Fourier Phase Retrieval With Extended Support Estimation via Deep Neural Network

We consider the problem of sparse phase retrieval from Fourier transform magnitudes to recover the
$k$
-sparse signal vector and its support
$\mathcal {T}$
. We exploit extended support estimate
$\mathcal {E}$
with size larger than
$k$
satisfying
$\mathcal {E} \supseteq \mathcal {T}$
and obtained by a trained deep neural network (DNN). To make the DNN learnable, it provides
$\mathcal {E}$
as the union of equivalent solutions of
$\mathcal {T}$
by utilizing modulo Fourier invariances. Set
$\mathcal {E}$
can be estimated with short running time via the DNN, and support
$\mathcal {T}$
can be determined from the DNN output rather than from the full index set by applying hard thresholding to
$\mathcal {E}$
. Thus, the DNN-based extended support estimation improves the reconstruction performance of the signal with a low complexity burden dependent on
$k$
. Numerical results verify that the proposed scheme has a superior performance with lower complexity compared to local search-based greedy sparse phase retrieval and a state-of-the-art variant of the Fienup method.