  The recovery of a signal from the magnitude of its Fourier transform, also
known as phase retrieval, is of fundamental importance in many scientific
fields. It is well known that due to the loss of Fourier phase the problem in
1D is ill-posed. Without further constraints, there is no unique solution to
the problem. In contrast, uniqueness up to trivial ambiguities very often
exists in higher dimensions, with mild constraints on the input. In this paper
we focus on the 2D phase retrieval problem and provide insight into this
uniqueness property by exploring the connection between the 2D and 1D
formulations. In particular, we show that 2D phase retrieval can be cast as a
1D problem with additional constraints, which limit the solution space. We then
prove that only one additional constraint is sufficient to reduce the many
feasible solutions in the 1D setting to a unique solution for almost all
signals. These results allow to obtain an analytical approach (with
combinatorial complexity) to solve the 2D phase retrieval problem when it is
unique.
