# Homotopy Measures for Representative Trajectories

## Abstract

An important task in trajectory analysis is defining a meaningful representative for a cluster of similar trajectories. Formally defining and computing such a representative $$r$$ is a challenging problem. We propose and discuss two new definitions, both of which use only the geometry of the input trajectories. The definitions are based on the homotopy area as a measure of similarity between two curves, which is a minimum area swept by all possible deformations of one curve into the other. In the first definition we wish to minimize the maximum homotopy area between $$r$$ and any input trajectory, whereas in the second definition we wish to minimize the sum of the homotopy areas between $$r$$ and the input trajectories. For both definitions computing an optimal representative is NP-hard. However, for the case of minimizing the sum of the homotopy areas, an optimal representative can be found efficiently in a natural class of restricted inputs, namely, when the arrangement of trajectories forms a directed acyclic graph.

## Corresponding Publications

• ### Homotopy Measures for Representative Trajectories

Proc. 24th European Symposium on Algorithms, 2016
@inproceedings{homotopymeasures2016,
author = {Chambers, Erin and Kostitsyna, Irina and L{\"o}ffler, Maarten
and Staals, Frank},
title = {Homotopy Measures for Representative Trajectories},
booktitle = {Proc. 24th European Symposium on Algorithms},
year = {2016},
location = {Aarhus, Denmark},
numpages = {17},
pages = {27:1--27:17},
keywords = {trajectories, representative, median, homotopy area},
category = {trajectories},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
volume = {57},
editor = {Piotr Sankowski and Christos Zaroliagis},
doi = {10.4230/LIPIcs.ESA.2016.27},
url = {https://dx.doi.org/10.4230/LIPIcs.ESA.2016.27},
}

• ### Homotopy Measures for Representative Trajectories

Abstr. 31th European Workshop on Computational Geometry (EuroCG), 2015
@article{homotopymeasures_eurocg2015,
author = {Chambers Erin and Kostitsyna, Irina and L{\"o}ffler, Maarten and Staals, Frank},
title = {Homotopy Measures for Representative Trajectories},
journal = {Abstr. 31th European Workshop on Computational Geometry (EuroCG)},
year = {2015},
location = {Ljubljana, Slovenia},
numpages = {4},
pages = {125 -- 128},
url = {http://eurocg15.fri.uni-lj.si/pub/eurocg15-book-of-abstracts.pdf},
category = {trajectories},
project = {homotopymeasures2016},
}