Metric View Planning Problem with Traveling Cost and Visibility Range

In this paper, we consider the problem where a point robot in a 2D or 3D environment equipped with an omnidirectional range sensor of finite range D is asked to inspect a set of surface patches, while minimizing the sum of view cost, proportional to the number of viewpoints planned, and the travel cost, proportional to the length of path traveled. We call it the metric view planning problem with traveling cost and visibility range or metric TVPP in short. Via an L-reduction from the set covering problem to a two-dimensional metric TVPP, we show that the metric TVPP cannot be approximated within O(log m) ratio by any polynomial algorithm, where m is the number of surface patches to cover. We then analyze the natural two-level algorithm, presented by Danner and Kavraki (2002), of solving first the view planning problem to get an approximate solution, and then solving, again using an approximation algorithm, the Metric traveling salesman problem to connect the planned viewpoints. We show this greedy algorithm has the approximation ratio of O(log m). Thus, it asymptotically achieves the best approximation ratio one can hope for.