be some (abstract) domain in which the image is sparse. This sparsity can possibly be exploited by compressive sampling techniques, which is a topic of growing interest in the field of signal processing. Reconstruction of the underlying signals is done via constrained l_1-optimization, rather than the more traditional l_2 (Least Squares) optimization. Progress in convex optimization has made this feasible. Can the ideas from compressive sensing, in particular l_1-optimization, be exploited in astronomical imaging?

It is known that l_1-optimization provides better results than l_2-optimization when the data is sparse. Sparsity implies that the image contains a number of discrete sources and is mostly empty. This may be a reasonable assumption for high-resolution survey maps in which most diffuse emission is resolved, but is certainly not true for maps made to study the properties of our own Galaxy and the EoR signal. These maps contain information in every pixel. This information may be extracted from the raw data by parameterizing the image as a grid of pixels and solving for them in the least squares sense. Although a general framework has been laid down, there are still many open issues concerning the application to actual data, such as an optimal choice for the grid of pixel, trade-offs between resolution and noise in the image and computational efficiency.

This work package has the following objectives:

• Investigate how ideas from compressive sampling can be used in astronomical imaging.
• Make a trade-off between l_1- and l_2-optimized imaging in different types of imaging problems.
• Demonstrate the performance in theory, simulations and on experimental data.