Image Extrapolation for the Time Discrete Metamorphosis Model - Existence and Applications

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.