Approximate solutions to several visibility optimization problems

Abstract

The visibility level set function introduced by Tsai et al. allows for gradient based and variational formulations of many classical visibility optimization problems. In this work we propose solutions to two such problems. The first asks where to position n-observers such that the area visible to these observers is maximized. The second problem is to determine the shortest route an observer should take through a map such that every point in the map is visible from at least one vantage point on the route. These problems are similar to the "art gallery" and "watchman route" problems, respectively. We propose a greedy iterative algorithm, formulated in the level set framework as the solution to the art gallery problem. We also propose a variational solution to the watchman route problem which achieves complete visibility coverage of the domain while attaining a local minimum of path length. © 2011 International Press.