  Many methods for processing scalar and vector valued images, volumes and
other data in the context of inverse problems are based on variational
formulations. Such formulations require appropriate regularization functionals
that model expected properties of the object to reconstruct. Prominent examples
of regularization functionals in a vector-space context are the total variation
(TV) and the Mumford-Shah functional, as well as higher-order schemes such as
total generalized variation models. Driven by applications where the signals or
data live in nonlinear manifolds, there has been quite some interest in
developing analogous methods for nonlinear, manifold-valued data recently. In
this chapter, we consider various variational regularization methods for
manifold-valued data. In particular, we consider TV minimization as well as
higher order models such as total generalized variation (TGV). Also, we discuss
(discrete) Mumford-Shah models and related methods for piecewise constant data.
We develop discrete energies for denoising and report on algorithmic approaches
to minimize them. Further, we also deal with the extension of such methods to
incorporate indirect measurement terms, thus addressing the inverse problem
setup. Finally, we discuss wavelet sparse regularization for manifold-valued
data.
