Fourier Phase Retrieval with Extended Support Estimation via Deep Neural Network
This work addresses the problem of efficiently recovering sparse signals from Fourier magnitudes, which is important for applications like imaging and signal processing, though it is incremental as it builds on existing support estimation techniques.
The paper tackles sparse phase retrieval from Fourier magnitudes by using a deep neural network to estimate an extended support set that contains the true support, then applying hard thresholding to recover the signal. This approach achieves superior performance with lower complexity compared to existing methods like greedy sparse phase retrieval and a state-of-the-art Fienup variant.
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.