  Recently variational models with priors involving first and second order
derivatives resp. differences were successfully applied for image restoration.
There are several ways to incorporate the derivatives of first and second order
into the prior, for example additive coupling or using infimal convolution
(IC), as well as the more general model of total generalized variation (TGV).
The later two methods give also decompositions of the restored images into
image components with distinct "smoothness" properties which are useful in
applications.
  This paper is the first attempt to generalize these models to manifold-valued
images. We propose both extrinsic and intrinsic approaches. The extrinsic
approach is based on embedding the manifold into an Euclidean space of higher
dimension. Models following this approach can be formulated within the
Euclidean space with a constraint restricting them to the manifold. Then
alternating direction methods of multipliers can be employed for finding
minima. However, the components within the infimal convolution or total
generalized variation decomposition live in the embedding space rather than on
the manifold which makes their interpretation difficult. Therefore we also
investigate two intrinsic approaches. For manifolds which are Lie groups we
propose three priors which exploit the group operation, an additive one,
another with IC coupling and a third TGV like one. For computing the minimizers
of the intrinsic models we apply gradient descent algorithms. For general
Riemannian manifolds we further define a model for infimal convolution based on
the recently developed second order differences.
  We demonstrate by numerical examples that our approaches works well for the
circle, the 2-sphere, the rotation group, and the manifold of positive definite
matrices with the affine invariant metric.

Numerical Analysis