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.

在本文中，我们介绍了一种计算框架，用于通过合并广义采样和函数主成分分析的框架，从未知函数的低分辨率间接测量和高分辨率训练观察中恢复未知函数的高分辨率近似值。特别是，我们通过数据驱动的方法提高了信号分辨率，该方法将感兴趣的函数建模为随机场的实现，并利用通过相同的基础随机过程生成的一组观察训练。我们研究了由此产生的估计程序的性能，并表明如果适当的低秩和角度条件成立，并且训练集相对于所需的分辨率足够大，那么高分辨率恢复确实是可能的。而且，我们表明，可以通过利用函数主成分的稀疏表示来减小训练集的大小。此外，各种数值示例说明了所提出的重建程序的有效性。