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

A New Upper Bound for the VC-Dimension of Visibility Regions

In this paper we are proving the following fact. Let P be an arbitrary simple polygon, and let S be an arbitrary set of 15 points inside P. Then there exists a subset T of S that is not "visually discernible", that is, T is not equal to the intersection of S with the visibility region vis(v) of any point v in P. In other words, the VC-dimension d of visibility regions in a simple polygon cannot exceed 14. Since Valtr proved in 1998 that d \in [6,23] holds, no progress has been made on this bound. By epsilon-net theorems our reduction immediately implies a smaller upper bound to the number of guards needed to cover P.Comment: 25 pages, 18 Figures. An extended abstract of this paper appeared at SoCG '1

Visibility Domains and Complexity

Two problems in discrete and computational geometry are considered that are related to questions about the combinatorial complexity of arrangements of visibility domains and about the hardness of path planning under cost measures defined using visibility domains. The first problem is to estimate the VC-dimension of visibility domains. The VC-dimension is a fundamental parameter of every range space that is typically used to derive upper bounds on the size of hitting sets. Better bounds on the VC-dimension directly translate into better bounds on the size of hitting sets. Estimating the VC-dimension of visibility domains has proven to be a hard problem. In this thesis, new tools to tackle this problem are developed. Encircling arguments are combined with decomposition techniques of a new kind. The main ingredient of the novel approach is the idea of relativization that makes it possible to replace in the analysis of intersections the complicated visibility domains by simpler geometric ranges. The main result here is the new upper bound of 14 on the VC-dimension of visibility polygons in simple polygons that improves significantly upon the previously known best upper bound of 23. For the VC-dimension of perimeter visibility domains, the new techniques yield an upper bound of 7 that leaves only a very small gap to the best known lower bound of 5. The second problem considered is to compute the barrier resilience of visibility domains. In barrier resilience problems, one is given a set of barriers and two points s and t in R^d. The task is to find the minimum number of barriers one has to remove such that there is a way between s and t that does not cross a barrier. In the field of sensor networks, the barriers are interpreted as sensor ranges and the barrier resilience of a network is a measure for its vulnerability. In this thesis the very natural special case where the barriers are visibility domains is investigated. It can also be formulated in terms of finding a so-called minimum witness path. For visibility domains in simple polygons it is shown that one can find an optimal path efficiently. For polygons with holes an approximation hardness result is shown that is stronger than previous hardness results in geometric settings. Two different three-dimensional settings are considered and their respective relations to the Minimum Neighborhood Path problem and the Minimum Color Path problem in graphs are demonstrated. For one of the three-dimensional problems a 2-approximation algorithm is designed. For the general problem of finding minimum witness paths among polyhedral obstacles it turns out that it is not approximable in a strong sense

Transforming triangulations on non planar-surfaces

We consider whether any two triangulations of a polygon or a point set on a non-planar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinder.Comment: 19 pages, 17 figures. This version has been accepted in the SIAM Journal on Discrete Mathematics. Keywords: Graph of triangulations, triangulations on surfaces, triangulations of polygons, edge fli

Residue formulae for vector partitions and Euler-MacLaurin sums

Given a finite set of vectors spanning a lattice and lying in a halfspace of a real vector space, to each vector $a$ in this vector space one can associate a polytope consisting of nonnegative linear combinations of the vectors in the set which sum up to $a$. This polytope is called the partition polytope of $a$. If $a$ is integral, this polytope contains a finite set of lattice points corresponding to nonnegative integral linear combinations. The partition polytope associated to an integral $a$ is a rational convex polytope, and any rational convex polytope can be realized canonically as a partition polytope. We consider the problem of counting the number of lattice points in partition polytopes, or, more generally, computing sums of values of exponential-polynomial functions on the lattice points in such polytopes. We give explicit formulae for these quantities using a notion of multi-dimensional residue due to Jeffrey-Kirwan. We show, in particular, that the dependence of these quantities on $a$ is exponential-polynomial on "large neighborhoods" of chambers. Our method relies on a theorem of separation of variables for the generating function, or, more generally, for periodic meromorphic functions with poles on an arrangement of affine hyperplanes.Comment: Latex, 44 pages, eepic picture file

