2017/05/12 by Alexander Effland, Martin Rumpf, Effland, Alexander +3
Computer Science · Engineering · #37L65 #3D Shape Modeling and Analysis #49M25 #65D18 #65L20 #Advanced Numerical Analysis Techniques #Advanced Vision and Imaging #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1705.04490
openalex publication_date 2017/05/12 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
The space of images can be equipped with a Riemannian metric measuring both\nthe cost of transport of image intensities and the variation of image\nintensities along motion lines. The resulting metamorphosis model was\nintroduced and analyzed by Trouv 'e and Younes, and a variational time\ndiscretization for the geodesic interpolation was proposed by Berkels et al. In\nthis paper, this time discrete model is expanded and an image extrapolation via\na discrete exponential map is consistently derived for the variational time\ndiscretization. For a given weakly differentiable initial image and an initial\nimage variation, the exponential map allows to compute a discrete geodesic\nextrapolation path in the space of images. It is shown that a time step of this\nshooting method can be formulated in the associated deformations only. For\nsufficiently small time steps local existence and uniqueness are proved using a\nsuitable fixed point formulation and the implicit function theorem. A spatial\nGalerkin discretization with cubic splines on coarse meshes for the deformation\nand piecewise bilinear finite elements on fine meshes for the image intensities\nare used to derive a fully practical algorithm. Different applications\nunderline the efficiency and stability of the proposed approach.\n