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Lifting Methods for Manifold-Valued Variational Problems

In: Handbook of Variational Methods for Nonlinear Geometric Data

Author

Listed:
  • Thomas Vogt

    (University of Lübeck, Institute of Mathematics and Image Computing)

  • Evgeny Strekalovskiy

    (Technical University Munich
    Google Germany GmbH)

  • Daniel Cremers

    (Technical University Munich)

  • Jan Lellmann

    (University of Lübeck, Institute of Mathematics and Image Computing)

Abstract

Lifting methods allow to transform hard variational problems such as segmentation and optical flow estimation into convex problems in a suitable higher-dimensional space. The lifted models can then be efficiently solved to a global optimum, which allows to find approximate global minimizers of the original problem. Recently, these techniques have also been applied to problems with values in a manifold. We provide a review of such methods in a refined framework based on a finite element discretization of the range, which extends the concept of sublabel-accurate lifting to manifolds. We also generalize existing methods for total variation regularization to support general convex regularization.

Suggested Citation

  • Thomas Vogt & Evgeny Strekalovskiy & Daniel Cremers & Jan Lellmann, 2020. "Lifting Methods for Manifold-Valued Variational Problems," Springer Books, in: Philipp Grohs & Martin Holler & Andreas Weinmann (ed.), Handbook of Variational Methods for Nonlinear Geometric Data, chapter 0, pages 95-119, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-31351-7_3
    DOI: 10.1007/978-3-030-31351-7_3
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