SIAM Journal on Imaging Sciences, Volume 13, Issue 1, Page 176-213, January 2020.
This paper deals with tensor completion for the solution of multidimensional inverse problems arising in nuclear magnetic resonance (NMR) relaxometry. We study the problem of reconstructing an approximately low-rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, nonuniform sampling strategies, and parameter selection methods are developed in this context. In particular, we derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property-based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by nonuniform sampling from a Parseval tight frame. The proposed algorithm is then applied to the setting of nuclear magnetic resonance relaxometry, for both simulated and experimental data. We compare our results with basis pursuit as well as with the state-of-the-art nonsubsampled data acquisition and reconstruction approach. Our experiments indicate that tensor recovery promises to significantly accelerate $N$-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments to be performed. Our methods could also be applied to other similar inverse problems arising in machine learning, signal and image processing, and computer vision.

中文翻译：

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核磁共振弛豫法的$ N $维张量完成

SIAM影像科学杂志，第13卷，第1期，第176-213页，2020年1月。
本文涉及张量完成，以解决核磁共振弛豫法中产生的多维逆问题。我们研究了从少量嘈杂的线性测量中重建近似低秩张量的问题。在这种情况下，开发了新的恢复保证，数值算法，非均匀采样策略和参数选择方法。特别地，我们推导了用于张量完成的不动点连续算法，并证明了其收敛性。证明了基于等轴特性的有限张量恢复保证。为亚高斯测量算子和通过从Parseval紧框架进行非均匀采样获得的测量获得了概率恢复保证。然后将所提出的算法应用于模拟和实验数据的核磁共振弛豫法的设置。我们将结果与基础追踪以及最新的非下采样数据采集和重建方法进行比较。我们的实验表明，张量恢复有望显着加速N维NMR弛豫和相关实验，从而使以前无法进行的实验得以实现。我们的方法还可以应用于机器学习，信号和图像处理以及计算机视觉中出现的其他类似逆问题。我们的实验表明，张量恢复有望显着加速N维NMR弛豫和相关实验，从而使以前无法进行的实验得以实现。我们的方法还可以应用于机器学习，信号和图像处理以及计算机视觉中出现的其他类似逆问题。我们的实验表明，张量恢复有望显着加速N维NMR弛豫和相关实验，从而使以前无法进行的实验得以实现。我们的方法还可以应用于机器学习，信号和图像处理以及计算机视觉中出现的其他类似逆问题。