  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.
