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Functional Lifting for Variational Problems with Higher-Order Regularization

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Imaging, Vision and Learning Based on Optimization and PDEs (IVLOPDE 2016)

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

Variational approaches are an established paradigm in the field of image processing . The non-convexity of the functional can be addressed by functional lifting and convex relaxation techniques, which aim to solve a convex approximation of the original energy on a larger space. However, so far these approaches have been limited to first-order , gradient-based regularizers such as the total variation . In this work, we propose a way to extend functional lifting to a second-order regularizer derived from the Laplacian. We prove that it can be represented efficiently and thus allows numerical optimization. We experimentally demonstrate the usefulness on a synthetic convex denoising problem and on synthetic as well as real-world image registration problems.

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Notes

  1. 1.

    See [28] and http://github.com/tum-vision/prost for the most recent version.

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Acknowledgements

This work was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—394737018 “Functional Lifting 2.0: Efficient Convexifications for Imaging and Vision”. We would like to thank Emanuel Laude and Thomas Möllenhoff for providing their library prost, which was used to solve the saddle-point formulation of the problems.

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Authors and Affiliations

  1. Institute of Mathematics and Image Computing (MIC), University of Lübeck, Lübeck, Germany

    Benedikt Loewenhauser

  2. Technical University of Munich, Garching, Germany

    Benedikt Loewenhauser

  3. Institute of Mathematics and Image Computing (MIC), University of Lübeck, Lübeck, Germany

    Jan Lellmann

Authors
  1. Benedikt Loewenhauser
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  2. Jan Lellmann
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Correspondence to Jan Lellmann .

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Editors and Affiliations

  1. Department of Mathematics, University of Bergen, Bergen, Norway

    Xue-Cheng Tai

  2. Norwegian Defence Research Establishment, Kjeller, Norway

    Egil Bae

  3. University of South-Eastern Norway, Porsgrunn, Norway

    Marius Lysaker

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Loewenhauser, B., Lellmann, J. (2018). Functional Lifting for Variational Problems with Higher-Order Regularization. In: Tai, XC., Bae, E., Lysaker, M. (eds) Imaging, Vision and Learning Based on Optimization and PDEs. IVLOPDE 2016. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-91274-5_5

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