  The reconstruction of images from their corresponding noisy Radon transform
is a typical example of an ill-posed linear inverse problem as arising in the
application of computerized tomography (CT). As the (naive) solution does not
depend on the measured data continuously, regularization is needed to
re-establish a continuous dependence. In this work, we investigate simple, but
yet still provably convergent approaches to learning linear regularization
methods from data. More specifically, we analyze two approaches: One generic
linear regularization that learns how to manipulate the singular values of the
linear operator in an extension of our previous work, and one tailored approach
in the Fourier domain that is specific to CT-reconstruction. We prove that such
approaches become convergent regularization methods as well as the fact that
the reconstructions they provide are typically much smoother than the training
data they were trained on. Finally, we compare the spectral as well as the
Fourier-based approaches for CT-reconstruction numerically, discuss their
advantages and disadvantages and investigate the effect of discretization
errors at different resolutions.
