Searching Polyhedra by Rotating Half-Planes

The Searchlight Scheduling Problem was first studied in 2D polygons, where the goal is for point guards in fixed positions to rotate searchlights to catch an evasive intruder. Here the problem is extended to 3D polyhedra, with the guards now boundary segments who rotate half-planes of illumination. After carefully detailing the 3D model, several results are established. The first is a nearly direct extension of the planar one-way sweep strategy using what we call exhaustive guards, a generalization that succeeds despite there being no well-defined notion in 3D of planar "clockwise rotation". Next follow two results: every polyhedron with r>0 reflex edges can be searched by at most r^2 suitably placed guards, whereas just r guards suffice if the polyhedron is orthogonal. (Minimizing the number of guards to search a given polyhedron is easily seen to be NP-hard.) Finally we show that deciding whether a given set of guards has a successful search schedule is strongly NP-hard, and that deciding if a given target area is searchable at all is strongly PSPACE-hard, even for orthogonal polyhedra. A number of peripheral results are proved en route to these central theorems, and several open problems remain for future work.Comment: 45 pages, 26 figure

Guarding and Searching Polyhedra

Guarding and searching problems have been of fundamental interest since the early years of Computational Geometry. Both are well-developed areas of research and have been thoroughly studied in planar polygonal settings. In this thesis we tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra. We solve the Art Gallery Problem with reflex edge guards in orthogonal polyhedra having reflex edges in just two directions: generalizing a classic theorem by O'Rourke, we prove that r/2 + 1 reflex edge guards are sufficient and occasionally necessary, where r is the number of reflex edges. We also show how to compute guard locations in O(n log n) time. Then we investigate the Art Gallery Problem with mutually parallel edge guards in orthogonal polyhedra with e edges, showing that 11e/72 edge guards are always sufficient and can be found in linear time, improving upon the previous state of the art, which was e/6. We also give tight inequalities relating e with the number of reflex edges r, obtaining an upper bound on the guard number of 7r/12 + 1. We further study the Art Gallery Problem with edge guards in polyhedra having faces oriented in just four directions, obtaining a lower bound of e/6 - 1 edge guards and an upper bound of (e+r)/6 edge guards. All the previously mentioned results hold for polyhedra of any genus. Additionally, several guard types and guarding modes are discussed, namely open and closed edge guards, and orthogonal and non-orthogonal guarding. Next, we model the Searchlight Scheduling Problem, the problem of searching a given polyhedron by suitably turning some half-planes around their axes, in order to catch an evasive intruder. After discussing several generalizations of classic theorems, we study the problem of efficiently placing guards in a given polyhedron, in order to make it searchable. For general polyhedra, we give an upper bound of r^2 on the number of guards, which reduces to r for orthogonal polyhedra. Then we prove that it is strongly NP-hard to decide if a given polyhedron is entirely searchable by a given set of guards. We further prove that, even under the assumption that an orthogonal polyhedron is searchable, approximating the minimum search time within a small-enough constant factor to the optimum is still strongly NP-hard. Finally, we show that deciding if a specific region of an orthogonal polyhedron is searchable is strongly PSPACE-hard. By further improving our construction, we show that the same problem is strongly PSPACE-complete even for planar orthogonal polygons. Our last results are especially meaningful because no similar hardness theorems for 2-dimensional scenarios were previously known

Algorithms for Optimizing Search Schedules in a Polygon

In the area of motion planning, considerable work has been done on guarding problems, where "guards", modelled as points, must guard a polygonal space from "intruders". Different variants of this problem involve varying a number of factors. The guards performing the search may vary in terms of their number, their mobility, and their range of vision. The model of intruders may or may not allow them to move. The polygon being searched may have a specified starting point, a specified ending point, or neither of these. The typical question asked about one of these problems is whether or not certain polygons can be searched under a particular guarding paradigm defined by the types of guards and intruders. In this thesis, we focus on two cases of a chain of guards searching a room (polygon with a specific starting point) for mobile intruders. The intruders must never be allowed to escape through the door undetected. In the case of the two guard problem, the guards must start at the door point and move in opposite directions along the boundary of the polygon, never crossing the door point. At all times, the guards must be able to see each other. The search is complete once both guards occupy the same spot elsewhere on the polygon. In the case of a chain of three guards, consecutive guards in the chain must always be visible. Again, the search starts at the door point, and the outer guards of the chain must move from the door in opposite directions. These outer guards must always remain on the boundary of the polygon. The search is complete once the chain lies entirely on a portion of the polygon boundary not containing the door point. Determining whether a polygon can be searched is a problem in the area of visibility in polygons; further to that, our work is related to the area of planning algorithms. We look for ways to find optimal schedules that minimize the distance or time required to complete the search. This is done by finding shortest paths in visibility diagrams that indicate valid positions for the guards. In the case of the two-guard room search, we are able to find the shortest distance schedule and the quickest schedule. The shortest distance schedule is found in O(n^2) time by solving an L_1 shortest path problem among curved obstacles in two dimensions. The quickest search schedule is found in O(n^4) time by solving an L_infinity shortest path problem among curved obstacles in two dimensions. For the chain of three guards, a search schedule minimizing the total distance travelled by the outer guards is found in O(n^6) time by solving an L_1 shortest path problem among curved obstacles in two dimensions

