17743318
The Centroid of Points with Approximate Weights
Let S be a set of points in ℝd, each with a weight that is not known precisely, only known to fall within some range. What is the locus of the centroid of S? We prove that this locus is a convex polytope, the projection of a zonotope in ℝd+1. We derive complexity bounds and algorithms for the construction of these “centroid polytopes”.
11907465
Deformable Free-Space Tilings for Kinetic Collision Detection†
We present kinetic data structures for detecting collisions between a set of polygons that are moving continuously. Unlike classical collision detection methods that rely on bounding volume hierarchies, our method is based on deformable tilings of the free space surrounding the polygons. The basic shape of our tiles is that of a pseudo-triangle, a shape sufficiently flexible to allow extensive deformation, yet structured enough to make detection of self-collisions easy. We show different schemes for maintaining pseudo-triangulations as a kinetic data structure, and we analyze their performance. Specifically, we first describe an algorithm for maintaining a pseudo-triangulation of a point set, and show that the pseudo-triangulation changes only quadratically many times if points move along algebraic arcs of constant degree. In addition, by refining the pseudo-triangulation, we show triangulations of points that only change about O(n 7 / 3 ) times for linear motion. We then describe an algorithm for maintaining a pseudo-triangulation of a set of convex polygons. Finally, we extend our algorithm to the general case of maintaining a pseudo-triangulation of a set of moving or deforming simple polygons.
10626633
Cylindrical static and kinetic binary space partitions
We describe the first known algorithm for efficiently maintaining a Binary Space Partition (BSP) for n continuously moving segments in the plane, whose interiors remain disjoint throughout the motion. Under reasonable assumptions on the motion, we show that the total number of times this BSP changes is O.n 2 / ,a nd that we can update the BSP in O.logn/ expected time per change. Throughout the motion, the expected size of the BSP is O.n logn/. We also consider the problem of constructing a BSP for n static triangles with pairwise-disjoint interiors in R 3 . We present a randomized algorithm that constructs a BSP of size O.n 2 / in O.n 2 log 2 n/ expected time. We also describe a deterministic algorithm that constructs a BSP of size O..nCk/log 2 n/ and height O.logn/ in O..nCk/log 3 n/ time, where k is the number of intersection points between the edges of the projections of the triangles onto the xy-plane. This is the first known algorithm that constructs a BSP of O.logn/ height for disjoint triangles inR 3 . © 2000 Elsevier Science B.V. All rights reserved.
2060671
Landmark Selection and Greedy Landmark-Descent Routing for Sensor Networks
We study the problem of landmark selection for landmark-based routing in a network of fixed wireless communication nodes. We present a distributed landmark selection algorithm that does not rely on global clock synchronization, and a companion local greedy landmark-based routing scheme. We assume no node location information, and that each node can communicate with some of its geographic neighbors. Each node is named by its hop count distances to a small number of nearby landmarks. Greedy routing at a node is performed to equalize its vector of landmark distances to that of the destination. This is done by following the shortest path to the landmark that maximizes the ratio of its distances to the source and the destination. In addition, we propose a method to alleviate the difficulty in routing to destinations near the boundaries by virtually expanding the network boundaries. The greedy routing, when combined with our landmark selection scheme, has a provable bounded path stretch relative to the best path possible, and guarantees packet delivery in the continuous domain. In the discrete domain, our simulations show that the landmark selection scheme is effective, and the companion routing scheme performs well under realistic settings. Both the landmark selection and greedy routing assumes no specific communication model and works with asymmetric links. Although some of the analysis are non-trivial, the algorithms are simple, flexible and cost-effective enough to warrant a real-world deployment.
