Consensus and Sectioning-Based ADMM With Norm-1 Regularization for Imaging With a Compressive Reflector Antenna

This paper presents three distributed techniques to find a sparse solution of the under-determined linear problem ${\bf g}={\bf Hu}$ with a norm-1 regularization, based on the Alternating Direction Method of Multipliers (ADMM). Each one of these techniques divide the matrix ${\bf H}$ into submatrices by rows, columns, or both rows and columns, leading to the so-called consensus-based ADMM, sectioning-based ADMM, and consensus and sectioning-based ADMM, respectively. They are validated for a particular millimeter-wave imaging problem based on the use of a Compressive Reflector Antenna (CRA). The CRA is a hardware designed to increase the sensing capacity of an imaging system and reduce the mutual information among measurements, allowing an effective imaging of sparse targets with the use of Compressive Sensing (CS) techniques. In previous works, the consensus-based ADMM has been proved to accelerate the imaging process, and the sectioning-based ADMM has shown the ability to potentially reduce the amount of information to be exchanged among the computational nodes, based on the system configuration. In this paper, the mathematical formulation and graphical interpretation of these two techniques, together with the consensus and sectioning-based ADMM approach, are presented. The imaging quality, the imaging time, the convergence, the communication efficiency among the computational nodes, and the computational complexity are analyzed and compared. The distributed capabilities of the ADMM-based approaches, together with the high sensing capacity of the CRA, allow the imaging of metallic targets in a 3D domain in quasi-real time with a reduced amount of information exchanged among the nodes.