Sparse interferometric Stokes imaging under the polarization constraint (Polarized SARA)

MNRAS 478, 4442–4463 (2018) doi:10.1093/mnras/sty1182 Sparse interferometric Stokes imaging under the polarization constraint (Polarized SARA) Jasleen Birdi,‹ Audrey Repetti and Yves Wiaux Institute of Sensors, Signals and Systems, Heriot-Watt University, Edinburgh EH14 4AS, UK Accepted 2018 April 25. Received 2018 March 29; in original form 2017 December 31 We develop a novel algorithm for sparse imaging of Stokes parameters in radio interferometry under the polarization constraint. The latter is a physical non-linear relation between the Stokes parameters, imposing the polarization intensity as 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 we 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 method is dubbed Polarized SARA. Using simulated observations of M87 with the Event Horizon Telescope, we demonstrate that imposing the polarization constraint leads to superior image quality. For the considered data sets, the results also indicate better performance of the average sparsity prior in comparison with the widely used Cotton–Schwab CLEAN algorithm and other total variation based priors for polarimetric imaging. Our MATLAB code is available online on GitHub. Key words: techniques: high angular resolution – techniques: image processing – techniques: interferometric – techniques: polarimetric. 1 I N T RO D U C T I O N The study of the polarized emissions from various astrophysical sources in the Universe provides invaluable information about the origin as well as the medium of propagation of these emissions. In many cases, these sources generate appreciable linearly polarized radiations and only negligible circularly polarized radiations. Thus, the study of linearly polarized emissions is of particular interest. These radiations can be generated, for instance, due to the synchrotron emission from the electrons in high-energy objects (Ginzburg & Syrovatskii 1965). Analysis of these polarized emissions gives insight into the strength and orientation of the magnetic field in the sources. Moreover, while traversing, the interaction with the magnetized plasma along the line of sight to the source can modify the polarization state of these radiations via processes such as Faraday rotation (Pacholczyk 1970; Simard-Normandin, Kronberg & Button 1981). As a result, the polarized emissions also characterize the magnetic field distributions of these plasmas (Dreher, Carilli & Perley 1987; Brentjens & De Bruyn 2005). This all indicates the importance of imaging these polarized emissions, which is referred to as polarimetric imaging.  E-mail: In the context of polarimetric imaging for radio interferometry (RI), the intensity distribution of the sky image of interest is characterized by the Stokes parameters, I, Q, U and V, which are all real valued. While I represents the total intensity of the radio emissions, Q, U and V describe the polarization state of the electromagnetic radiations coming from the target area of the sky. In particular, Q and U refer to the linear polarization, and V denotes the circular polarization. Furthermore, the linear polarization image P is given by P = Q + i U. The magnitude of this complex valued quantity provides the linear polarization intensity, while the electric vector polarization angle (EVPA) can be obtained from its phase. Importantly, the Stokes parameters are not completely independent but are constrained by a physical non-linear relation imposing that the polarization intensity is a lower bound on the total intensity:  Q2 + U 2 + V 2 ≤ I . We can also see this constraint, namely the polarization constraint, as the generalization of the more simple positivity constraint on the intensity image in the context of unpolarized imaging. In order to produce linear polarimetric images at very high angular resolutions, one of the possibilities is to leverage the technique of very long baseline interferometry (VLBI; Roberts, Wardle & Brown 1994). VLBI consists of a collection of radio antennas, spread all across the Earth or even in space (space VLBI), with the aim of producing images of the target sources in the sky at very high angular resolutions. Very recently, the Event Horizon Tele-  C 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society ABSTRACT Joint Stokes imaging for RI 1 http://eventhorizontelescope.org Carrillo et al. (2014), Garsden et al. (2015) and Onose et al. (2016), to name a few. In particular, these techniques reconstruct the image of interest by leveraging the sparsity of the sought image, either in the image domain or in a transformed domain. Though applied only for Stokes I image reconstruction, the quality of reconstruction obtained by these techniques has been shown to outperform that obtained by CLEAN on simulated as well as on a few real data sets (Carrillo, McEwen & Wiaux 2014; Onose et al. 2016; Pratley et al. 2016; Onose, Dabbech & Wiaux 2017; Dabbech et al. 2017a). Very recently, the first application of these sparsity regularized methods for polarimetric imaging has been developed by Akiyama et al. (2017a). In this case, the authors promote the sparsity of the underlying images using the 1 norm along with the total variation (TV) regularization (Rudin, Osher & Fatemi 1992; Chambolle & Lions 1997), and they solve the resultant problem using a monotonic version of fast iterative shrinkage/thresholding algorithm (FISTA; Beck & Teboulle 2009a,b). The authors validate their technique on simulated EHT data and obtain super-resolved Stokes images. The resolution of the reconstructed images is much higher than that obtained by CLEAN. However, similar to CLEAN, this sparsity-based approach also solves independently for the Stokes images. In practice, the Stokes images are physically linked. As previously discussed, due to the polarization constraint, the intensity in each pixel of the total intensity image cannot be smaller than the corresponding polarization intensity. Nevertheless, to the best of our knowledge, none of the previously mentioned methods takes this constraint explicitly into account. It is worth mentioning then that in the absence of this constraint, non-physical reconstructions may be produced. One way to reconstruct the images with physical meaning is to make use of maximum entropy methods (MEMs). These methods aim to find an image that maximizes the entropy function while being consistent with the acquired data (Cornwell & Evans 1985). As an extension to polarimetric MEM, a special entropy function incorporating this polarization constraint is used (Narayan & Nityananda 1986; Holdaway & Wardle 1990; Coughlan (...truncated)