Sparse Phase Retrieval for Phaseless Fourier Measurement Based on Riemannian Optimization

Given the inherent challenges of measuring phase in numerous scenarios, Phase Retrieval (PR)—the task of reconstructing the original signal from phaseless measurements—stands as paramount. The absence of phase often renders prior knowledge about the signal and the structure of phaseless measurements crucial for effective solutions. This paper tackles the Fourier Transform (FT) PR problem for sparse signals. We recast the FT PR as a novel optimization problem on the Riemannian manifold by leveraging the sparsity and structural properties of the measurement. Then, an effective iterative algorithm is developed to address this problem using Riemannian optimization techniques. Numerical simulations validate the effectiveness of the proposed algorithm and demonstrate its superior accuracy compared to the existing methods.