In this article, we provide a detailed review of recent advances in the recovery of continuous-domain multidimensional signals from their few nonuniform (multichannel) measurements using structured low-rank (SLR) matrix completion formulation. This framework is centered on the fundamental duality between the compactness (e.g., sparsity) of the continuous signal and the rank of a structured matrix, whose entries are functions of the signal. This property enables the reformulation of the signal recovery as an SLR matrix completion problem, which includes performance guarantees. We also review fast algorithms that are comparable in complexity to current compressed sensing (CS) methods, which enable the framework's application to large-scale magnetic resonance (MR) recovery problems. The remarkable flexibility of the formulation can be used to exploit signal properties that are difficult to capture by current sparse and low-rank optimization strategies. We demonstrate the utility of the framework in a wide range of MR imaging (MRI) applications, including highly accelerated imaging, calibration-free acquisition, MR artifact correction, and ungated dynamic MRI.

中文翻译：

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结构化低阶算法：理论、磁共振应用以及机器学习的链接


在本文中，我们详细回顾了使用结构化低秩（SLR）矩阵完成公式从少数非均匀（多通道）测量中恢复连续域多维信号的最新进展。该框架以连续信号的紧凑性（例如，稀疏性）和结构化矩阵的秩之间的基本对偶性为中心，其条目是信号的函数。此属性使得能够将信号恢复重新表述为 SLR 矩阵完成问题，其中包括性能保证。我们还回顾了复杂性与当前压缩感知（CS）方法相当的快速算法，这使得该框架能够应用于大规模磁共振（MR）恢复问题。该公式的显着灵活性可用于利用当前稀疏和低秩优化策略难以捕获的信号属性。我们展示了该框架在各种 MR 成像 (MRI) 应用中的实用性，包括高度加速成像、免校准采集、MR 伪影校正和非门控动态 MRI。