# Measurement Matrix Design for Phase Retrieval Based on Mutual Information

## TL;DR

This paper investigates the design of deterministic measurement matrices for phase retrieval by maximizing mutual information, providing theoretical insights and practical algorithms that outperform random Gaussian matrices in noisy conditions.

## Contribution

It introduces a mutual information-based framework for designing measurement matrices, including optimal solutions in low SNR and methods for structured measurements.

## Key findings

- Optimal matrices derived in low SNR regime
- Proposed design methods outperform random Gaussian matrices
- Simulation results show improved phase recovery performance

## Abstract

In phase retrieval problems, a signal of interest (SOI) is reconstructed based on the magnitude of a linear transformation of the SOI observed with additive noise. The linear transform is typically referred to as a measurement matrix. Many works on phase retrieval assume that the measurement matrix is a random Gaussian matrix, which, in the noiseless scenario with sufficiently many measurements, guarantees invertability of the transformation between the SOI and the observations, up to an inherent phase ambiguity. However, in many practical applications, the measurement matrix corresponds to an underlying physical setup, and is therefore deterministic, possibly with structural constraints. In this work we study the design of deterministic measurement matrices, based on maximizing the mutual information between the SOI and the observations. We characterize necessary conditions for the optimality of a measurement matrix, and analytically obtain the optimal matrix in the low signal-to-noise ratio regime. Practical methods for designing general measurement matrices and masked Fourier measurements are proposed. Simulation tests demonstrate the performance gain achieved by the proposed techniques compared to random Gaussian measurements for various phase recovery algorithms.