text: We study the problem of visibility-based exploration, reconstruction and surveillance in the context of supervised learning. Using a level set representation of data and information, we train a convolutional neural network to determine vantage points that maximize visibility. We show that this method drastically reduces the on-line computational cost and determines a small set of vantage points that solve the problem. This enables us to efficiently produce highly-resolved and topologically accurate maps of complex 3D environments. We present realistic simulations on 2D and 3D urban environments.
  @cite_25: The visibility level set function introduced by 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.
  @cite_6: We propose a computational method to optimally position a sensor system. Each sensor has limited range and viewing angle, and it may fail with a certain failure rate. The goal is to find the optimal locations as well as the viewing directions of all the sensors and achieve the maximal surveillance of the known environment. We set up the problem using the level set framework. Both the environment and the viewing range of the sensors are represented by level set functions. Then we solve a system of ordinary differential equations (ODEs) to find the optimal viewing directions and locations of all sensors together. Furthermore, we use the intermittent diffusion, which converts the ODEs into stochastic differential equations, to find the global maximum of the total surveillance area. The numerical examples include various failure rates of sensors, different rates of importance of the surveillance region, and 3-dimensional setups. They show the effectiveness of the proposed method.
summary: propose an alternating minimization scheme for optimizing the visibility of @math observers @cite_25 . use a system of differential equations to optimize the location and orientation of @math sensors to maximize surveillance @cite_6 . Both works assume the number of sensors is given.