Cai Jian-Feng, Qu Xiaobo, Xu Weiyu, Ye Gui-Bo
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.
Department of Electronic Science, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, Xiamen University, P.O. Box 979, Xiamen 361005, China.
Appl Comput Harmon Anal. 2016 Sep;41(2):470-490. doi: 10.1016/j.acha.2016.02.003. Epub 2016 Mar 2.
This paper explores robust recovery of a superposition of distinct complex exponential functions with or without damping factors from a few random Gaussian projections. We assume that the signal of interest is of 2 - 1 dimensions and < 2 - 1. This framework covers a large class of signals arising from real applications in biology, automation, imaging science, etc. To reconstruct such a signal, our algorithm is to seek a low-rank Hankel matrix of the signal by minimizing its nuclear norm subject to the consistency on the sampled data. Our theoretical results show that a robust recovery is possible as long as the number of projections exceeds (ln). No incoherence or separation condition is required in our proof. Our method can be applied to spectral compressed sensing where the signal of interest is a superposition of complex sinusoids. Compared to existing results, our result here does not need any separation condition on the frequencies, while achieving better or comparable bounds on the number of measurements. Furthermore, our method provides theoretical guidance on how many samples are required in the state-of-the-art non-uniform sampling in NMR spectroscopy. The performance of our algorithm is further demonstrated by numerical experiments.
本文探讨了从少量随机高斯投影中稳健恢复带有或不带有阻尼因子的不同复指数函数叠加的问题。我们假设感兴趣的信号是二维的且维度小于2 - 1。该框架涵盖了生物学、自动化、成像科学等实际应用中出现的一大类信号。为了重建这样一个信号，我们的算法是通过在采样数据一致性的约束下最小化其核范数来寻找信号的低秩汉克尔矩阵。我们的理论结果表明，只要投影数量超过(ln)，就有可能实现稳健恢复。我们的证明中不需要任何不相干性或分离条件。我们的方法可应用于频谱压缩感知，其中感兴趣的信号是复正弦波的叠加。与现有结果相比，我们的结果在此处不需要对频率有任何分离条件，同时在测量数量上实现了更好或相当的界。此外，我们的方法为核磁共振光谱学中最先进的非均匀采样需要多少样本提供了理论指导。数值实验进一步证明了我们算法的性能。
