Stable Phaseless Sampling and Reconstruction of Real-Valued Signals with Finite Rate of Innovation
Stable Phaseless Sampling and Reconstruction of Real-Valued Signals with Finite Rate of Innovation
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DOI：
10.1007/s10440-020-00371-5
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发表时间：
2018-01
影响因子：
1.6
通讯作者：
Cheng Cheng-Cheng;Qiyu Sun
中科院分区：
文献类型：
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作者：
Cheng Cheng-Cheng;Qiyu Sun
空间信号是由其在整个域上的评价来定义的。在这篇文章中，我们考虑了有限新息率(FRI)的实值信号的稳定重构，从它们在整个域上的幅度测量或它们在离散子集上的无相样本，一直到符号。红外信号广泛存在于磁共振波谱、超宽带通信、心电等工程应用中。对于FRI信号，我们引入一个无向图来描述它的拓扑结构，通过在整个域上的点评估度量来建立它的图连通性和它的相位可恢复性之间的等价性，应用图的连通分量分解来寻找它唯一的风景分解和具有相同幅度度量的FRI信号集。我们构造了具有有限密度的离散集，使得FRI信号在整个区域上的幅度测量由其在这些离散子集上的无相采样来确定，并且我们证明了相应的无相采样过程关于信号空间上的新的诱导度量和采样数据集上的标准度量具有双Lipschitz性质。在本文中，我们还提出了一种具有线性复杂度的算法，该算法从上述采样集上的(未受污染的)无相移样本重建FRI信号，而不受噪声水平和原始FRI信号是否可相位恢复的先验信息的限制。该算法在理论上是稳定的，并在数值上证明了该算法在幅值测量中逼近原始FRI信号。
A spatial signal is defined by its evaluations on the whole domain. In this paper, we consider stable reconstruction of real-valued signals with finite rate of innovation (FRI), up to a sign, from their magnitude measurements on the whole domain or their phaseless samples on a discrete subset. FRI signals appear in many engineering applications such as magnetic resonance spectrum, ultra wide-band communication and electrocardiogram. For an FRI signal, we introduce an undirected graph to describe its topological structure, establish the equivalence between its graph connectivity and its phase retrievability by point evaluation measurements on the whole domain, apply the graph connected component decomposition to find its unique landscape decomposition and the set of FRI signals that have the same magnitude measurements. We construct discrete sets with finite density so that magnitude measurements of an FRI signal on the whole domain are determined by its phaseless samples taken on those discrete subsets, and we show that the corresponding phaseless sampling procedure has bi-Lipschitz property with respect to a new induced metric on the signal space and the standard-metric on the sampling data set. In this paper, we also propose an algorithm with linear complexity to reconstruct an FRI signal from its (un)corrupted phaseless samples on the above sampling set without restriction on the noise level and apriori information whether the original FRI signal is phase retrievable. The algorithm is theoretically guaranteed to be stable, and numerically demonstrated to approximate the original FRI signal in magnitude measurements.