Measurement Matrix Design for Phase Retrieval Based on Mutual Information

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.

arXiv commentThis paper was presented in part at the 2017 International Symposium on Information Theory

References in corpus (8)

• Phase Retrieval via Wirtinger Flow: Theory and Algorithms
• Reference-less measurement of the transmission matrix of a highly scattering material using a DMD and phase retrieval techniques
• Quantum Tomography under Prior Information
• STFT Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms
• Phase Retrieval from 1D Fourier Measurements: Convexity, Uniqueness, and Algorithms
• Communications-Inspired Projection Design with Application to Compressive Sensing
• Phase Recovery, MaxCut and Complex Semidefinite Programming
• Intensity-only optical compressive imaging using a multiply scattering material and a double phase retrieval approach

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