  We study the problem of registering images. The framework we use is
metamorphosis and we construct a variational Eulerian space-time setting and
pose the registration problem as an infinite-dimensional optimisation problem.
The geodesic equations correspond to a system of advection and continuity
equations and are solved analytically. Well-posedness of a primal conforming
finite element method is established and its convergence is investigated
numerically. This provides a discrete forward operator for the matching
parameterized by a space-time velocity field. We propose a gradient descent
method on this control variable and show several promising numerical results
for this approach.
