### Abstract

Original language | English |
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Title of host publication | Mathematical Methods for Signal and Image Analysis and Representation |

Editors | L.M.J. Florack, R. Duits, G. Jongbloed, M.N.M. Lieshout, van, P.L. Davies |

Place of Publication | London |

Publisher | Springer |

Pages | 137-158 |

ISBN (Print) | 978-1-4471-2352-1 |

DOIs | |

Publication status | Published - 2012 |

### Publication series

Name | Computational Imaging and Vision |
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Volume | 41 |

ISSN (Print) | 1381-6446 |

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### Cite this

*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

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*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.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic

TY - CHAP

T1 - Left invariant evolution equations on Gabor transforms

AU - Duits, R.

AU - Führ, H.

AU - Janssen, B.J

PY - 2012

Y1 - 2012

N2 - 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.

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.

U2 - 10.1007/978-1-4471-2353-8_8

DO - 10.1007/978-1-4471-2353-8_8

M3 - Chapter

SN - 978-1-4471-2352-1

T3 - Computational Imaging and Vision

SP - 137

EP - 158

BT - Mathematical Methods for Signal and Image Analysis and Representation

A2 - Florack, L.M.J.

A2 - Duits, R.

A2 - Jongbloed, G.

A2 - Lieshout, van, M.N.M.

A2 - Davies, P.L.

PB - Springer

CY - London

ER -