  In this paper, we propose a novel overlapping domain decomposition method
that can be applied to various problems in variational imaging such as total
variation minimization. Most of recent domain decomposition methods for total
variation minimization adopt the Fenchel--Rockafellar duality, whereas the
proposed method is based on the primal formulation. Thus, the proposed method
can be applied not only to total variation minimization but also to those with
complex dual problems such as higher order models. In the proposed method, an
equivalent formulation of the model problem with parallel structure is
constructed using a custom overlapping domain decomposition scheme with the
notion of essential domains. As a solver for the constructed formulation, we
propose a decoupled augmented Lagrangian method for untying the coupling of
adjacent subdomains. Convergence analysis of the decoupled augmented Lagrangian
method is provided. We present implementation details and numerical examples
for various model problems including total variation minimizations and higher
order models.
