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.
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.
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.
36092896
Fractionally cascaded information in a sensor network
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.
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.
1985585
A Computational Framework for Handling Motion
We present a framework for implementing geometric algorithms involving motion. It is written in C++ and modeled after and makes extensive use of CGAL (Computational Geometry Algorithms Library) [4]. The framework allows easy implementation of kinetic data structure style geometric algorithms—ones in which the combinatorial structure changes only at discrete times corresponding to roots of functions of the motions of the primitives. This paper discusses the architecture of the framework and how to use it. We also briefly present a polynomial package we wrote, that supports exact and filtered comparisons of real roots of polynomials and is extensively used in the framework. We plan to include our framework in the next release of CGAL.
1832504
Metropolis light transport
We present a new Monte Carlo method for solving the light transport problem, inspired by the Metropolis sampling method in computational physics. To render an image, we generate a sequence of light transport paths by randomly mutating a single current path (e.g. adding a new vertex to the path). Each mutation is accepted or rejected with a carefully chosen probability, to ensure that paths are sampled according to the contribution they make to the ideal image. We then estimate this image by sampling many paths, and recording their locations on the image plane. Our algorithm is unbiased, handles general geometric and scattering models, uses little storage, and can be orders of magnitude more efficient than previous unbiased approaches. It performs especially well on problems that are usually considered difficult, e.g. those involving bright indirect light, small geometric holes, or glossy surfaces. Furthermore, it is competitive with previous unbiased algorithms even for relatively simple scenes. The key advantage of the Metropolis approach is that the path space is explored locally, by favoring mutations that make small changes to the current path. This has several consequences. First, the average cost per sample is small (typically only one or two rays). Second, once an important path is found, the nearby paths are explored as well, thus amortizing the expense of finding such paths over many samples. Third, the mutation set is easily extended. By constructing mutations that preserve certain properties of the path (e.g. which light source is used) while changing others, we can exploit various kinds of coherence in the scene. It is often possible to handle difficult lighting problems efficiently by designing a specialized mutation in this way. CR Categories: I.3.7 [Computer Graphics]: ThreeDimensional Graphics and Realism; I.3.3 [Computer Graphics]: Picture/Image Generation; G.1.9 [Numerical Analysis]: Integral Equations—Fredholm equations.
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.
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.
45865813
Persistence Barcodes for Shapes
In this paper, we initiate a study of shape description and classification via the application of persistent homology to 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 parameterizing 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 parameterized families of mathematical shapes.