Using mutual information for designing the measurement matrix in phase retrieval problems

In the phase retrieval problem, the observations consist of the magnitude of a linear transformation of the signal of interest (SOI) with additive noise, where the linear transformation is typically referred to as measurement matrix. The objective is then to reconstruct the SOI from the observations up to an inherent phase ambiguity. 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 uniqueness of the mapping between the SOI and the observations. However, in many applications, e.g., optical imaging, the measurement matrix corresponds to the underlying physical setup, and is therefore a deterministic matrix with structure constraints. In this work we study the design of deterministic measurement matrices, aimed at maximizing the mutual information between the SOI and the observations. We characterize necessary conditions for the optimal measurement matrix, and propose a practical design method for measurement matrices corresponding to masked Fourier measurements. Simulation tests of the proposed method show that it achieves the same performance as random Gaussian matrices for various phase recovery algorithms.