Compressive Radio-Interferometric Sensing With Random Beamforming as Rank-One Signal Covariance Projections

Radio-interferometry (RI) observes the sky at unprecedented angular resolutions, enabling the study of several far-away galactic objects such as galaxies and black holes. In RI, an array of antennas probes cosmic signals coming from the observed region of the sky. The covariance matrix of the vector gathering all these antenna measurements offers, by leveraging the Van Cittert-Zernike theorem, an incomplete and noisy Fourier sensing of the image of interest. The number of noisy Fourier measurements—or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">visibilities</i>—scales as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(Q^{2}B)$</tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Q$</tex-math></inline-formula> antennas and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$B$</tex-math></inline-formula> short-time integration (STI) intervals. We address the challenges posed by this vast volume of data, which is anticipated to increase significantly with the advent of large antenna arrays, by proposing a compressive sensing technique applied directly at the level of the antenna measurements. First, this paper shows that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">beamforming</i>—a common technique of dephasing antenna signals—usually used to focus some region of the sky, is equivalent to sensing a rank-one projection (ROP) of the signal covariance matrix. We build upon our recent work [1] to propose a compressive sensing scheme relying on random beamforming, trading the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Q^{2}$</tex-math></inline-formula>-dependence of the data size for a smaller number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P$</tex-math></inline-formula> of ROPs. We provide image recovery guarantees for sparse image reconstruction. Secondly, the data size is made independent of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$B$</tex-math></inline-formula> by applying <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> random modulations of the ROP vectors obtained for the STI. The resulting sample complexities, theoretically derived in a simpler case without modulations and numerically obtained in phase transition diagrams, are shown to scale as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(K)$</tex-math></inline-formula> where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> is the image sparsity. This illustrates the potential of the approach.