Planet Four: Terrains - Discovery of Araneiforms Outside of the South Polar Layered Deposits

We present the results of a systematic mapping of seasonally sculpted terrains on the South Polar region of Mars with the Planet Four: Terrains (P4T) online citizen science project. P4T enlists members of the general public to visually identify features in the publicly released Mars Reconnaissance Orbiter CTX images. In particular, P4T volunteers are asked to identify: 1) araneiforms (including features with a central pit and radiating channels known as 'spiders'); 2) erosional depressions, troughs, mesas, ridges, and quasi-circular pits characteristic of the South Polar Residual Cap (SPRC) which we collectively refer to as 'Swiss cheese terrain', and 3) craters. In this work we present the distributions of our high confidence classic spider araneiforms and Swiss cheese terrain identifications. We find no locations within our high confidence spider sample that also have confident Swiss cheese terrain identifications. Previously spiders were reported as being confined to the South Polar Layered Deposits (SPLD). Our work has provided the first identification of spiders at locations outside of the SPLD, confirmed with high resolution HiRISE imaging. We find araneiforms on the Amazonian and Hesperian polar units and the Early Noachian highland units, with 75% of the identified araneiform locations in our high confidence sample residing on the SPLD. With our current coverage, we cannot confirm whether these are the only geologic units conducive to araneiform formation on the Martian South Polar region. Our results are consistent with the current CO2 jet formation scenario with the process exploiting weaknesses in the surface below the seasonal CO2 ice sheet to carve araneiform channels into the regolith over many seasons. These new regions serve as additional probes of the conditions required for channel creation in the CO2 jet process. (Abridged)Comment: accepted to Icarus - Supplemental data files are available at https://www.zooniverse.org/projects/mschwamb/planet-four-terrains/about/results - Icarus print version available at http://www.sciencedirect.com/science/article/pii/S001910351730055

Connectivity Constraints in Network Analysis

This dissertation establishes mathematical foundations of connectivity requirements arising in both abstract and geometric network analysis. Connectivity constraints are ubiquitous in network design and network analysis. Aside from the obvious applications in communication and transportation networks, they have also appeared in forest planning, political distracting, activity detection in video sequences and protein-protein interaction networks. Theoretically, connectivity constraints can be analyzed via polyhedral methods, in which we investigate the structure of (vertex)-connected subgraph polytope (CSP). One focus of this dissertation is on performing an extensive study of facets of CSP. We present the first systematic study of non-trivial facets of CSP. One advantage to study facets is that a facet-defining inequality is always among the tightest valid inequalities, so applying facet-defining inequalities when imposing connectivity constraints can guarantee good performance of the algorithm. We adopt lifting techniques to provide a framework to generate a wide class of facet-defining inequalities of CSP. We also derive the necessary and sufficient conditions when a vertex separator inequality, which plays a critical role in connectivity constraints, induces a facet of CSP. Another advantage to study facets is that CSP is uniquely determined by its facets, so full understanding of CSP's facets indicates full understanding of CSP itself. We are able to derive a full description of CSP for a wide class of graphs, including forest and several types of dense graphs, such as graphs with small independence number, s-plex with small s and s-defective cliques with small s. Furthermore, we investigate the relationship between lifting techniques, maximum weight connected subgraph problem and node-weight Steiner tree problem and study the computational complexity of generation of facet-defining inequalities. Another focus of this dissertation is to study connectivity in geometric network analysis. In geometric applications like wireless networks and communication networks, the concept of connectivity can be defined in various ways. In one case, connectivity is imposed by distance, which can be modeled by unit disk graphs (UDG). We create a polytime algorithm to identify large 2-clique in UDG; in another case when connectivity is based on visibility, we provide a generalization of the two-guard problem

Efficient Image Segmentation and Segment-Based Analysis in Computer Vision Applications

This dissertation focuses on efficient image segmentation and segment-based object recognition in computer vision applications. Special attention is devoted to analyzing shape, of particular importance for our two applications: plant species identification from leaf photos, and object classification in remote sensing images. Additionally, both problems are bound by efficiency, constraining the choice of applicable methods: leaf recognition results are to be used within an interactive system, while remote sensing image analysis must scale well over very large image sets. Leafsnap was the first mobile app to provide automatic recognition of tree species, currently counting with over 1.7 million downloads. We present an overview of the mobile app and corresponding back end recognition system, as well as a preliminary analysis of user-submitted data. More than 1.7 million valid leaf photos have been uploaded by users, 1.3 million of which are GPS-tagged. We then focus on the problem of segmenting photos of leaves taken against plain light-colored backgrounds. These types of photos are used in practice within Leafsnap for tree species recognition. A good segmentation is essential in order to make use of the distinctive shape of leaves for recognition. We present a comparative experimental evaluation of several segmentation methods, including quantitative and qualitative results. We then introduce a custom-tailored leaf segmentation method that shows superior performance while maintaining computational efficiency. The other contribution of this work is a set of attributes for analysis of image segments. The set of attributes is designed for use in knowledge-based systems, so they are selected to be intuitive and easily describable. The attributes can also be computed efficiently, to allow applicability across different problems. We experiment with several descriptive measures from the literature and encounter certain limitations, leading us to introduce new attribute formulations and more efficient computational methods. Finally, we experiment with the attribute set on our two applications: plant species identification from leaf photos and object recognition in remote sensing images