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A Variational Approach for Physically Based Image Interpolation Across Boundaries

  • Conference paper
Multiple Shooting and Time Domain Decomposition Methods

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 9))

Abstract

In this contribution we present an optimal control approach for physics-based optical flow estimation and image interpolation. The aim of the developed process is to identify appropriate boundary data of an underlying physical model describing the transport field, which reason the movement of an initial brightness distribution. Thereby, the flow field as solution of the time-dependent non-linear Navier-Stokes equations is coupled to a transport dominant convection-diffusion equation describing the brightness intensity. Thus, we have to deal with a weakly coupled PDE system as state equation of a PDE constrained optimisation problem. The data is given in form of consecutive images, with a sparse temporal resolution, representing the brightness distribution at different time points. We will present the mathematical theory of the resulting optimisation problem, which is based on a Robin-type boundary control. We describe the numerical solution process and present by means of synthetical test cases the functionality of the method. Finally we discuss the application of multiple shooting techniques for the considered problem, since we observed that the employed Newton-type method is very sensitive with respect to the chosen initial value.

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Acknowledgements

This article is based on collaboration with Prof. Dr. Dr. h.c. R. Rannacher (IAM, Heidelberg University) and Priv.-Doz. Dr. C.S. Garbe (IPM, Heidelberg University). Furthermore M. Geiger and Dr. T. Carraro (both IAM, Heidelberg University) supported the author to connect the content to the time decomposition topic in the conclusions. The author gratefully acknowledges all mentioned persons for many fruitful discussions and for their support.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Im Neuenheimer Feld 294, 69120, Heidelberg, Germany

    Matthias Klinger

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  1. Matthias Klinger
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Correspondence to Matthias Klinger .

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Editors and Affiliations

  1. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Thomas Carraro

  2. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Michael Geiger

  3. Heidelberg University, Interdisciplinary Center for Scientific Computing, Heidelberg, Germany

    Stefan Körkel

  4. Heidelberg University, Institute for Applied Mathematics, Heidelberg, Germany

    Rolf Rannacher

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Klinger, M. (2015). A Variational Approach for Physically Based Image Interpolation Across Boundaries. In: Carraro, T., Geiger, M., Körkel, S., Rannacher, R. (eds) Multiple Shooting and Time Domain Decomposition Methods. Contributions in Mathematical and Computational Sciences, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-23321-5_13

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