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Left Invariant Evolution Equations on Gabor Transforms

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Mathematical Methods for Signal and Image Analysis and Representation

Part of the book series: Computational Imaging and Vision ((CIVI,volume 41))

Abstract

By means of the unitary Gabor transform one can relate operators on signals to operators on the space of Gabor transforms. In order to obtain a translation and modulation invariant operator on the space of signals, the corresponding operator on the reproducing kernel space of Gabor transforms must be left invariant, i.e. it should commute with the left regular action of the reduced Heisenberg group H r. By using the left invariant vector fields on H r and the corresponding left-invariant vector fields on phase space in the generators of our transport and diffusion equations on Gabor transforms we naturally employ the essential group structure on the domain of a Gabor transform. Here we mainly restrict ourselves to non-linear adaptive left-invariant convection (reassignment), while maintaining the original signal.

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Notes

  1. 1.

    As we explain in [118, Appendices B and C] the Gabor domain is a principal fiber bundle equipped with the Cartan connection form \(\omega_{g}(X_{g})= \langle\mathrm{d}s + \frac{1}{2}(p\,\mathrm{d}q -q\, \mathrm{d}p) , X_{g}\rangle\), or equivalently, it is a contact manifold, cf. [54, p. 6], [118, Appendix B, Definition B.14], .

  2. 2.

    The metric tensor is degenerate on H r, but we consider a contact manifold where tangent vectors along horizontal curves do not have an -component.

  3. 3.

    The induced frame operator can be efficiently diagonalized by Zak-transform, [235], boiling down to diagonalization of inverse Fourier transform on H r, [118, Chap. 2.3]. We used this in our algorithms.

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Acknowledgements

The Netherlands Organisation for Scientific Research (NWO) is gratefully acknowledged for financial support.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science & Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, 5600 MB, Eindhoven, The Netherlands

    Remco Duits

  2. Lehrstuhl A für Mathematik, RWTH Aachen, 52056, Aachen, Germany

    Hartmut Führ

  3. Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, 5600 MB, Eindhoven, The Netherlands

    Bart Janssen

Authors
  1. Remco Duits
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  2. Hartmut Führ
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  3. Bart Janssen
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Correspondence to Remco Duits .

Editor information

Editors and Affiliations

  1. Dept. Mathematics & Computer Science, Eindhoven University of Technology, Eindhoven, 5600 MB, The Netherlands

    Luc Florack

  2. Dept. Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, Eindhoven, 5600 MB, The Netherlands

    Remco Duits

  3. Dept. Applied Mathematics, Delft University of Technology, Delft, 2628 CD, The Netherlands

    Geurt Jongbloed

  4. Probability & Stochastic Networks (PNA), Centrum Wiskunde & Informatica, Amsterdam, 1098 XG, The Netherlands

    Marie-Colette van Lieshout

  5. Fakultät für Mathematik, Universität Duisburg-Essen, Essen, 45141, Germany

    Laurie Davies

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Duits, R., Führ, H., Janssen, B. (2012). Left Invariant Evolution Equations on Gabor Transforms. In: Florack, L., Duits, R., Jongbloed, G., van Lieshout, MC., Davies, L. (eds) Mathematical Methods for Signal and Image Analysis and Representation. Computational Imaging and Vision, vol 41. Springer, London. https://doi.org/10.1007/978-1-4471-2353-8_8

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