Fourier Phase Retrieval With Arbitrary Reference Signal

Fourier phase retrieval problem aims at recovering a signal from its Fourier amplitude measurements. A good initialization and prior information about the sparsity or support of the target signal is critical for robust recovery. Holographic phase retrieval is a related problem in which the presence of a reference signal makes the signal recovery problem linear. The existing methods, however, only work if the support of the reference and target signals are sufficiently separated. In this paper, we present a Fourier phase retrieval algorithm in the presence of a known (reference) signal at arbitrary location in the scene. We assume that a small part of the target signal is known without any other assumption about the support and separation of the known and unknown parts of the signal. Our recovery algorithm is based on alternating minimization and gradient descent. We demonstrate that our proposed method significantly improves the Fourier phase retrieval for natural images and synthetic images with multiple objects.