  Interpolating between measures supported by polygonal or polyhedral domains
is a problem that has been recently addressed by the semi-discrete optimal
transport framework. Within this framework, one of the domains is discretized
with a set of samples, while the other one remains continuous. In this paper we
present a method to introduce some symmetry into the solution using coupled
power diagrams. This symmetry is key to capturing the discontinuities of the
transport map reflected in the geometry of the power cells. We design our
method as a fixed-point algorithm alternating between computations of
semi-discrete transport maps and recentering of the sites. The resulting
objects are coupled power diagrams with identical geometry, allowing us to
approximate displacement interpolation through linear interpolation of the
meshes vertices. Through these coupled power diagrams, we have a natural way of
jointly sampling measures.
