The problem of phase retrieval (PR) involves recovering an unknown image from limited amplitude measurement data and is a challenge nonlinear inverse problem in computational imaging and image processing. However, many of the PR methods are based on black-box network models that lack interpretability and plug-and-play (PnP) frameworks that are computationally complex and require careful parameter tuning. To address this, we have developed PRISTA-Net, a deep unfolding network (DUN) based on the first-order iterative shrinkage thresholding algorithm (ISTA). This network utilizes a learnable nonlinear transformation to address the proximal-point mapping sub-problem associated with the sparse priors, and an attention mechanism to focus on phase information containing image edges, textures, and structures. Additionally, the fast Fourier transform (FFT) is used to learn global features to enhance local information, and the designed logarithmic-based loss function leads to significant improvements when the noise level is low. All parameters in the proposed PRISTA-Net framework, including the nonlinear transformation, threshold parameters, and step size, are learned end-to-end instead of being manually set. This method combines the interpretability of traditional methods with the fast inference ability of deep learning and is able to handle noise at each iteration during the unfolding stage, thus improving recovery quality. Experiments on Coded Diffraction Patterns (CDPs) measurements demonstrate that our approach outperforms the existing state-of-the-art methods in terms of qualitative and quantitative evaluations. Our source codes are available at \emph{https://github.com/liuaxou/PRISTA-Net}.

翻译：相位恢复（PR）问题涉及从有限的幅度测量数据中恢复未知图像，是计算成像与图像处理领域中的非线性逆问题挑战。然而，许多相位恢复方法基于缺乏可解释性的黑盒网络模型，或采用计算复杂且需要精细参数调节的即插即用（PnP）框架。为解决此问题，我们提出了PRISTA-Net——一种基于一阶迭代收缩阈值算法（ISTA）的深度展开网络（DUN）。该网络利用可学习非线性变换处理与稀疏先验相关的近端点映射子问题，并通过注意力机制聚焦包含图像边缘、纹理和结构的相位信息。此外，采用快速傅里叶变换（FFT）学习全局特征以增强局部信息，设计的基于对数函数的损失函数在低噪声水平下可显著提升性能。所提出的PRISTA-Net框架中所有参数（包括非线性变换、阈值参数和步长）均通过端到端学习而非人工设定。该方法融合了传统方法的可解释性与深度学习的快速推理能力，能够在展开阶段每次迭代中处理噪声，从而提高恢复质量。在编码衍射图样（CDPs）测量上的实验表明，我们的方法在定性和定量评估上均优于现有最先进方法。源代码已发布于 \emph{https://github.com/liuaxou/PRISTA-Net}。