论文标题
高分辨率信号通过广义采样和功能性主成分分析恢复
High-resolution signal recovery via generalized sampling and functional principal component analysis
论文作者
论文摘要
在本文中，我们引入了一个计算框架，用于从其低分辨率间接测量值以及通过合并通用采样和功能性主体组件分析的框架来恢复未知函数的高分辨率近似。特别是，我们通过数据驱动的方法增加了信号分辨率，该方法将感兴趣的功能建模为实现随机场的实现，并利用通过相同的基础随机过程生成的一组训练的观测值。我们研究了由此产生的估计程序的性能，并表明，如果相对于所需的分辨率，训练集适当的训练集足够大，则确实可以恢复高分辨率恢复。此外，我们表明训练集的大小可以通过利用功能性主成分的稀疏表示来减少。此外，各种数值示例说明了提出的重建程序的有效性。
In this paper, we introduce a computational framework for recovering a high-resolution approximation of an unknown function from its low-resolution indirect measurements as well as high-resolution training observations by merging the frameworks of generalized sampling and functional principal component analysis. In particular, we increase the signal resolution via a data driven approach, which models the function of interest as a realization of a random field and leverages a training set of observations generated via the same underlying random process. We study the performance of the resulting estimation procedure and show that high-resolution recovery is indeed possible provided appropriate low-rank and angle conditions hold and provided the training set is sufficiently large relative to the desired resolution. Moreover, we show that the size of the training set can be reduced by leveraging sparse representations of the functional principal components. Furthermore, the effectiveness of the proposed reconstruction procedure is illustrated by various numerical examples.
