arXiv subjects

Frank Staals

Publications and source records attributed to Frank Staals.

40 records · Page 3Linked to original sources

arXiv2015

Mixed Map Labeling

Point feature map labeling is a geometric problem, in which a set of input points must be labeled with a set of disjoint rectangles (the bounding boxes of the label texts). Typically, labeling models either use internal labels, which must touch their feature point, or external (boundary) labels, which are placed on one of the four sides of the input points' bounding box and which are connected to their feature points by crossing-free leader lines. In this paper we study polynomial-time algorithms for maximizing the number of internal labels in a mixed labeling model that combines internal and external labels. The model requires that all leaders are parallel to a given orientation $θ\in [0,2π)$, whose value influences the geometric properties and hence the running times of our algorithms.

cs.CG↗

arXiv2015

Central Trajectories

An important task in trajectory analysis is clustering. The results of a clustering are often summarized by a single representative trajectory and an associated size of each cluster. We study the problem of computing a suitable representative of a set of similar trajectories. To this end we define a central trajectory $\mathcal{C}$, which consists of pieces of the input trajectories, switches from one entity to another only if they are within a small distance of each other, and such that at any time $t$, the point $\mathcal{C}(t)$ is as central as possible. We measure centrality in terms of the radius of the smallest disk centered at $\mathcal{C}(t)$ enclosing all entities at time $t$, and discuss how the techniques can be adapted to other measures of centrality. We first study the problem in $\mathbb{R}^1$, where we show that an optimal central trajectory $\mathcal{C}$ representing $n$ trajectories, each consisting of $τ$ edges, has complexity $Θ(τn^2)$ and can be computed in $O(τn^2 \log n)$ time. We then consider trajectories in $\mathbb{R}^d$ with $d\geq 2$, and show that the complexity of $\mathcal{C}$ is at most $O(τn^{5/2})$ and can be computed in $O(τn^3)$ time.

cs.CG↗

arXiv2014

Terrain visibility with multiple viewpoints

We study the problem of visibility in polyhedral terrains in the presence of multiple viewpoints. We consider a triangulated terrain with $m>1$ viewpoints (or guards) located on the terrain surface. A point on the terrain is considered \emph{visible} if it has an unobstructed line of sight to at least one viewpoint. We study several natural and fundamental visibility structures: (1) the visibility map, which is a partition of the terrain into visible and invisible regions; (2) the \emph{colored} visibility map, which is a partition of the terrain into regions whose points have exactly the same visible viewpoints; and (3) the Voronoi visibility map, which is a partition of the terrain into regions whose points have the same closest visible viewpoint. We study the complexity of each structure for both 1.5D and 2.5D terrains, and provide efficient algorithms to construct them. Our algorithm for the visibility map in 2.5D terrains improves on the only existing algorithm in this setting. To the best of our knowledge, the other structures have not been studied before.

cs.CG↗

arXiv2013

Trajectory Grouping Structures

The collective motion of a set of moving entities like people, birds, or other animals, is characterized by groups arising, merging, splitting, and ending. Given the trajectories of these entities, we define and model a structure that captures all of such changes using the Reeb graph, a concept from topology. The trajectory grouping structure has three natural parameters that allow more global views of the data in group size, group duration, and entity inter-distance. We prove complexity bounds on the maximum number of maximal groups that can be present, and give algorithms to compute the grouping structure efficiently. We also study how the trajectory grouping structure can be made robust, that is, how brief interruptions of groups can be disregarded in the global structure, adding a notion of persistence to the structure. Furthermore, we showcase the results of experiments using data generated by the NetLogo flocking model and from the Starkey project. The Starkey data describe the movement of elk, deer, and cattle. Although there is no ground truth for the grouping structure in this data, the experiments show that the trajectory grouping structure is plausible and has the desired effects when changing the essential parameters. Our research provides the first complete study of trajectory group evolvement, including combinatorial, algorithmic, and experimental results.

cs.CG↗