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In a variety of fields, in particular those involving imaging and optics, weoften measure signals whose phase is missing or has been irremediablydistorted. Phase retrieval attempts the recovery of the phase information of asignal from the magnitude of its Fourier transform to enable the reconstructionof the original signal. A fundamental question then is: "Under which conditionscan we uniquely recover the signal of interest from its measured magnitudes?"In this paper, we assume the measured signal to be sparse. This is a naturalassumption in many applications, such as X-ray crystallography, speckle imagingand blind channel estimation. In this work, we derive a sufficient conditionfor the uniqueness of the solution of the phase retrieval (PR) problem for bothdiscrete and continuous domains, and for one and multi-dimensional domains.More precisely, we show that there is a strong connection between PR and theturnpike problem, a classic combinatorial problem. We also prove that theexistence of collisions in the autocorrelation function of the signal maypreclude the uniqueness of the solution of PR. Then, assuming the absence ofcollisions, we prove that the solution is almost surely unique on 1-dimensionaldomains. Finally, we extend this result to multi-dimensional signals by solvinga set of 1-dimensional problems. We show that the solution of themulti-dimensional problem is unique when the autocorrelation function has nocollisions, significantly improving upon a previously known result.

Phase Retrieval for Sparse Signals: Uniqueness Conditions