## Thursday, February 06, 2014

### Phase Retrieval for Sparse Signals: Uniqueness Conditions

Robin Scheibler also mentioned to me that [his] colleague Juri Ranieri [had a new preprint out]: Phase Retrieval for Sparse Signals: Uniqueness Conditions (http://arxiv.org/abs/1308.3058). He then goes on to describe it:

"The following paper studies the well-known problem of phase retrieval under interesting and novel assumption. In fact, while most of the literature focused on the discrete case, where the signal of interest is a vector, here the authors obtain uniqueness conditions for the continuous sparse case. More precisely, they assume that the original signal is a linear combinations of Dirac's deltas and they study when the reconstruction is unique. It turns out that the uniqueness of the reconstruction for the 1D case depends on the support of the autocorrelation function of the signal. Under similar conditions, they show that the phase retrieval of 2+D signals is always unique."
Thanks Robin.

In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts the recovery of the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. A fundamental question then is: "Under which conditions can 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 natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. In this work, we derive a sufficient condition for the uniqueness of the solution of the phase retrieval (PR) problem for both discrete and continuous domains, and for one and multi-dimensional domains. More precisely, we show that there is a strong connection between PR and the turnpike problem, a classic combinatorial problem. We also prove that the existence of collisions in the autocorrelation function of the signal may preclude the uniqueness of the solution of PR. Then, assuming the absence of collisions, we prove that the solution is almost surely unique on 1-dimensional domains. Finally, we extend this result to multi-dimensional signals by solving a set of 1-dimensional problems. We show that the solution of the multi-dimensional problem is unique when the autocorrelation function has no collisions, significantly improving upon a previously known result.

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