Terrain visibility and guarding problems
Watchtowers are located on terrains to detect fires, military units are deployed to watch the terrain to prevent infiltration, and relay stations are placed such that no dead zone is present on the terrain to maintain uninterrupted communication. In this thesis, any entity that is capable of observing or sensing a piece of land or an object on the land is referred to as a guard. Thus, watchtowers, military units and relay stations are guards and so are sensors, observers (human beings), cameras and the like. Observing, seeing, covering and guarding will mean the same. The viewshed of a given guard on a terrain is defined to be those portions of the terrain visible to the guard and the calculation of the viewshed of the guard is referred to as the viewshed problem. Locating minimum number of guards on a terrain (T) such that every point on the terrain is guarded by at least one of the guards is known as terrain guarding problem (TGP). Terrains are generally represented as regular square grids (RSG) or triangulated irregular networks (TIN). In this thesis, we study the terrain guarding problem and the viewshed problem on both representations. The first problem we deal with is the 1.5 dimensional terrain guarding problem (1.5D TGP). 1.5D terrain is a cross-section of a TIN and is characterized by a piecewise linear curve. The problem has been shown to be NP-Hard. To solve the problem to optimality, a finite dominating set (FDS) of size O(n2) and a witness set of size O(n2) have been presented earlier, where n is the number of vertices on T. An FDS is a finite set of points that contains an optimal solution to an optimization problem possibly with an uncountable feasible set. A witness set is a discretization of the terrain, and thus a finite set, such that guarding of the elements of the witness set implies guarding of T. We show that there exists an FDS, composed of convex points and dip points, with cardinality O(n). We also prove that there exist witness sets of cardinality O(n), which are smaller than O(n2) found earlier. The existence of smaller FDSs and witness sets leads to the reduction of decision variables and constraints respectively in the zero-one integer programming (ZOIP) formulation of the problem. Next, we discuss the viewshed problem and TGP on TINs, also known as 2.5D terrain guarding problem. No FDS has been proposed for this problem yet. To solve the problem to optimality the viewshed problem must also be solved. Hidden surface removal algorithms that claim to solve the viewshed problem do not provide analytical solutions and present some ambiguities regarding implementation. Other studies that make use of the horizon information of the terrain to calculate viewshed do so by projecting the vertices of the horizon onto the supporting plane of the triangle of interest and then by connecting the projections of the vertices to find the visible region on the triangle. We show that this approach is erroneous and present an alternative projection model in 3D space. The invisible region on a given triangle caused by another traingle is shown to be characterized by a system of nonlinear equations, which are linearized to obtain a polyhedral set. Finally, a realistic example of the terrain guarding problem is studied, which involves the surveillance of a rugged geographical terrain approximated by RSG by means of thermal cameras. A number of issues related to the terrain-guarding problem on RSGs are addressed with integer-programming models proposed to solve the problem. Next, a sensitivity analysis is carried out in which two fictitious terrains are created to see the effect of the resolution of a terrain, and of terrain characteristics, on coverage optimization. Also, a new problem, called the blocking path problem, is introduced and solved by an integer-programming formulation based on a network paradigm.