We introduce a framework for the reconstruction and representation of functions in a setting where these objects cannot be directly observed, but only indirect and noisy measurements are available, namely an inverse problem setting. The proposed methodology can be applied either to the analysis of indirectly observed functional images or to the associated covariance operators, representing second-order information, and thus lying on a non-Euclidean space. To deal with the ill-posedness of the inverse problem, we exploit the spatial structure of the sample data by introducing a flexible regularizing term embedded in the model. Thanks to its efficiency, the proposed model is applied to MEG data, leading to a novel approach to the investigation of functional connectivity.

中文翻译：

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线性逆问题中协方差算子的表示与重构

我们引入了一个框架，用于在无法直接观察这些对象的环境中重建和表示函数，但只有间接和嘈杂的测量可用，即逆问题设置。所提出的方法可以应用于分析间接观察到的功能图像或关联的协方差算子，表示二阶信息，因此位于非欧几里得空间。为了处理逆问题的不适定性，我们通过在模型中引入一个灵活的正则化项来利用样本数据的空间结构。由于其效率，所提出的模型适用于 MEG 数据，从而为研究功能连接提供了一种新方法。