Robust Compressive Phase Retrieval via L1 Minimization With Application to Image Reconstruction

TL;DR

This paper introduces an L1 minimization approach for compressive phase retrieval that leverages signal sparsity, providing a fast, accurate, and noise-robust method for real-valued image reconstruction from Fourier magnitude data.

Contribution

It formulates L1 minimization problems for compressive phase retrieval and develops alternating direction algorithms, demonstrating effectiveness for real-valued, nonnegative image reconstruction.

Findings

The method is fast and accurate.

It is robust to measurement noise.

Optimal solutions are achieved in noise-free cases.

Abstract

Phase retrieval refers to a classical nonconvex problem of recovering a signal from its Fourier magnitude measurements. Inspired by the compressed sensing technique, signal sparsity is exploited in recent studies of phase retrieval to reduce the required number of measurements, known as compressive phase retrieval (CPR). In this paper, l1 minimization problems are formulated for CPR to exploit the signal sparsity and alternating direction algorithms are presented for problem solving. For real-valued, nonnegative image reconstruction, the image of interest is shown to be an optimal solution of the formulated l1 minimization in the noise free case. Numerical simulations demonstrate that the proposed approach is fast, accurate and robust to measurements noises.