### Abstract

Inspired by our own visual system, we consider the construction of and reconstruction from an ori-
entation bundle function (OBF) Uf : 2 S1 as a local orientation score of an image, f : 2 , via a
wavelet transform corresponding to a representation of the Euclidean motion group onto 2(2) and ori-
ented wavelet 2(2). This wavelet transform is a unitary mapping with stable inverse, which allows us to
directly relate each operation on OBFs to an operation on images in a robust manner. We examine the
geometry of the domain of an OBF and show that the only sensible operations on OBFs are nonlinear and shift-
twist invariant. As an example, we consider all linear second-order shift-twist invariant evolution equations on
OBFs corresponding to stochastic processes on the Euclidean motion group in order to construct nonlinear
shift-twist invariant operations on OBFs. Given two such stochastic processes, we derive the probability density
that particles of the different processes collide. As an application, we detect elongated structures in images and
automatically close the gaps between them

Original language | English |
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Pages (from-to) | 151-156 |

Journal | Pattern Recognition and Image Analysis |

Volume | 15 |

Issue number | 1 |

Publication status | Published - 2005 |

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## Cite this

Duits, R., Almsick, van, M. A., Duits, M., Franken, E. M., & Florack, L. M. J. (2005). Image processing via shift-twist invariant operations on orientation bundle functions.

*Pattern Recognition and Image Analysis*,*15*(1), 151-156.