### Abstract

We consider left-invariant diusion processes on DTI data by embedding the data into the space R3 o S2 of 3D positions and orientations.
We then define and solve the diffusion equation in a moving frame of reference dened using left-invariant derivatives. The diffusion process is made adaptive to the data in order to do Perona-Malik-like edge preserving smoothing, which is necessary to handle ber structures near regions of large isotropic diusion such as the ventricles of the brain.
The corresponding partial dierential systems are solved using nite difference stencils. We include experiments both on synthetic data and on DTI-images of the brain.

Original language | English |
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Pages (from-to) | 138-168 |

Journal | Numerical Mathematics: Theory, Methods and Applications |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 |