Robust Phase Retrieval from Quantized and Noisy Measurements

The phase retrieval (PR) problem involves recovering a signal from amplitude-only measurements and has been extensively studied in various physical measurement and signal processing systems. Recently, there has been increasing interest in the quantized phase retrieval (QPR) problem, where the observed amplitude is quantized to binary values. The existing studies have focused on the noiseless case. In this paper, we address a robust QPR problem that accounts for both quantization and noise. We formulate the robust QPR problem using maximum-likelihood estimation (MLE), which results in a non-convex and challenging optimization problem due to the complexity of its likelihood function. To obtain a high-quality solution, we develop an expectation-maximization (EM) algorithm combined with the Wirtinger flow, allowing us to convert the problem into a more manageable form at each iteration. Our method is implemented on both randomly generated synthetic data and image reconstruction tasks. The results demonstrate that our approach significantly enhances signal recovery performance under the combined effects of quantization and noise. Additionally, the proposed method exhibits better runtime efficiency compared to existing approaches.