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.