  We propose a variational regularisation approach for the problem of
template-based image reconstruction from indirect, noisy measurements as given,
for instance, in X-ray computed tomography. An image is reconstructed from such
measurements by deforming a given template image. The image registration is
directly incorporated into the variational regularisation approach in the form
of a partial differential equation that models the registration as either mass-
or intensity-preserving transport from the template to the unknown
reconstruction. We provide theoretical results for the proposed variational
regularisation for both cases. In particular, we prove existence of a
minimiser, stability with respect to the data, and convergence for vanishing
noise when either of the abovementioned equations is imposed and more general
distance functions are used. Numerically, we solve the problem by extending
existing Lagrangian methods and propose a multilevel approach that is
applicable whenever a suitable downsampling procedure for the operator and the
measured data can be provided. Finally, we demonstrate the performance of our
method for template-based image reconstruction from highly undersampled and
noisy Radon transform data. We compare results for mass- and
intensity-preserving image registration, various regularisation functionals,
and different distance functions. Our results show that very reasonable
reconstructions can be obtained when only few measurements are available and
demonstrate that the use of a normalised cross correlation-based distance is
advantageous when the image intensities between the template and the unknown
image differ substantially.
