  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 to recover the phase information of a
signal from the magnitude of its Fourier transform to enable the reconstruction
of the original signal. Solving the phase retrieval problem is equivalent to
recovering a signal from its auto-correlation function. In this paper, we
assume the original signal to be sparse; this is a natural assumption in many
applications, such as X-ray crystallography, speckle imaging and blind channel
estimation. We propose an algorithm that resolves the phase retrieval problem
in three stages: i) we leverage the finite rate of innovation sampling theory
to super-resolve the auto-correlation function from a limited number of
samples, ii) we design a greedy algorithm that identifies the locations of a
sparse solution given the super-resolved auto-correlation function, iii) we
recover the amplitudes of the atoms given their locations and the measured
auto-correlation function. Unlike traditional approaches that recover a
discrete approximation of the underlying signal, our algorithm estimates the
signal on a continuous domain, which makes it the first of its kind.
  Along with the algorithm, we derive its performance bound with a theoretical
analysis and propose a set of enhancements to improve its computational
complexity and noise resilience. Finally, we demonstrate the benefits of the
proposed method via a comparison against Charge Flipping, a notable algorithm
in crystallography.