2650108
Lightweight sensing and communication protocols for target enumeration and aggregation
The development of lightweight sensing andcommunication protocols is a key requirement for designing resource constrained sensor networks. This paper introduces a set of efficient protocols and algorithms, DAM, EBAM, and EMLAM, for constructing and maintaining sensor aggregates that collectively monitor target activity in the environment. A sensor aggregate comprises those nodes in a network that satisfy a grouping predicate for a collaborative processing task. The parameters of the predicate depend on the task and its resource requirements. Since the foremost purpose of a sensor network is to selectively gather information about the environment, the formation of appropriate sensor aggregates is crucial for optimally allocating resources to sensing and communication tasks.This paper makes minimal assumptions about node onboard processing and communication capabilities so as to allow possible implementations on resource-constrained hardware. Factors affecting protocol performance are discussed. The paper presents simulation results showing how the protocol performance varies as key network and task parameters are varied. It also provides probabilistic analyses of network behavior consistent with the simulation results. The protocols have been experimentally validated on a sensor network testbed comprising 25 Berkeley MICA sensor motes.
36092896
Fractionally cascaded information in a sensor network
34799453
Estimating surface normals in noisy point cloud data
In this paper we describe and analyze a method based on local least square fitting for estimating the normals at all sample points of a point cloud data (PCD) set, in the presence of noise. We study the effects of neighborhood size, curvature, sampling density, and noise on the normal estimation when the PCD is sampled from a smooth curve in ℝ2 or a smooth surface in ℝ3, and noise is added. The analysis allows us to find the optimal neighborhood size using other local information from the PCD. Experimental results are also provided.
9104347
Reconstruction Using Witness Complexes
We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions.
2998093
An empirical comparison of techniques for updating Delaunay triangulations
The computation of Delaunay triangulations from static point sets has been extensively studied in computational geometry. When the points move with known trajectories, kinetic data structures can be used to maintain the triangulation. However, there has been little work so far on how to maintain the triangulation when the points move without explicit motion plans, as in the case of a physical simulation. In this paper we examine how to update Delaunay triangulations after small displacements of the defining points, as might be provided by a physics-based integrator. We have implemented a variety of update algorithms, many new, toward this purpose. We ran these algorithms on a corpus of data sets to provide running time comparisons and determined that updating Delaunay can be significantly faster than recomputing.
5682640
A dual-space approach to tracking and sensor management in wireless sensor networks
Wireless ad hoc sensor networks have the advantage of spanning a large geographical region and being able to collaboratively detect and track non-local spatio-temporal events. This paper presents a dual-space approach to event tracking and sensor resource management in sensor networks. The dual-space transformation maps a non-local phenomenon, e.g., the edge of a half-plane shadow, to a single point in the dual space, and maps locations of distributed sensor nodes to a set of lines that partitions the dual space. The detection problem becomes finding and tracking the cell that contains the point in the arrangement defined by these lines. This mechanism can be effectively used for power management of the sensor network - nodes that will not be immediately visited by an event can be turned off to save energy required for sensing, processing, and communication. The approach has been successfully demonstrated on a laboratory testbed built using the UC Berkeley motes sensors. An implemented application of detecting and tracking light shadow edges moving over a sensor field is described.
207156516
Registration of point cloud data from a geometric optimization perspective
We propose a framework for pairwise registration of shapes represented by point cloud data (PCD). We assume that the points are sampled from a surface and formulate the problem of aligning two PCDs as a minimization of the squared distance between the underlying surfaces. Local quadratic approximants of the squared distance function are used to develop a linear system whose solution gives the best aligning rigid transform for the given pair of point clouds. The rigid transform is applied and the linear system corresponding to the new orientation is build. This process is iterated until it converges. The point-to-point and the point-to-plane Iterated Closest Point (ICP) algorithms can be treated as special cases in this framework. Our algorithm can align PCDs even when they are placed far apart, and is experimentally found to be more stable than point-to-plane ICP. We analyze the convergence behavior of our algorithm and of point-to-point and point-to-plane ICP under our proposed framework, and derive bounds on their rate of convergence. We compare the stability and convergence properties of our algorithm with other registration algorithms on a variety of scanned data.