Subdivision Directional Fields

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure-preserving: it reproduces curl-free fields precisely, and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover's distance computation, for efficient and robust computation

Coordinated Schematization for Visualizing Mobility Patterns on Networks

GPS trajectories of vehicles moving on a road network are a valuable source of traffic information. However, the sheer volume of available data makes it challenging to identify and visualize salient patterns. Meaningful visual summaries of trajectory collections require that both the trajectories and the underlying network are aggregated and simplified in a coherent manner. In this paper we propose a coordinated fully-automated pipeline for computing a schematic overview of mobility patterns from a collection of trajectories on a street network. Our pipeline utilizes well-known building blocks from GIS, automated cartography, and trajectory analysis: map matching, road selection, schematization, movement patterns, and metro-map style rendering. We showcase the results of our pipeline on two real-world trajectory collections around The Hague and Beijing

A sampling-based strategy for distributing taxis in a road network for occupancy maximization (GIS Cup)

We present a weighted sampling strategy for distributing a system of taxi agents on a road network. We consider a setting, in which each agent operates independently, following a prescribed strategy based on historical data. Furthermore, customer requests appear dynamically and are assigned to the closest unoccupied taxi agent.\u3cbr/\u3e\u3cbr/\u3eWe demonstrate that in this setting a simple sampling strategy based on the spatial distribution of historical data performs well in minimizing the average time that agents are unoccupied. The strategy is evaluated on taxi trip data in Manhattan and compared to various, more complex strategies

Segment Visibility Counting Queries in Polygons

Let $P$ be a simple polygon with $n$ vertices, and let $A$ be a set of $m$ points or line segments inside $P$. We develop data structures that can efficiently count the number of objects from $A$ that are visible to a query point or a query segment. Our main aim is to obtain fast, $O(\mathop{\textrm{polylog}} nm$), query times, while using as little space as possible. In case the query is a single point, a simple visibility-polygon-based solution achieves $O(\log nm)$ query time using $O(nm^2)$ space. In case $A$ also contains only points, we present a smaller, $O(n + m^{2 + \varepsilon}\log n)$-space, data structure based on a hierarchical decomposition of the polygon. Building on these results, we tackle the case where the query is a line segment and $A$ contains only points. The main complication here is that the segment may intersect multiple regions of the polygon decomposition, and that a point may see multiple such pieces. Despite these issues, we show how to achieve $O(\log n\log nm)$ query time using only $O(nm^{2 + \varepsilon} + n^2)$ space. Finally, we show that we can even handle the case where the objects in $A$ are segments with the same bounds.Comment: 27 pages, 13 figure

Subdivision Directional Fields

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the mixed finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure preserving: it reproduces curl-free fields precisely and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover's distance computation, for efficient and robust computation

Maximum physically consistent trajectories

We study the problem of detecting outlying measurements in a GPS trajectory. Our method considers the physical possibility for the tracked object to visit combinations of measurements,using simplified physics models. We aim to compute the maximum subsequence of the measurements that is consistent with a given physics model. We give an O(n log³ n) time algorithm for 2D-trajectories in a model with unbounded acceleration but bounded velocity, and an output-sensitive algorithm for any model where consistency checks can be done in O(1) time and consistency is transitive

Maximum Physically Consistent Trajectories

Trajectories are usually collected with physical sensors, which are prone to errors and cause outliers in the data. We aim to identify such outliers via the physical properties of the tracked entity, that is, we consider its physical possibility to visit combinations of measurements. We describe optimal algorithms to compute maximum subsequences of measurements that are consistent with (simplified) physics models. Our results are output-sensitive with respect to the number k of outliers in a trajectory of n measurements. Specifically, we describe an O(n log n log2 k) time algorithm for 2D trajectories using a model with unbounded acceleration but bounded velocity, and an O(nk) time algorithm for any model where consistency is "concatenable": a consistent subsequence that ends where another begins together form a consistent sequence. We also consider acceleration-bounded models which are not concatenable. We show how to compute the maximum subsequence for such models in O(nk2 log k) time, under appropriate realism conditions. Finally, we experimentally explore the performance of our algorithms on several large real-world sets of trajectories. Our experiments show that we are generally able to retain larger fractions of noisy trajectories than previous work and simpler greedy approaches. We also observe that the speed-bounded model may in practice approximate the acceleration-bounded model quite well, though we observed some variation between datasets