Phase Retrieval via Matrix Completion

TL;DR

This paper introduces a new convex optimization framework for phase retrieval that uses structured illuminations and matrix completion techniques, demonstrating stable and unique recovery of complex objects from diffraction data.

Contribution

It combines structured illumination with convex programming for phase retrieval, providing theoretical guarantees and empirical stability in noisy conditions.

Findings

Complex objects recoverable from few diffraction patterns

Noise-aware algorithms degrade gracefully with noise

Structured illumination patterns ensure unique phase determination

Abstract

This paper develops a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that any complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we introduce some theory showing that one can design very simple structured illumination patterns such that three diffracted figures uniquely determine the phase of the object we wish to recover.