8275001
Zonotopes as bounding volumes
Zonotopes are centrally symmetric polytopes with a very special structure: they are Minkowski sums of line segments. In this paper we propose to use zonotopes as bounding volumes for geometry in collision detection and other applications where the spatial relationship between two pieces of geometry is important. We show how to construct optimal, or approximately optimal zonotopes enclosing given set of points or other geometry. We also show how zonotopes can be used for efficient collision testing, based on their representation via their defining line segments --- without ever building their explicit description as polytopes. This implicit representation adds flexibility, power, and economy to the use of zonotopes as bounding volumes.
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.
1266304
Static and kinetic geometric spanners with applications
It is well known that the Delaunay Triangulation is a spanner graph of its vertices. In this paper we show that any bounded aspect ratio triangulation in two and three dimensions is a spanner graph of its vertices as well. We extend the notion of spanner graphs to environments with obstacles and show that both the Constrained Delaunay Triangulation and bounded aspect ratio conforming triangulations are spanners with respect to the corresponding visibility graph. We also show how to kinetize the Constrained Delaunay Triangulation. Using such time-varying triangulations we describe how to maintain sets of near neighbors for a set of moving points in both unconstrained and constrained environments. Such nearest neighbor maintenance is needed in many virtual environments where nearby agents interact. Finally, we show how to use the Constrained Delaunay Triangulation in order to maintain the relative convex hull of a set of points moving inside a simple polygon.
36092896
Fractionally cascaded information in a sensor network
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.
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.
456712
Persistence barcodes for shapes
In this paper, we initiate a study of shape description and classification via the application of persistent homology to two tangential constructions on geometric objects. Our techniques combine the differentiating power of geometry with the classifying power of topology. The homology of our first construction, the tangent complex, can distinguish between topologically identical shapes with different "sharp" features, such as corners. To capture "soft" curvature-dependent features, we define a second complex, the filtered tangent complex, obtained by parametrizing a family of increasing subcomplexes of the tangent complex. Applying persistent homology, we obtain a shape descriptor, called a barcode, that is a finite union of intervals. We define a metric over the space of such intervals, arriving at a continuous invariant that reflects the geometric properties of shapes. We illustrate the power of our methods through a number of detailed studies of parametrized families of mathematical shapes.
9789536
Robust global registration
We present an algorithm for the automatic alignment of two 3D shapes (data and model), without any assumptions about their initial positions. The algorithm computes for each surface point a descriptor based on local geometry that is robust to noise. A small number of feature points are automatically picked from the data shape according to the uniqueness of the descriptor value at the point. For each feature point on the data, we use the descriptor values of the model to find potential corresponding points. We then develop a fast branch-and-bound algorithm based on distance matrix comparisons to select the optimal correspondence set and bring the two shapes into a coarse alignment. The result of our alignment algorithm is used as the initialization to ICP (iterative closest point) and its variants for fine registration of the data to the model. Our algorithm can be used for matching shapes that overlap only over parts of their extent, for building models from partial range scans, as well as for simple symmetry detection, and for matching shapes undergoing articulated motion.
31972087
Sparse data aggregation in sensor networks
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.
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.