Development of an EVA systems cost model. Volume 1: Design guides synopsis-EVA equipment

EVA equipment design guides and crewman interfaces are provided. A summary presents data on suited crewman mobility capabilities and on off-the-shelf Skylab hardware for economy planning

Indoor Semantic Modelling for Routing:

Humans perform many activities indoors and they show a growing need for indoor navigation, especially in unfamiliar buildings such as airports, museums and hospitals. Complexity of such buildings poses many challenges for building managers and visitors. Indoor navigation services play an important role in supporting these indoor activities. Indoor navigation covers extensive topics such as: 1) indoor positioning and localization; 2) indoor space representation for navigation model generation; 3) indoor routing computation; 4) human wayfinding behaviours; and 5) indoor guidance (e.g., textual directories). So far, a large number of studies of pedestrian indoor navigation have presented diverse navigation models and routing algorithms/methods. However, the major challenge is rarely referred to: how to represent the complex indoor environment for pedestrians and conduct routing according to the different roles and sizes of users. Such complex buildings contain irregular shapes, large open spaces, complicated obstacles and different types of passages. A navigation model can be very complicated if the indoors are accurately represented. Although most research demonstrates feasible indoor navigation models and related routing methods in regular buildings, the focus is still on a general navigation model for pedestrians who are simplified as circles. In fact, pedestrians represent different sizes, motion abilities and preferences (e.g., described in user profiles), which should be reflected in navigation models and be considered for indoor routing (e.g., relevant Spaces of Interest and Points of Interest). In order to address this challenge, this thesis proposes an innovative indoor modelling and routing approach – two-level routing. It specially targets the case of routing in complex buildings for distinct users. The conceptual (first) level uses general free indoor spaces: this is represented by the logical network whose nodes represent the spaces and edges stand for their connectivity; the detailed (second) level focuses on transition spaces such as openings and Spaces of Interest (SOI), and geometric networks are generated regarding these spaces. Nodes of a geometric network refers to locations of doors, windows and subspaces (SOIs) inside of the larger spaces; and the edges represent detailed paths among these geometric nodes. A combination of the two levels can represent complex buildings in specified spaces, which avoids maintaining a largescale complete network. User preferences on ordered SOIs are considered in routing on the logical network, and preferences on ordered Points of Interest (POI) are adopted in routing on geometric networks. In a geometric network, accessible obstacle-avoiding paths can be computed for users with different sizes. To facilitate automatic generation of the two types of network in any building, a new data model named Indoor Navigation Space Model (INSM) is proposed to store connectivity, semantics and geometry of indoor spaces for buildings. Abundant semantics of building components are designed in INSM based on navigational functionalities, such as VerticalUnit(VU) and HorizontalConnector(HC) as vertical and horizontal passages for pedestrians. The INSM supports different subdivision ways of a building in which indoor spaces can be assigned proper semantics. A logical and geometric network can be automatically derived from INSM, and they can be used individually or together for indoor routing. Thus, different routing options are designed. Paths can be provided by using either the logical network when some users are satisfied with a rough description of the path (e.g., the name of spaces), or a geometric path is automatically computed for a user who needs only a detailed path which shows how obstacles can be avoided. The two-level routing approach integrates both logical and geometric networks to obtain paths, when a user provides her/his preferences on SOIs and POIs. For example, routing results for the logical network can exclude unrelated spaces and then derive geometric paths more efficiently. In this thesis, two options are proposed for routing just on the logical network, three options are proposed for routing just on the geometric networks, and seven options for two-level routing. On the logical network, six routing criteria are proposed and three human wayfinding strategies are adopted to simulate human indoor behaviours. According to a specific criterion, space semantics of logical nodes is utilized to assign different weights to logical nodes and edges. Therefore, routing on the logical network can be accomplished by applying the Dijkstra algorithm. If multiple criteria are adopted, an order