  We consider the problem of reconstructing a function $f\in L^2(\mathbb{R})$
given phase-less samples of its Gabor transform, which is defined by
$$\mathcal{G} f(x,\omega) := 2^{\frac14} \int_{\mathbb{R}} f(t) e^{-\pi
(t-x)^2} e^{-2\pi i y t}\,\mbox{d}t,\quad (x,y)\in\mathbb{R}^2.$$More
precisely, given sampling positions $\Omega\subseteq \mathbb{R}^2$ the task is
to reconstruct $f$ (up to global phase) from measurements $\{|\mathcal{G}
f(\omega)|: \,\omega\in\Omega\}$. This non-linear inverse problem is known to
suffer from severe ill-posedness. As for any other phase retrieval problem,
constructive recovery is a notoriously delicate affair due to the lack of
convexity. One of the fundamental insights in this line of research is that the
connectivity of the measurements is both necessary and sufficient for
reconstruction of phase information to be theoretically possible. In this
article we propose a reconstruction algorithm which is based on solving two
convex problems and, as such, amenable to numerical analysis. We show,
empirically as well as analytically, that the scheme accurately reconstructs
from noisy data within the connected regime.Moreover, to emphasize the
practicability of the algorithm we argue that both convex problems can actually
be reformulated as semi-definite programs for which efficient solvers are
readily available. The approach is based on ideas from complex analysis, Gabor
frame theory as well as matrix completion.
