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 priors. However, the requirement for the background information may be challenging when moving to the 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 required to ensure a unique solution can be decreased by nearly $1/2$ 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 $(2^{\frac{d+1}{d}}-1)$ folds of the signal is sufficient to ensure the uniqueness. At the same time, we also analyze the stability and robustness of the model when measurements and background information are corrupted by the 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, CBDR method uses the techniques of convexification and can be proven to own 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.

翻译：傅立叶相位恢复（PR）是各种应用中出现的严重病态逆问题。为了确保唯一解并减轻对初始化的依赖，可以利用背景信息作为结构先验。然而，当转向高分辨率成像时，对背景信息的要求可能具有挑战性。同时，先前提出的投影梯度下降（PGD）方法也需求大量背景信息。本文提出了关于背景信息需求的改进理论结果，以及两种基于Douglas-Rachford（DR）的方法。分析表明，对于二维信号，确保唯一解所需的背景信息相比一维信号可减少近$1/2$。通过将结果推广到$d$维，我们证明背景信息长度超过信号长度的$(2^{\frac{d+1}{d}}-1)$倍足以确保唯一性。同时，我们还分析了当测量和背景信息受噪声污染时模型的稳定性和鲁棒性。此外，提出了两种方法：背景Douglas-Rachford（BDR）和凸背景Douglas-Rachford（CBDR）。BDR是一种非凸方法，在温和假设下被证明具有局部R线性收敛速度。CBDR方法则采用凸化技术，并证明只要背景信息充足，就能保证全局收敛。为支持这一点，我们建立了一个称为F-RIP的新性质。我们通过仿真和实际实验测量测试了所提方法的性能，并证明与PGD方法相比，它们在更少的背景信息下实现了更高的恢复率。