  Phase retrieval consists in the recovery of an unknown signal from phaseless
measurements of its usually complex-valued Fourier transform. Without further
assumptions, this problem is notorious to be severe ill posed such that the
recovery of the true signal is nearly impossible. In certain applications like
crystallography, speckle imaging in astronomy, or blind channel estimation in
communications, the unknown signal has a specific, sparse structure. In this
paper, we exploit these sparse structure to recover the unknown signal uniquely
up to inevitable ambiguities as global phase shifts, transitions, and
conjugated reflections. Although using a constructive proof essentially based
on Prony's method, our focus lies on the derivation of a recovery guarantee for
multivariate signals using an adaptive sampling scheme. Instead of sampling the
entire multivariate Fourier intensity, we only employ Fourier samples along
certain adaptively chosen lines. For bivariate signals, an analogous result can
be established for samples in generic directions. The number of samples here
scales quadratically to the sparsity level of the unknown signal.
