### Samenvatting

In the standard scale space approach one obtains a scale space representation u:R of an image by means of an evolution equation on the additive group (R d ,¿+¿). However, it is common to apply a wavelet transform (constructed via a representation of a Lie-group G and admissible wavelet ¿) to an image which provides a detailed overview of the group structure in an image. The result of such a wavelet transform provides a function on a group G (rather than (R d ,¿+¿)), which we call a score. Since the wavelet transform is unitary we have stable reconstruction by its adjoint. This allows us to link operators on images to operators on scores in a robust way. To ensure -invariance of the corresponding operator on the image the operator on the wavelet transform must be left-invariant. Therefore we focus on left-invariant evolution equations (and their resolvents) on the Lie-group G generated by a quadratic form Q on left invariant vector fields. These evolution equations correspond to stochastic processes on G and their solution is given by a group convolution with the corresponding Green’s function, for which we present an explicit derivation in two particular image analysis applications. In this article we describe a general approach how the concept of scale space can be extended by replacing the additive group R d by a Lie-group with more structure.
The Dutch Organization for Scientific Research is gratefully acknowledged for financial support
This article provides the theory and general framework we applied in [9],[5],[8].

Originele taal-2 | Engels |
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Titel | Proceedings of the First International Conference on Scale Space and Variational Methods in Computer Vision (SSVM 2007) 30 May - 2 June 2007, Ischia, Italy |

Redacteuren | F. Sgallari, A. Murli, N. Paragios |

Plaats van productie | Berlin |

Uitgeverij | Springer |

Pagina's | 300-312 |

ISBN van geprinte versie | 978-3-540-72822-1 |

DOI's | |

Status | Gepubliceerd - 2007 |

Evenement | conference; SSVM 2007, Ischia, Italy; 2007-05-30; 2007-06-02 - Duur: 30 mei 2007 → 2 jun 2007 |

### Publicatie series

Naam | Lecture Notes in Computer Science |
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Volume | 4485 |

ISSN van geprinte versie | 0302-9743 |

### Congres

Congres | conference; SSVM 2007, Ischia, Italy; 2007-05-30; 2007-06-02 |
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Periode | 30/05/07 → 2/06/07 |

Ander | SSVM 2007, Ischia, Italy |

## Vingerafdruk Duik in de onderzoeksthema's van 'Scale spaces on Lie groups'. Samen vormen ze een unieke vingerafdruk.

## Citeer dit

Duits, R., & Burgeth, B. (2007). Scale spaces on Lie groups. In F. Sgallari, A. Murli, & N. Paragios (editors),

*Proceedings of the First International Conference on Scale Space and Variational Methods in Computer Vision (SSVM 2007) 30 May - 2 June 2007, Ischia, Italy*(blz. 300-312). (Lecture Notes in Computer Science; Vol. 4485). Springer. https://doi.org/10.1007/978-3-540-72823-8_26, https://doi.org/10.1007/978-3-540-72823-8