We develop a novel algorithm for sparse Stokes parameters imaging in radio interferometry under the polarization constraint. The latter is a physical non-linear relation between the Stokes parameters, imposing that the polarization intensity is a lower bound on the total intensity. To solve the joint inverse Stokes imaging problem including this bound, we leverage epigraphical projection techniques in convex optimization and design a primal-dual method offering a highly flexible and parallelizable structure. In addition, we propose to regularize each Stokes parameter map through an average sparsity prior in the context of a reweighted analysis approach (SARA). The resulting approach is dubbed Polarized SARA. We demonstrate on simulated observations of M87 with the Event Horizon Telescope that imposing the polarization constraint leads to superior image quality. The results also confirm that the performance of the average sparsity prior surpasses the alternative state-of-the-art priors for polarimetric imaging.

Radio interferometric imaging aims to estimate an unknown sky intensity image from degraded observations, acquired through an antenna array. In the theoretical case of a perfectly calibrated array, it has been shown that solving the corresponding imaging problem by iterative algorithms based on convex optimization and compressive sensing theory can be competitive with classical algorithms such as CLEAN. However, in practice, antenna-based gains are unknown and have to be calibrated. Future radio telescopes, such as the SKA, aim at improving imaging resolution and sensitivity by orders of magnitude. At this precision level, the direction-dependency of the gains must be accounted for, and radio interferometric imaging can be understood as a blind deconvolution problem. In this context, the underlying minimization problem is non-convex, and adapted techniques have to be designed. In this work, leveraging recent developments in non-convex optimization, we propose the first joint calibration and imaging method in radio interferometry, with proven convergence guarantees. Our approach, based on a block-coordinate forward-backward algorithm, jointly accounts for visibilities and suitable priors on both the image and the direction-dependent effects (DDEs). As demonstrated in recent works, sparsity remains the prior of choice for the image, while DDEs are modelled as smooth functions of the sky, i.e. spatially band-limited. Finally, we show through simulations the efficiency of our method, for the reconstruction of both images of point sources and complex extended sources. MATLAB code is available on GitHub.

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New generation of radio interferometers are envisaged to produce high quality, high dynamic range Stokes images of the observed sky from the corresponding undersampled Fourier domain measurements. In practice, these measurements are contaminated by the instrumental and atmospheric effects that are well represented by Jones matrices, and are most often varying with observation direction and time. These effects, usually unknown, act as a limiting factor in achieving the required imaging performance and thus, their calibration is crucial. To address this issue, we develop a global algorithm, named Polca SARA, aiming to perform full polarization, direction-dependent calibration, and sparse imaging by employing a non-convex optimization technique. In contrast with the existing approaches, the proposed method offers global convergence guarantees and flexibility to incorporate sophisticated priors to regularize the imaging as well as the calibration problem. Thus, we adapt a polarimetric imaging specific method, enforcing the physical polarization constraint along with a sparsity prior for the sought images. We perform extensive simulation studies of the proposed algorithm. The results indicate the superior performance of polarization constraint based imaging when combined with the calibration of the direction-dependent effects for full Jones matrices, including their off-diagonal terms (denoting polarization leakage). The chosen priors in the proposed approach are also shown to handle the unitary ambiguity problem to a good extent.

We examine the nature of the reconstructed 3D image as obtained by replay (or back-propagation) of the object wave from the hologram recording plane to the original object volume. While recording of a hologram involves transferring information from a 3D volume to a 2D detector, the replay of the hologram involves creating information in a set of 3D voxels from a much smaller number of 2D detector pixels, which on a first look appears to be surprising. We point out that the hologram replay process is a Hermitian transpose (and not inverse) of the hologram formation process and therefore only provides an approximation to the original 3D object function. With the knowledge of this Hermitian transpose property, we show how one may realize true 3D image reconstruction via a regularized optimization algorithm. The numerical illustrations of this optimization approach as presented here show excellent slice-by-slice tomographic 3D reconstruction of the original object under the weak scattering approximation. In particular, the reconstructed 3D image field has near-zero numerical values at voxels where the original object did not exist. We note that 3D image reconstruction of this kind cannot be achieved by the traditional physical replay process. In this sense, the proposed methodology for digital holographic image reconstruction goes beyond numerically mimicking the physical process involved in traditional film based holographic replay. The reconstruction approach may find potential applications in a number of digital holographic imaging systems.