Tensor Completion From Regular Sub-Nyquist Samples
Tensor Completion From Regular Sub-Nyquist Samples
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DOI：
10.1109/tsp.2019.2952044
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发表时间：
2020-01-01
影响因子：
5.4
通讯作者：
Akcakaya, Mehmet
中科院分区：
文献类型：
--
作者：
Kanatsoulis, Charilaos I.;Fu, Xiao;Akcakaya, Mehmet
信号采样和重构是信号处理的核心，是一项基本的工程任务。著名的Shannon-Nyquist定理保证了从均匀样本中以两倍于信号中存在的最大频率的速率获得完美的信号重建。不幸的是，大量感兴趣的信号远不是频带有限的。这推动了对亚奈奎斯特样本重建的研究，这主要取决于随机/非相干抽样程序的使用。然而，统一或定期抽样在实践中和从系统设计的角度来看更有吸引力，因为它的实施要简单得多，而且由于系统的限制，往往是必要的。在这项工作中，我们研究了三维或更高维信号(张量)的规则采样和重构。我们证明了从规则样本重建张量信号是可行的。在所提出的框架下，样本复杂度由张量秩来决定，而不是由信号带宽来决定。这一结果为为自然为张量的信号(例如图像和视频)设计实用的规则采样模式和系统提供了新的视角。作为一个具体的应用，我们证明了功能磁共振成像(FMRI)加速是一个张量采样问题，并设计了实用的采样方案和算法框架来处理它。数值结果表明，我们的张量采样策略在不牺牲重建精度的情况下显著加快了fMRI采样过程。
Signal sampling and reconstruction is a fundamental engineering task at the heart of signal processing. The celebrated Shannon-Nyquist theorem guarantees perfect signal reconstruction from uniform samples, obtained at a rate twice the maximum frequency present in the signal. Unfortunately a large number of signals of interest are far from being band-limited. This motivated research on reconstruction from sub-Nyquist samples, which mainly hinges on the use of random/incoherent sampling procedures. However, uniform or regular sampling is more appealing in practice and from the system design point of view, as it is far simpler to implement, and often necessary due to system constraints. In this work, we study regular sampling and reconstruction of three- or higher-dimensional signals (tensors). We show that reconstructing a tensor signal from regular samples is feasible. Under the proposed framework, the sample complexity is determined by the tensor rank-rather than the signal bandwidth. This result offers new perspectives for designing practical regular sampling patterns and systems for signals that are naturally tensors, e.g., images and video. For a concrete application, we show that functional magnetic resonance imaging (fMRI) acceleration is a tensor sampling problem, and design practical sampling schemes and an algorithmic framework to handle it. Numerical results show that our tensor sampling strategy accelerates the fMRI sampling process significantly without sacrificing reconstruction accuracy.