  In this paper the optimal transport and the metamorphosis perspectives are
combined. For a pair of given input images geodesic paths in the space of
images are defined as minimizers of a resulting path energy. To this end, the
underlying Riemannian metric measures the rate of transport cost and the rate
of viscous dissipation. Furthermore, the model is capable to deal with strongly
varying image contrast and explicitly allows for sources and sinks in the
transport equations which are incorporated in the metric related to the
metamorphosis approach by Trouv\'e and Younes. In the non-viscous case with
source term existence of geodesic paths is proven in the space of measures. The
proposed model is explored on the range from merely optimal transport to
strongly dissipative dynamics. For this model a robust and effective
variational time discretization of geodesic paths is proposed. This requires to
minimize a discrete path energy consisting of a sum of consecutive image
matching functionals. These functionals are defined on corresponding pairs of
intensity functions and on associated pairwise matching deformations. Existence
of time discrete geodesics is demonstrated. Furthermore, a finite element
implementation is proposed and applied to instructive test cases and to real
images. In the non-viscous case this is compared to the algorithm proposed by
Benamou and Brenier including a discretization of the source term. Finally, the
model is generalized to define discrete weighted barycentres with applications
to textures and objects.
