Polygon Exploration with Time-Discrete Vision

With the advent of autonomous robots with two- and three-dimensional scanning capabilities, classical visibility-based exploration methods from computational geometry have gained in practical importance. However, real-life laser scanning of useful accuracy does not allow the robot to scan continuously while in motion; instead, it has to stop each time it surveys its environment. This requirement was studied by Fekete, Klein and Nuechter for the subproblem of looking around a corner, but until now has not been considered in an online setting for whole polygonal regions. We give the first algorithmic results for this important algorithmic problem that combines stationary art gallery-type aspects with watchman-type issues in an online scenario: We demonstrate that even for orthoconvex polygons, a competitive strategy can be achieved only for limited aspect ratio A (the ratio of the maximum and minimum edge length of the polygon), i.e., for a given lower bound on the size of an edge; we give a matching upper bound by providing an O(log A)-competitive strategy for simple rectilinear polygons, using the assumption that each edge of the polygon has to be fully visible from some scan point.Comment: 28 pages, 17 figures, 2 photographs, 3 tables, Latex. Updated some details (title, figures and text) for final journal revision, including explicit assumption of full edge visibilit

Online Searching with an Autonomous Robot

We discuss online strategies for visibility-based searching for an object hidden behind a corner, using Kurt3D, a real autonomous mobile robot. This task is closely related to a number of well-studied problems. Our robot uses a three-dimensional laser scanner in a stop, scan, plan, go fashion for building a virtual three-dimensional environment. Besides planning trajectories and avoiding obstacles, Kurt3D is capable of identifying objects like a chair. We derive a practically useful and asymptotically optimal strategy that guarantees a competitive ratio of 2, which differs remarkably from the well-studied scenario without the need of stopping for surveying the environment. Our strategy is used by Kurt3D, documented in a separate video.Comment: 16 pages, 8 figures, 12 photographs, 1 table, Latex, submitted for publicatio

Engineering Art Galleries

The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this survey, we describe this work, show the chronology of developments, and compare current algorithms, including two unpublished versions, in an exhaustive experiment. Furthermore, we show what core algorithmic ingredients have led to recent successes

Computing Approximate Solutions to the Art Gallery Problem and Watchman Route Problem by Means of Photon Mapping

Wireless sensor networks (WSNs) can be partitioned component sensor nodes (SNs) who are meant to operate and sense information arriving from multiple spectra in their environment. Determining where to place a single SN or multiple SNs such that the amount of information gained is maximized while the number of SNs used to gain that information is minimized is an instance of solving the art gallery problem (AGP). In order to solve the AGP, we present the Sensor Placement Optimization via Queries (SPOQ) algorithm that uses level sets populated by queries to a photon map in order to find observation points that sense as many photons as possible. Since we are using photon mapping as our means of modeling how information is conveyed, SPOQ can then take into account static or dynamic environmental conditions and can use exploratory or precomputed sensing. Unmanned vehicles can be designated more generally as UxVs where “x” indicates the environment they are expected to operate – either in the air, on the ground, underwater or on the water’s surface. Determining how to plan an optimal route by a single UxV or multiple UxVs operating in their environment such that the amount of information gained is maximized while the cost of gaining that information is minimized is an instance of solving the watchman route problem (WRP). In order to solve the WRP, we present the Photon-mapping-Informed active-Contour Route Designator (PICRD) algorithm. PICRD heuristically solves the WRP by utilizing SPOQ’s AGP-solving vertices and connecting them with the high visibility vertices provided by a photon-mapping informed Chan-Vese segmentation mesh using a shortest-route path-finding algorithm. Since we are using photon-mapping as our foundation for determining sensor coverage by the PICRD algorithm, we can then take into account the behavior of photons as they propagate through the various environmental conditions that might be encountered by a single or multiple UxVs

Detours admitting short paths

Finding shortest paths between two vertices in a weighted graph is a well explored problem and several efficient algorithms for solving it have been reported. We propose a new variation of this problem which we call the Detour Admitting Shortest Path Problem (DASPP).We present an efficient algorithm for solving DASPP. This is the first algorithm that constructs a shortest path such that each edge of the shortest path admits a detour with no more than k−hops. This algorithm has important applications in transportation networks. We also present implementation issues for the detour admitting shortest path algorithm