Multiple illumination phaseless super-resolution (MIPS) with applications to phaseless DoA estimation and diffraction imaging

Phaseless super-resolution is the problem of recovering an unknown signal from measurements of the "magnitudes" of the "low frequency" Fourier transform of the signal. This problem arises in applications where measuring the phase, and making high-frequency measurements, are either too costly or altogether infeasible. The problem is especially challenging because it combines the difficult problems of phase retrieval and classical super-resolution. Recently, the authors in [1] demonstrated that by making three phaseless low-frequency measurements, obtained by appropriately "masking" the signal, one can uniquely and robustly identify the phase using convex programming and obtain the same super-resolution performance reported in [2]. However, the masks proposed in [1] are very specific and in many applications cannot be directly implemented. In this paper, we broadly extend the class of masks that can be used to recover the phase and show how their effect can be emulated in coherent diffraction imaging using multiple illuminations, as well as in direction-of-arrival (DoA) estimation using multiple sources to excite the environment. We provide numerical simulations to demonstrate the efficacy of the method and approach.