Solving Fourier Phase Retrieval with a Reference Image as a Sequence of Linear Inverse Problems

Fourier phase retrieval problem is equivalent to the recovery of a two-dimensional image from its autocorrelation measurements. This problem is generally nonlinear and nonconvex. Good initialization and prior information about the support or sparsity of the target image are often critical for a robust recovery. In this paper, we show that the presence of a known reference image can help us solve the nonlinear phase retrieval problem as a sequence of small linear inverse problems. Instead of recovering the entire image at once, our sequential method recovers a small number of rows or columns by solving a linear deconvolution problem at every step. Existing methods for the reference-based (holographic) phase retrieval either assume that the reference and target images are sufficiently separated so that the recovery problem is linear or recover the image via nonlinear optimization. In contrast, our proposed method does not require the separation condition. We performed an extensive set of simulations to demonstrate that our proposed method can successfully recover images from autocorrelation data under different settings of reference placement and noise.