Coded Aperture Design for Super-Resolution Phase Retrieval

Phase retrieval is an inverse problem which consists on estimating a complex signal from intensity-only measurements. Recent works have studied the problem of retrieving the phase of a high-resolution image from low-resolution phaseless measurements, under a setup that records coded diffraction patterns. However, the attainable resolution of the image depends on the sensor characteristics, whose cost increases in proportion to the resolution. Also, this methodology lacks theoretical analysis. Hence, this work derives a super-resolution model from low-resolution coded phaseless measurements, that in contrast with prior contributions, the attainable resolution of the image directly depends on the resolution of the coded aperture. For this model we establish that an image can be recovered (up to a global unimodular constant) with high probability. Also, the theoretical result states that the image reconstruction quality directly depends on the design of the coded aperture. Therefore, a strategy that designs the spatial distribution of the coded aperture is developed. Simulation results show that reconstruction quality using designed coded aperture is higher than the non-designed ensembles.