Abstract
Cubic splines are a classical tool for higher order interpolation of points in Euclidean space known to minimize the integral of the squared acceleration along the interpolation path. This paper transfers this method to the smooth interpolation of key frames in the space of images. To this end the metamorphosis model based on a simultaneous transport of image intensities and a modulation of intensities along motion trajectories is generalized. The proposed spline energy combines quadratic functionals of the Eulerian motion acceleration and of the second material derivative representing an acceleration in the change of intensities along motion paths. A variational time discretization of this spline model is proposed and the convergence to a suitably relaxed time continuous model is discussed using the tool of \(\varGamma \)-convergence. In particular, this also allows to establish the existence of an optimal spline path interpolating given key frame images. The spatial discretization is based on a finite difference and a stable spline interpolation. A variety of numerical examples demonstrates the robustness and versatility of the proposed method for real images using a variant of the iPALM algorithm for the minimization of the fully discrete energy functional.
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The comprehensive proof is available for review on request and will appear elsewhere.
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Acknowledgements
This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via project 211504053 - Collaborative Research Center 1060 and project 390685813 - Hausdorff Center for Mathematics.
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Justiniano, J., Rajković, M., Rumpf, M. (2021). Splines for Image Metamorphosis. In: Elmoataz, A., Fadili, J., Quéau, Y., Rabin, J., Simon, L. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2021. Lecture Notes in Computer Science(), vol 12679. Springer, Cham. https://doi.org/10.1007/978-3-030-75549-2_37
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