Terrain Visibility with Multiple Viewpoints

Abstract

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 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 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.

Terrain prickliness: theoretical grounds for low complexity viewsheds

“…It is conceivable that one could achieve this assuming, for instance, realistic terrain conditions [17], or a slope condition [21]. Finally, extending our study to viewsheds of multiple viewpoints, whose complexity was studied recently for the first time [7], is another interesting direction for further research.…”

“…5). Vertices of the viewshed of p can have three types [7]. A vertex of type 1 is a vertex of T , of which there are clearly only n. A vertex of type 2 is the intersection of an edge of T and a vase.…”

An Efficient Algorithm for the 1D Total Visibility-Index Problem and Its Parallelization

“…Despite its simplicity and disappointing quadratic performance, to the best of our knowledge, this is the best known solution for this problem to date. e 1.5D terrain-guarding problem (TGP), which is highly related to the 1D total visibility-index problem, has been extensively studied [14,15,18,22]. e terrain-guarding problem involves nding the minimum number of points needed to view an entire 1-dimensional set of vertices.…”

“…Hurtado et. al [18] present a solution to a variant of TGP by considering the viewshed (list of visible points) of m viewpoints (or guards). By sweeping a one-dimensional terrain (an x-monotone polyline of vertices) and maintaining a list of vertices that are dominated, they nd the viewshed of m points in O(n + m log m) time.…”

“…While Hurtado et. al [18] nd the list of points visible from m viewpoints, we are concerned with nding the number of points on the terrain visible, for each point. erefore, it is not clear how their techniques can be used to solve the total visibility-index problem e ciently.…”