  We demonstrate that numerically computed approximations of Koopman
eigenfunctions and eigenvalues create a natural framework for data fusion in
applications governed by nonlinear evolution laws. This is possible because the
eigenvalues of the Koopman operator are invariant to invertible transformations
of the system state, so that the values of the Koopman eigenfunctions serve as
a set of intrinsic coordinates that can be used to map between different
observations (e.g., measurements obtained through different sets of sensors) of
the same fundamental behavior. The measurements we wish to merge can also be
nonlinear, but must be "rich enough" to allow (an effective approximation of)
the state to be reconstructed from a single set of measurements. This approach
requires independently obtained sets of data that capture the evolution of the
heterogeneous measurements and a single pair of "joint" measurements taken at
one instance in time. Computational approximations of eigenfunctions and their
corresponding eigenvalues from data are accomplished using Extended Dynamic
Mode Decomposition. We illustrate this approach on measurements of
spatio-temporal oscillations of the FitzHugh-Nagumo PDE, and show how to fuse
point measurements with principal component measurements, after which either
set of measurements can be used to estimate the other set.
