Communications in Mathematical Sciences
Volume 9 (2011)
Approximate solutions to several visibility optimization problems
Pages: 535 – 550
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.
visibility, optimization, level set method
2010 Mathematics Subject Classification
35-xx, 49-xx, 65-xx