Left invariant evolution equations on Gabor transforms

R. Duits, H. Führ, B.J Janssen

Research output: Chapter in Book/Report/Conference proceedingChapterAcademic

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.
Original languageEnglish
Title of host publicationMathematical Methods for Signal and Image Analysis and Representation
EditorsL.M.J. Florack, R. Duits, G. Jongbloed, M.N.M. Lieshout, van, P.L. Davies
Place of PublicationLondon
PublisherSpringer
Pages137-158
ISBN (Print)978-1-4471-2352-1
DOIs
Publication statusPublished - 2012

Publication series

NameComputational Imaging and Vision
Volume41
ISSN (Print)1381-6446

Fingerprint

Gabor Transform
Evolution Equation
Invariant
Vector Field
Operator
Reproducing Kernel Space
Invariant Operator
Heisenberg Group
Commute
Transport Equation
Diffusion equation
Convection
Phase Space
Modulation
Generator

Cite this

Duits, R., Führ, H., & Janssen, B. J. (2012). Left invariant evolution equations on Gabor transforms. In L. M. J. Florack, R. Duits, G. Jongbloed, M. N. M. Lieshout, van, & P. L. Davies (Eds.), Mathematical Methods for Signal and Image Analysis and Representation (pp. 137-158). (Computational Imaging and Vision; Vol. 41). London: Springer. https://doi.org/10.1007/978-1-4471-2353-8_8
Duits, R. ; Führ, H. ; Janssen, B.J. / Left invariant evolution equations on Gabor transforms. Mathematical Methods for Signal and Image Analysis and Representation. editor / L.M.J. Florack ; R. Duits ; G. Jongbloed ; M.N.M. Lieshout, van ; P.L. Davies. London : Springer, 2012. pp. 137-158 (Computational Imaging and Vision).
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Duits, R, Führ, H & Janssen, BJ 2012, Left invariant evolution equations on Gabor transforms. in LMJ Florack, R Duits, G Jongbloed, MNM Lieshout, van & PL Davies (eds), Mathematical Methods for Signal and Image Analysis and Representation. Computational Imaging and Vision, vol. 41, Springer, London, pp. 137-158. https://doi.org/10.1007/978-1-4471-2353-8_8

Left invariant evolution equations on Gabor transforms. / Duits, R.; Führ, H.; Janssen, B.J.

Mathematical Methods for Signal and Image Analysis and Representation. ed. / L.M.J. Florack; R. Duits; G. Jongbloed; M.N.M. Lieshout, van; P.L. Davies. London : Springer, 2012. p. 137-158 (Computational Imaging and Vision; Vol. 41).

Research output: Chapter in Book/Report/Conference proceedingChapterAcademic

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AB - 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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Duits R, Führ H, Janssen BJ. Left invariant evolution equations on Gabor transforms. In Florack LMJ, Duits R, Jongbloed G, Lieshout, van MNM, Davies PL, editors, Mathematical Methods for Signal and Image Analysis and Representation. London: Springer. 2012. p. 137-158. (Computational Imaging and Vision). https://doi.org/10.1007/978-1-4471-2353-8_8