Fannjiang

Compressed Sensing in Astronomical Imaging

Background

Interferometry is the definitive imaging tool of radio astronomy, allowing us to image finely structured radio sources such as galaxies, nebulas, and supernova remnants by using an array of many antennas to emulate a single lens. The aperture of the array is the greatest pairwise distance between antennas—which, at tens of kilometers for interferometers like the Very Large Array (VLA) in New Mexico, gives us far higher imaging resolution than a single lens could. Radio interferometry works by measuring the visibility function, or the interference fringes of the radio signal, at every pair of antennas in the array. The van Cittert-Zernike theorem gives the visibility function, as measured by one pair of antennas in the array, over the viewing window of the sky P as

$V(𝐛)=\underset{𝑃}{\int }I(𝐩)e^{-2\pi i𝐛\cdot 𝐩}𝑑𝐩.$,

where :<math> \mathbf{b} <\math> is the displacement vector between the two antennas, called the baseline, and :<math> I(\mathbf{p}) <\math> is the intensity of the radiation from direction :<math> \mathbf{p}<\math>.

In effect, the array samples the two-dimensional Fourier transform of the spatial intensity distribution :<math> I(\mathbf{p}) <\math> of the radio source. Ideally, if we thoroughly sample the Fourier plane, we can invert the transform to reconstruct :<math> I(\mathbf{p}) <\math>, the image of the radio source. However, the data acquisition quickly gets expensive, as we need to capture one Fourier coefficient per desired pixel in the image. The need to capture so much data has motivated a new generation of ambitious interferometers, including the Atacama Large Millimetre/submillimetre Array (ALMA), which will use 66 antennas, and the Square Kilometre Array (SKA), which will use several thousand antennas to probe the Fourier plane. Meanwhile, smaller interferometers like the 27-antenna VLA often sample over a period of time, allowing the rotation of the earth to synthesize new baselines between pairs of antennas.

Despite such efforts, there are always irregular holes on the Fourier plane where sampling of the visibility function is thin or simply nonexistent. This data deficiency is currently managed by interpolating or filling in zeros for unknown visibility values, and applying deconvolution algorithms such as CLEAN to the resulting “dirty images”. However, it may not be necessary to collect the extensive data set in the first place.

Among imaging's most promising developments in recent years is the theory of compressed sensing (CS), which has shown that the information of a signal can be preserved even when sampling does not fulfill the fundamental Nyquist rate (Donoho 2006; Candès et al. 2006a,b). The theory revolves around a priori knowledge that the signal is sparse or compressible in some basis, in which case its information naturally resides in a relatively small number of coefficients. Instead of directly sampling the signal, whereby full sampling would be inevitable in finding every non-zero or significant coefficient, CS allows us to compute just a few inner products of the signal along measurement vectors of certain favorable characteristics. The novelty of CS over image compression is that it takes advantage of signal compressibility to alleviate data acquisition, not just data storage.

It is well established that objects typical of astronomic study are sparse or compressible (Polygiannakis 2003; Dollet 2004); indeed, they are often sparse in the natural pixel basis. Recent studies have recognized this innate agreeability with CS, examining its potential in radio interferometry compared to traditional deconvolution methods (Wiaux et al. 2009; Li et. al. 2011).