  We introduce the transport-transform (TT) and the relative
transport-transform (RTT) metrics between finite point patterns on a general
space, which provide a unified framework for earlier point pattern metrics, in
particular the generalized spike time and the normalized and unnormalized OSPA
metrics. Our main focus is on barycenters, i.e. minimizers of a $q$-th order
Fr\'echet functional with respect to these metrics.
  We present a heuristic algorithm that terminates in a local minimum and is
shown to be fast and reliable in a simulation study. The algorithm serves as an
umbrella method that can be applied on any state space where an appropriate
algorithm for solving the location problem for individual points is available.
We present applications to geocoded data of crimes in Euclidean space and on a
street network, illustrating that barycenters serve as informative summary
statistics. Our work is a first step towards statistical inference in
covariate-based models of repeated point pattern observations.
