Interactive Exploration of the Temporal $α$-Shape

Abstract: Shape is a powerful tool to understand point sets. A formal notion of shape is given by $\alpha$-shapes, which generalize the convex hull and provide adjustable level of detail. Many real-world point sets have an inherent temporal property as natural processes often happen over time, like lightning strikes during thunderstorms or moving animal swarms. To explore such point sets, where each point is associated with one timestamp, interactive applications may utilize $\alpha$-shapes and allow the user to specify different time windows and $\alpha$-values. We show how to compute the temporal $\alpha$-shape $\alpha_T$, a minimal description of all $\alpha$-shapes over all time windows, in output-sensitive linear time. We also give complexity bounds on $|\alpha_T|$. We use $\alpha_T$ to interactively visualize $\alpha$-shapes of user-specified time windows without having to constantly compute requested $\alpha$-shapes. Experimental results suggest that our approach outperforms an existing approach by a factor of at least $\sim$52 and that the description we compute has reasonable size in practice. The basis for our algorithm is an existing algorithm which computes all Delaunay triangles over all time windows using $\mathcal{O}(1)$ time per triangle. Our approach generalizes to higher dimensions with the same runtime for fixed $d$.