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

Abstract

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 in [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61--84; A. Trouvé and L. Younes, Found. Comput. Math., 5 (2005), pp. 173--198], and a variational time discretization for the geodesic interpolation was proposed in [B. Berkels, A. Effland, and M. Rumpf, SIAM J. Imaging Sci., 8 (2015), pp. 1457--1488]. In this paper, this time discrete model is expanded and an image extrapolation via a discretization of the geometric 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 one 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 deformations 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.

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