交点检测中轨迹中正方形或圆的临界位置
Critical Placements of a Square or Circle amidst Trajectories for Junction Detection
摘要：
出于自动路口识别跟踪数据，我们研究了一个问题，放置一个正方形或固定大小的磁盘在一个排列的线或线段在平面上。我们让直线和线段与正方形或圆形的交点之间的距离定义一个聚类，并研究了这种聚类的\emph{critical}放置的复杂性。在这里，临界意味着位置的任意小的移动改变了聚类。参数$\varepsilon$定义聚类的粒度。在没有任何关于$\varepsilon$的假设的情况下，临界放置具有平凡的$O(n^4)$上界。当正方形或圆形具有单位大小且$0 < \varepsilon < 1$给定时，我们给出了一个改进的$O(n^2/\varepsilon^2)$界，在最坏情况下，该界是紧的。我们使用我们的组合边界来设计有效的算法来计算连接。作为我们的算法的概念证明，我们有一个原型实现，展示了他们的应用程序中的一组$n$轨迹和他们的$k$最重要的路口的基本可视化。
英文摘要：
Motivated by automated junction recognition in tracking data, we study a problem of placing a square or disc of fixed size in an arrangement of lines or line segments in the plane. We let distances among the intersection points of the lines and line segments with the square or circle define a clustering, and study the complexity of \emph{critical} placements for this clustering. Here critical means that arbitrarily small movements of the placement change the clustering. A parameter $\varepsilon$ defines the granularity of the clustering. Without any assumptions on $\varepsilon$, the critical placements have a trivial $O(n^4)$ upper bound. When the square or circle has unit size and $0 < \varepsilon < 1$ is given, we show a refined $O(n^2/\varepsilon^2)$ bound, which is tight in the worst case. We use our combinatorial bounds to design efficient algorithms to compute junctions. As a proof of concept for our algorithms we have a prototype implementation that showcases their application in a basic visualization of a set of $n$ trajectories and their $k$ most important junctions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
