  We consider so-called branched transport and variants thereof in two space
dimensions. In these models one seeks an optimal transportation network for a
given mass transportation task. In two space dimensions, they are closely
connected to Mumford--Shah-type image processing problems, which in turn can be
related to certain higher-dimensional convex optimization problems via
so-called functional lifting. We examine the relation between these different
models and exploit it to solve the branched transport model numerically via
convex optimization. To this end we develop an efficient numerical treatment
based on a specifically designed class of adaptive finite elements. This method
allows the computation of finely resolved optimal transportation networks
despite the high dimensionality of the convex optimization problem and its
complicated set of nonlocal constraints. In particular, by design of the
discretization the infinite set of constraints reduces to a finite number of
inequalities.
