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**1 - 3**of**3**### Trajectory Grouping Structure under Geodesic Distance

"... Abstract In recent years trajectory data has become one of the main types of geographic data, and hence algorithmic tools to handle large quantities of trajectories are essential. A single trajectory is typically represented as a sequence of time-stamped points in the plane. In a collection of traj ..."

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Abstract In recent years trajectory data has become one of the main types of geographic data, and hence algorithmic tools to handle large quantities of trajectories are essential. A single trajectory is typically represented as a sequence of time-stamped points in the plane. In a collection of trajectories one wants to detect maximal groups of moving entities and their behaviour (merges and splits) over time. This information can be summarized in the trajectory grouping structure. Significantly extending the work of Buchin et al. [WADS 2013] into a realistic setting, we show that the trajectory grouping structure can be computed efficiently also if obstacles are present and the distance between the entities is measured by geodesic distance. We bound the number of critical events: times at which the distance between two subsets of moving entities is exactly ε, where ε is the threshold distance that determines whether two entities are close enough to be in one group. In case the n entities move in a simple polygon along trajectories with τ vertices each we give an O(τ n 2 ) upper bound, which is tight in the worst case. In case of well-spaced obstacles we give an O(τ (n 2 + mλ 4 (n))) upper bound, where m is the total complexity of the obstacles, and λ s (n) denotes the maximum length of a Davenport-Schinzel sequence of n symbols of order s. In case of general obstacles we give an O(τ min{n 2 + m 3 λ 4 (n), n 2 m 2 }) upper bound. Furthermore, for all cases we provide efficient algorithms to compute the critical events, which in turn leads to efficient algorithms to compute the trajectory grouping structure.

### A Reeb Graph Approach to Tractography∗

"... We propose an efficient algorithm for discovering the high-level topological structure of a collection of 3-dimensional trajectories. Our algorithm computes a sparse graph repre-senting the latent “bundling ” and “unbundling ” structure of the trajectory data. This graph can serve both as a compact ..."

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We propose an efficient algorithm for discovering the high-level topological structure of a collection of 3-dimensional trajectories. Our algorithm computes a sparse graph repre-senting the latent “bundling ” and “unbundling ” structure of the trajectory data. This graph can serve both as a compact signature of the trajectory data set as well as a tool for ef-ficient comparison among different data sets. Our problem formulation and the algorithms are broadly applicable and general-purpose but we focus on a particular neuroscience application to highlight the key features. In particular, our motivation stems from the emerging area of brain tractog-raphy, which aims to construct the connectome of human brain white matter fibers. These fibers can be inferred non-invasively using magnetic resonance imaging (MRI) diffu-sion scans of the brain interior and modeled abstractly as a set of time-independent geometric trajectories in a three-dimensional brain space. Real neuronal fiber pathways ex-hibit complex but natural bundling structures, which elude existing MRI reconstruction techniques, but are easily cap-tured by our algorithm. We validate our algorithms both theoretically (uniqueness of the graph representation and provably efficient algorithms) and empirically (using both synthetic and real scanned brain data sets).