of criteria is applied for routing according to a specific user. In this way, logical paths can be computed as a sequence of indoor spaces with clear semantics. On geometric networks, this thesis proposes a new routing method to provide detailed paths avoiding indoor obstacles with respect to pedestrian sizes. This method allows geometric networks to be derived for individual users with different sizes for any specified spaces. To demonstrate the use of the two types of network, this thesis tests routing on one level (the logical or the geometric network). Four case studies about the logical network are presented in both simple and complex buildings. In the simple building, no multiple paths lie between spaces A and B, but in the complex buildings, multiple logical paths exist and the candidate paths can be reduced by applying these routing criteria in an order for a user. The relationships of these criteria to user profiles are assumed in this thesis. The proposed geometric routing regarding user sizes is tested with three case studies: 1) routing for pedestrians with two distinct sizes in one space; 2) routing for pedestrians with changed sizes in one space; and 3) a larger geometric network formed by the ones in a given sequence of spaces. The first case shows that a small increase of user size can largely change the accessible path; the second case shows different path segments for distinct sizes can be combined into one geometric path; the third case demonstrates a geometric network can be created ’on the fly’ for any specified spaces of a building. Therefore, the generation and routing of geometric networks are very flexible and fit to given users. To demonstrate the proposed two-level routing approach, this thesis designs five cases. The five cases are distinguished according to the method of model creation (pre-computed or ’on-the-fly’) and model storage (on the client or server). Two of them are realized in this thesis: 1) Case 1 just in the client pre-computes the logical network and derives geometric networks ’on the fly’; 2) Case 2 just in the client pre-computes and stores the logical and geometric networks for certain user sizes. Case 1 is implemented in a desktop application for building managers, and Case 2 is realized as a mobile mock-up for mobile users without an internet connection. As this thesis shows, two-level routing is powerful enough to effectively provide indicative logical paths and/or comprehensive geometric paths, according to different user requirements on path details. In the desktop application, three of the proposed routing options for two-level routing are tested for the simple OTB building and the complex Schiphol Airport building. These use cases demonstrate that the two-level routing approach includes the following merits: It supports routing in different abstraction forms of a building. The INSM model can describe different subdivision results of a building, and it allows two types of routing network to be derived – pure logical and geometric ones. The logical network contains the topology and semantics of indoor spaces, and the geometric network provides accurate geometry for paths. A consistent navigation model is formed with the two networks, i.e., the conceptual and detailed levels. On the conceptual level, it supports routing on a logical network and assists the derivation of a conceptual path (i.e., logical path) for a user in terms of space sequence. Routing criteria are designed based on the INSM semantics of spaces, which can generate logical paths similar to human wayfinding results such as minimizing VerticalUnit or HorizontalConnector. On the detailed level, it considers the size of users and results in obstacle-avoiding paths. By using this approach, geometric networks can be generated to avoid obstacles for the given users and accessible paths are flexibly provided for user demands. This approach can process changes of user size more efficiently, in contrast to routing on a complete geometric network. It supports routing on both the logical and the geometric networks, which can generate geometric paths based on user-specific logical paths, or re-compute logical paths when geometric paths are inaccessible. This computation method is very useful for complex buildings. The two-level routing approach can flexibly provide logical and geometric paths according to user preferences and sizes, and can adjust the generated paths in limited time. Based on the two-level routing approach, this thesis also provides a vision on possible cooperation with other methods. A potential direction is to design more routing options according to other indoor scenarios and user preferences. Extensions of the two-level routing approach, such as other types of semantics, multi-level networks and dynamic obstacles, will make it possible to deal with other routing cases. Last but not least, it is also promising to explore its relationships with indoor guidance, different building subdivisions and outdoor navigation