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.

本研究提出了一种基于正则采样的三或高维张量信号重建框架，使用张量秩替代信号带宽来决定采样复杂度，尤其适用于自然为张量的信号，例如图像和视频。同时，我们将功能性磁共振成像 (fMRI) 加速问题视为张量采样问题，并设计出实用的采样方案和算法框架，数值结果表明我们的张量采样策略能够大大加速 fMRI 采样过程而不影响重建精度。

正则子奈奎斯特采样的张量完成

Tensor Completion from Regular Sub-Nyquist Samples

Compressed Sensing (CS) is a novel technique for simultaneous signal sampling
and compression based on the existence of a sparse representation of signal and
a projected dictionary $PD$, where $P\in\mathbb{R}^{m\times d}$ is the
projection matrix and $D\in\mathbb{R}^{d\times n}$ is the dictionary. To
exactly recover the signal with a small number of measurements $m$, the
projected dictionary $PD$ is expected to be of low mutual coherence. Several
previous methods attempt to find the projection $P$ such that the mutual
coherence of $PD$ can be as low as possible. However, they do not minimize the
mutual coherence directly and thus their methods are far from optimal. Also the
solvers they used lack of the convergence guarantee and thus there has no
guarantee on the quality of their obtained solutions. This work aims to address
these issues. We propose to find an optimal projection by minimizing the mutual
coherence of $PD$ directly. This leads to a nonconvex nonsmooth minimization
problem. We then approximate it by smoothing and solve it by alternate
minimization. We further prove the convergence of our algorithm. To the best of
our knowledge, this is the first work which directly minimizes the mutual
coherence of the projected dictionary with a convergence guarantee. Numerical
experiments demonstrate that the proposed method can recover sparse signals
better than existing methods.

本研究提出通过直接最小化投影词典的互不一致性来找到最优投影，并证明了算法的收敛性。该算法在实验中表现优于现有方法，能更好地恢复稀疏信号。