Phase Retrieval With Background Information: Decreased References and Efficient Methods

Fourier phase retrieval (PR) is a severely ill-posed inverse problem that arises in various applications. To guarantee a unique solution and relieve the dependence on the initialization, background information can be exploited as a structural prior. However, the requirement for the background information may be challenging when moving to high-resolution imaging. At the same time, the previously proposed projected gradient descent (PGD) method also demands much background information. In this paper, we present an improved theoretical result about the demand for the background information, along with two Douglas Rachford (DR) based methods. Analytically, we demonstrate that the background information required to ensure a unique solution can be decreased by nearly <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1/2$ </tex-math></inline-formula> for the 2-D signals compared to the 1-D signals. By generalizing the results into d-dimension, we show that the length of the background information more than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\left ({{2^{\frac {d+1}{d}}-1}}\right)$ </tex-math></inline-formula> folds of the signal is sufficient to ensure uniqueness. At the same time, we also analyze the stability and robustness of the model when the measurements and background information are corrupted by noise. Furthermore, two methods called Background Douglas Rachford (BDR) and Convex Background Douglas Rachford (CBDR) are proposed. BDR, which is a kind of non-convex method, is proven to have the local R-linear convergence rate under mild assumptions. Instead, the CBDR method uses the techniques of convexification and can be proven to have a global convergence guarantee as long as the background information is sufficient. To support this, a new property called F-RIP is established. We test the performance of the proposed methods through simulations as well as real experimental measurements, and demonstrate that they achieve a higher recovery rate with less background information compared to the PGD method.