Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons

Given a triangulation of a simple polygonP, we present linear-time algorithms for solving a collection of problems concerning shortest paths and visibility withinP. These problems include calculation of the collection of all shortest paths insideP from a given source vertexS to all the other vertices ofP, calculation of the subpolygon ofP consisting of points that are visible from a given segment withinP, preprocessingP for fast "ray shooting" queries, and several related problems.Work on this paper by this author has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, the IBM Corporation, and from the U.S.-Israel Binational Science Foundation.Work on this paper by this author has been supported by National Science Foundation Grant DCR-86-05962.     

A linear-time algorithm for constructing a circular visibility diagram

To computer circular visibility inside a simple polygon, circular arcs that emanate from a given interior point are classified with respect to the edges of the polygon they first intersect. Representing these sets of circular arcs by their centers results in a planar partition called the circular visibility diagram. AnO(n) algorithm is given for constructing the circular visibility diagram for a simple polygon withn vertices.     

OPTIMAL MESH ALGORITHMS FOR PROXIMITY AND VISIBILITY PROBLEMS IN SIMPLE POLYGONS*

We present optimal parallel algorithms that run in time on mesh-connected computer for a number of fundamental problems concerning proximity and visibility in a simple polygon. These include computing shortest paths, shortest path trees, shortest path partitions, all-farthest neighbors, the visibility polygon of a point, the weak visibility polygon of an edge, and the ray shooting problem.     

Efficient Algorithms for Computing a Complete Visibility Region in Three-Dimensional Space

Given a set P of polygons in three-dimensional space, two points p and q are said to be visible from each other with respect to P if the line segment joining them does not intersect any polygon in P . A point p is said to be completely visible from an area source S if p is visible from every point in S . The completely visible region CV(S, P) from S with respect to P is defined as the set of all points in three-dimensional space that are completely visible from S . We present two algorithms for computing CV(S, P) for P with a total of n vertices and a convex polygonal source S with m vertices. Our first result is a divide-and-conquer algorithm which runs in O(m ² n ² α(mn)) time and space, where α(mn) is the inverse of Ackermann's function. We next give an incremental algorithm for computing CV(S,P) in O(m ² n+mn ² α(n)) time and O(mn+n ² ) space. We also prove that CV(S,P) consists of Θ(mn+n ² ) surface elements such as vertices, edges, and faces. Received November 16, 1995; revised November 11, 1996.     

Visibility of disjoint polygons

Consider a collection of disjoint polygons in the plane containing a total ofn edges. We show how to build, inO(n ²) time and space, a data structure from which inO(n) time we can compute the visibility polygon of a given point with respect to the polygon collection. As an application of this structure, the visibility graph of the given polygons can be constructed inO(n ²) time and space. This implies that the shortest path that connects two points in the plane and avoids the polygons in our collection can be computed inO(n ²) time, improving earlierO(n ² logn) results.     

A New Approach for the Geodesic Voronoi Diagram of Points in a Simple Polygon and Other Restricted Polygonal Domains

We introduce a new method for computing the geodesic Voronoi diagram of point sites in a simple polygon and other restricted polygonal domains. Our method combines a sweep of the polygonal domain with the merging step of a usual divide-and-conquer algorithm. The time complexity is O((n+k) log(n+k)) where n is the number of vertices and k is the number of points, improving upon previously known bounds. Space is O(n+k) . Other polygonal domains where our method is applicable include (among others) a polygonal domain of parallel disjoint line segments and a polygonal domain of rectangles in the L ₁ metric. Received February 15, 1996; revised November 2, 1996.     

An approximative solution to the Zookeeper's Problem

Consider a simple polygon P containing disjoint convex polygons each of which shares an edge with P. The Zookeeper's Problem then asks for the shortest route in P that visits all convex polygons without entering their interiors. Existing algorithms that solve this problem run in time super-linear in the size of P and the convex polygons. They also suffer from numerical problems.In this paper, we shed more light on the problem and present a simple linear time algorithm for computing an approximate solution. The algorithm mainly computes shortest paths and intersections between lines using basic data structures. It does not suffer from numerical problems. We prove that the computed approximation route is at most 6 times longer than the shortest route in the exact solution.     

A fast algorithm for computing sparse visibility graphs

AnO(¦E¦log² n) algorithm is presented to construct the visibility graph for a collection ofn nonintersecting line segments, where ¦E¦ is the number of edges in the visibility graph. This algorithm is much faster than theO(n ²)-time andO(n ²)-space algorithms by Asanoet al., and by Welzl, on sparse visibility graphs. Thus we partially resolve an open problem raised by Welzl. Further, our algorithm uses onlyO(n) working storage.     

