Summary: The Riemannian metamorphosis model introduced and analyzed in [M. I. Miller and L. Younes, Int. J. Comput. Vis. 41, No. 1–2, 61–84 (2001; Zbl 1012.68714); A. Trouvé and L. Younes, Found. Comput. Math. 5, No. 2, 173–198 (2005; Zbl 1099.68116)] is taken into account to develop an image extrapolation tool in the space of images. To this end, the variational time discretization for the geodesic interpolation proposed in [B. Berkels and the first and second authors, SIAM J. Imaging Sci. 8, No. 3, 1457–1488 (2015; Zbl 1325.65031)] is picked up to define a discrete exponential map. For a given weakly differentiable initial image and a sufficiently small initial image variation it is shown how to compute a discrete geodesic extrapolation path in the space of images. The resulting discrete paths are indeed local minimizers of the corresponding discrete path energy. A spatial Galerkin discretization with cubic splines on coarse meshes for image deformations and piecewise bilinear finite elements on fine meshes for image intensity functions is used to derive a fully practical algorithm. The method is applied to real images and image variations recorded with a digital camera.
For the entire collection see [Zbl 1362.68015].