### Abstract

We consider the problem of fitting a discrete curve to time-labeled data points on the set ℙ_{n} of all n-by-n symmetric positive-definite matrices. The quality of a curve is measured by a weighted sum of a term that penalizes its lack of fit to the data and a regularization term that penalizes speed and acceleration. The corresponding objective function depends on the choice of a Riemannian metric on ℙ_{n}. We consider the Euclidean metric, the Log-Euclidean metric and the affine-invariant metric. For each, we derive a numerical algorithm to minimize the objective function. We compare these in terms of reliability and speed, and we assess the visual appearance of the solutions on examples for n = 2. Notably, we find that the Log-Euclidean and the affine-invariant metrics tend to yield similar - and sometimes identical - results, while the former allows for much faster and more reliable algorithms than the latter.

Original language | English (US) |
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Title of host publication | 2011 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2011 - Proceedings |

Pages | 4232-4235 |

Number of pages | 4 |

DOIs | |

State | Published - Aug 18 2011 |

Externally published | Yes |

Event | 36th IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2011 - Prague, Czech Republic Duration: May 22 2011 → May 27 2011 |

### Other

Other | 36th IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2011 |
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Country | Czech Republic |

City | Prague |

Period | 5/22/11 → 5/27/11 |

### All Science Journal Classification (ASJC) codes

- Software
- Signal Processing
- Electrical and Electronic Engineering

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## Cite this

*2011 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2011 - Proceedings*(pp. 4232-4235). [5947287] https://doi.org/10.1109/ICASSP.2011.5947287