Approximative Terrain Guarding with Given Number of Guards

Guarding a surface is a well known optimization problem of the visibility site analysis and has many applications. The basic problem is searching for the minimum number of guards needed to guard (see) the entire surface. More realistic is the guarding where the number of guards is upward limited and the optimization problem is to search for their locations in order to guard as much surface as possible. In the paper this problem is treated in detail. Several known heuristics (greedy add, greedy add with swap and stingy drop) are revised and a new technique called solution improving technique is proposed. The technique improves the results of the known algorithms and is used in indirect solving of the problem. Tests on 44 DEMs from USGS DEM Repository showed that our technique yields comparative results for smaller number of guards and better results for higher number of guards

Terrain visibility optimization problems

Ankara : The Department of Industrial Engineering and the Institute of Engineering and Sciences of Bilkent University, 2001.Thesis (Master's) -- Bilkent University, 2001.Includes bibliographical references leaves 92-96The Art Gallery Problem is the problem of determining the number of observers necessary to cover an art gallery such that every point is seen by at least one observer. This problem is well known and has a linear time solution for the 2 dimensional case, but little is known about 3-D case. In this thesis, the dominance relationship between vertex guards and point guards is searched and found that a convex polyhedron can be constructed such that it can be covered by some number of point guards which is one third of the number of the vertex guards needed. A new algorithm which tests the visibility of two vertices is constructed for the discrete case. How to compute the visible region of a vertex is shown for the continuous case. Finally, several potential applications of geometric terrain visibility in geographic information systems and coverage problems related with visibility are presented.Düger, İbrahimM.S

VC-Dimension of Exterior Visibility

In this paper, we study the Vapnik-Chervonenkis (VC)-dimension of set systems arising in 2D polygonal and 3D polyhedral configurations where a subset consists of all points visible from one camera. In the past, it has been shown that the VC-dimension of planar visibility systems is bounded by 23 if the cameras are allowed to be anywhere inside a polygon without holes [1]. Here, we consider the case of exterior visibility, where the cameras lie on a constrained area outside the polygon and have to observe the entire boundary. We present results for the cases of cameras lying on a circle containing a polygon (VC-dimension= 2) or lying outside the convex hull of a polygon (VC-dimension= 5). The main result of this paper concerns the 3D case: We prove that the VC-dimension is unbounded if the cameras lie on a sphere containing the polyhedron, hence the term exterior visibility

The three-dimensional art gallery problem and its solutions

This thesis addressed the three-dimensional Art Gallery Problem (3D-AGP), a version of the art gallery problem, which aims to determine the number of guards required to cover the interior of a pseudo-polyhedron as well as the placement of these guards. This study exclusively focused on the version of the 3D-AGP in which the art gallery is modelled by an orthogonal pseudo-polyhedron, instead of a pseudo-polyhedron. An orthogonal pseudopolyhedron provides a simple yet effective model for an art gallery because of the fact that most real-life buildings and art galleries are largely orthogonal in shape. Thus far, the existing solutions to the 3D-AGP employ mobile guards, in which each mobile guard is allowed to roam over an entire interior face or edge of a simple orthogonal polyhedron. In many realword applications including the monitoring an art gallery, mobile guards are not always adequate. For instance, surveillance cameras are usually installed at fixed locations. The guard placement method proposed in this thesis addresses such limitations. It uses fixedpoint guards inside an orthogonal pseudo-polyhedron. This formulation of the art gallery problem is closer to that of the classical art gallery problem. The use of fixed-point guards also makes our method applicable to wider application areas. Furthermore, unlike the existing solutions which are only applicable to simple orthogonal polyhedra, our solution applies to orthogonal pseudo-polyhedra, which is a super-class of simple orthogonal polyhedron. In this thesis, a general solution to the guard placement problem for 3D-AGP on any orthogonal pseudo-polyhedron has been presented. This method is the first solution known so far to fixed-point guard placement for orthogonal pseudo-polyhedron. Furthermore, it has been shown that the upper bound for the number of fixed-point guards required for covering any orthogonal polyhedron having n vertices is (n3/2), which is the lowest upper bound known so far for the number of fixed-point guards for any orthogonal polyhedron. This thesis also provides a new way to characterise the type of a vertex in any orthogonal pseudo-polyhedron and has conjectured a quantitative relationship between the numbers of vertices with different vertex configurations in any orthogonal pseudo-polyhedron. This conjecture, if proved to be true, will be useful for gaining insight into the structure of any orthogonal pseudo-polyhedron involved in many 3-dimensional computational geometrical problems. Finally the thesis has also described a new method for splitting orthogonal polygon iv using a polyline and a new method for splitting an orthogonal polyhedron using a polyplane. These algorithms are useful in applications such as metal fabrication