Frequency subspace amplitude flow for phase retrieval

A novel approach, termed frequency subspace amplitude flow (FSAF), is proposed to reconstruct complex-valued signal from “phaseless” measurements. The proposed FSAF consists of two stages: the first stage approximates low-frequency coefficients of an unknown signal by the spectral method, and the second stage refines the results by the truncated conjugate gradient of amplitude-based nonconvex formulation. FSAF is easy to implement and applicable to natural images, where no additional constraint is needed. Extensive experiments with 1D signals, 2D images, and natural images corroborate significant improvements by using the proposed FSAF method over the state of the art. Especially for sample complexity, FSAF pushes the state of the art for exactly reconstructing complex natural signals (with a size of n) from 3.2n to 2.2n under the Gaussian model, and from 5n to 3n under the coherent diffraction pattern (CDP) model without increasing computational complexity. More importantly, the proposed method is highly flexible and can be easily adapted to the existing algorithms under different noise models.