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

Effland, Alexander and Rumpf, Martin and Schäfer, Florian (2018) Image Extrapolation for the Time Discrete Metamorphosis Model: Existence and Applications. SIAM Journal on Imaging Sciences, 11 (1). pp. 834-862. ISSN 1936-4954.

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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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Additional Information:© 2018, Society for Industrial and Applied Mathematics. Received by the editors May 9, 2017; accepted for publication (in revised form) January 10, 2018; published electronically March 15, 2018. A preliminary version of this paper appeared in Proceedings of the International Conference on Scale Space and Variational Methods in Computer Vision, Lecture Notes in Comput. Sci. 10302, Springer, Cham, 2017, pp. 473--485. The research of the first and second authors was supported by the Hausdorff Center for Mathematics and the Collaborative Research Center 1060 funded by the German Research Foundation.
Funding AgencyGrant Number
Hausdorff Center for MathematicsUNSPECIFIED
Deutsche Forschungsgemeinschaft (DFG)SFB 1060
Subject Keywords:shape space, metamorphosis, variational time discretization, image extrapolation
Issue or Number:1
Classification Code:AMS subject classifications. 65D18, 37L65, 49M25, 65L20
Record Number:CaltechAUTHORS:20180418-103622047
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Official Citation:Image Extrapolation for the Time Discrete Metamorphosis Model: Existence and Applications Alexander Effland, Martin Rumpf, and Florian Schäfer SIAM Journal on Imaging Sciences 2018 11:1, 834-862
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85942
Deposited By: George Porter
Deposited On:18 Apr 2018 21:21
Last Modified:03 Oct 2019 19:37

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