Quantization-Aware Phase Retrieval

We address the problem of phase retrieval (PR) from quantized measurements. The goal is to reconstruct a signal from quadratic measurements encoded with a finite precision, which is indeed the case in practical applications. We develop an iterative projected-gradient-type algorithm that recovers the signal subject to ensuring consistency with the measurement, meaning that the recovered signal, when encoded, must yield the same set of measurements that one started with. The algorithm involves rank-1 projection, which stems from the idea of lifting, originally proposed in the context of PhaseLift. The consistency criterion is enforced using a one-sided quadratic cost. We also determine the probability with which different vectors lead to the same set of quantized measurements, which makes it impossible to resolve them. Naturally, this probability depends on how correlated such vectors are, and how coarsely/finely the measurements are quantized. The proposed algorithm is also capable of incorporating a sparsity constraint on the signal. An analysis of the cost function reveals that it is bounded probabilistically, both above and below, by functions that are dependent on how well-correlated the estimate is with the ground-truth. We also derive the Cramér-Rao lower bound (CRB) on the achievable reconstruction accuracy. A comparison with the state-of-the-art algorithms shows that the proposed algorithm has a higher reconstruction accuracy and is about 2 to 3[Formula: see text]dB away from the CRB. The edge, in terms of the reconstruction signal-to-noise ratio, over the competing algorithms is higher (about 5 to 6[Formula: see text]dB) when the quantization is coarse, thereby making the proposed scheme particularly attractive in such scenarios. We also demonstrate a concrete application of the proposed method to frequency-domain optical-coherence tomography (FDOCT).