  The space of images can be equipped with a Riemannian metric measuring both
the cost of transport of image intensities and the variation of image
intensities along motion lines. The resulting metamorphosis model was
introduced and analyzed by Trouv\'e and Younes, and a variational time
discretization for the geodesic interpolation was proposed by Berkels et al. In
this paper, this time discrete model is expanded and an image extrapolation via
a discrete exponential map is consistently derived for the variational time
discretization. For a given weakly differentiable initial image and an initial
image variation, the exponential map allows to compute a discrete geodesic
extrapolation path in the space of images. It is shown that a time step of this
shooting method can be formulated in the associated deformations only. For
sufficiently small time steps local existence and uniqueness are proved using a
suitable fixed point formulation and the implicit function theorem. A spatial
Galerkin discretization with cubic splines on coarse meshes for the deformation
and piecewise bilinear finite elements on fine meshes for the image intensities
are used to derive a fully practical algorithm. Different applications
underline the efficiency and stability of the proposed approach.
