Robust Sparse Phase Retrieval from Differential Measurements Using Reweighted $L_{1}$ Minimization

We consider the problem of sparse Fourier phase retrieval where the goal is to recover a sparse signal from the magnitudes of its Fourier transform. In our earlier work, it was shown that using a certain differential measurement model, it is possible to perform sparse Fourier phase retrieval with near-minimal sample complexity. Such measurements are realizable using interferometric imaging techniques and can remove the ambiguities inherent in Fourier phase retrieval in the noiseless setting. However, in presence of noise, such a differential measurement model can potentially lead to noise amplification and degrade the performance as sparsity increases. To address this issue, we propose to use a reweighted $l_{1}$ minimization algorithm in conjunction with our differential measurement model. Reweighted $l_{1}$ minimization has been successfully adopted in compressed sensing where it shows good performance in presence of large sparsity. We demonstrate a sparse phase retrieval algorithm based on reweighted $l_{1}$ minimization which shows superior performance in low Signal to Noise ratio (SNR) and high sparsity regime. 11This work was supported in parts by NSF NCS-FO 1734940, and by the University of California, San Diego