Using Convex Optimization of Autocorrelation with Constrained Support and Windowing for Improved Phase Retrieval Accuracy

TL;DR

This paper introduces a convex optimization approach with support constraints and windowing techniques to improve phase retrieval accuracy in diffraction imaging, especially for noisy and sparse data.

Contribution

It presents a novel convex formulation for phase retrieval that relaxes support constraints and incorporates windowing, enhancing reconstruction quality over traditional non-convex methods.

Findings

Improved visual quality of reconstructed images.

Significant reduction in crystallographic R-factor from 0.4 to 0.1.

Effective handling of noisy and sparse diffraction data.

Abstract

In imaging modalities recording diffraction data, the original image can be reconstructed assuming known phases. When phases are unknown, oversampling and a constraint on the support region in the original object can be used to solve a non-convex optimization problem. Such schemes are ill-suited to find the optimum solution for sparse data, since the recorded image does not correspond exactly to the original wave function. We construct a convex optimization problem using a relaxed support constraint and a maximum-likelihood treatment of the recorded data as a sample from the underlying wave function. We also stress the need to use relevant windowing techniques to account for the sampled pattern being finite. On simulated data, we demonstrate the benefits of our approach in terms of visual quality and an improvement in the crystallographic R-factor from .4 to .1 for highly noisy data.