简单多边形可见点问题的快速求解算法

简单多边形可见点问题是计算几何的基本问题之一。在许多领域均有应用。本文在参考现有算法的基础上,提出了改进的方法,文中方法先用射线法求取第一个可见点,然后利用文中设定的规则搜索后续可见点。     

A Sampling-Based Motion Planning Approach to Maintain Visibility of Unpredictable Targets

This paper deals with the surveillance problem of computing the motions of one or more robot observers in order to maintain visibility of one or several moving targets. The targets are assumed to move unpredictably, and the distribution of obstacles in the workspace is assumed to be known in advance. Our algorithm computes a motion strategy by maximizing the shortest distance to escape—the shortest distance the target must move to escape an observer's visibility region. Since this optimization problem is intractable, we use randomized methods to generate candidate surveillance paths for the observers. We have implemented our algorithms, and we provide experimental results using real mobile robots for the single target case, and simulation results for the case of two targets-two observers. Rafael Murrieta-Cid received the B.S degree in Physics Engineering (1990), and the M.Sc. degree in Automatic Manufacturing Systems (1993), both from “Instituto Tecnológico y de Estudios Superiores de Monterrey” (ITESM) Campus Monterrey. He received his Ph.D. from the “Institut National Polytechnique” (INP) of Toulouse, France (1998). His Ph.D research was done in the Robotics and Artificial Intelligence group of the LAAS/CNRS. In 1998–1999, he was a postdoctoral researcher in the Computer Science Department at Stanford University. From January 2000 to July 2002 he was an assistant professor in the Electrical Engineering Department at ITESM Campus México City, México. In 2002–2004, he was working as a postdoctoral research associate in the Beckman Institute and Department of Electrical and Computer Engineering of the University of Illinois at Urbana-Champaign. Since August 2004, he is director of the Mechatronics Research Center in the ITESM Campus Estado de México, México. He is mainly interested in sensor-based robotics motion planning and computer vision. Benjamin Tovar received the B.S degree in electrical engineering from ITESM at Mexico City, Mexico, in 2000, and the M.S. in electrical engineering from University of Illinois, Urbana-Champaign, USA, in 2004. Currently (2005) he is pursuing the Ph.D degree in Computer Science at the University of Illinois. Prior to M.S. studies he worked as a research assistant at Mobile Robotics Laboratory at ITESM Mexico City. He is mainly interested in motion planning, visibility-based tasks, and minimal sensing for robotics. Seth Hutchinson received his Ph. D. from Purdue University in West Lafayette, Indiana in 1988. He spent 1989 as a Visiting Assistant Professor of Electrical Engineering at Purdue University. In 1990 Dr. Hutchinson joined the faculty at the University of Illinois in Urbana-Champaign, where he is currently a Professor in the Department of Electrical and Computer Engineering, the Coordinated Science Laboratory, and the Beckman Institute for Advanced Science and Technology. Dr. Hutchinson is currently a senior editor of the IEEE Transactions on Robotics and Automation. In 1996 he was a guest editor for a special section of the Transactions devoted to the topic of visual servo control, and in 1994 he was co-chair of an IEEE Workshop on Visual Servoing. In 1996 and 1998 he co-authored papers that were finalists for the King-Sun Fu Memorial Best Transactions Paper Award. He was co-chair of IEEE Robotics and Automation Society Technical Committee on Computer and Robot Vision from 1992 to 1996, and has served on the program committees for more than thirty conferences related to robotics and computer vision. He has published more than 100 papers on the topics of robotics and computer vision.     

A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs

In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks. Given an error parameter such that , our algorithm maintains approximate all-pairs shortest paths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a 1+ factor. The time bounds for both query and update for our algorithm is O( ^-1 n ^2/3 log ² n log D) , where n is the number of nodes in G and D is the sum of its edge lengths. The time bound for the queries is worst case, while that for the additions is amortized. Our approximation algorithm is based upon a novel technique for approximately representing all-pairs shortest paths among a selected subset of the nodes by a sparse substitute graph. Received January 1995; revised February 1997.     

A Ray-Testing Approach to Automatically Identify Surface Visibility in Parting Line Generation

It is critical to identify core/cavity and undercut surfaces of molds in partingline generation. A new Ray-Testing approach is presented to detect these surfaces by identifying thevisibility of surfaces. A simple, direct and efficient algorithm to identify surface visibility isdeveloped. Case studies show that the approach is practical and valuable in automated parting linegeneration.     

A Note on Point Location in Delaunay Triangulations of Random Points

This short note considers the problem of point location in a Delaunay triangulation of n random points, using no additional preprocessing or storage other than a standard data structure representing the triangulation. A simple and easy-to-implement (but, of course, worst-case suboptimal) heuristic is shown to take expected time O(n ^1/3 ) . Received November 27, 1997; revised February 15, 1998.     

A new linear algorithm for triangulating monotone polygons

Let P = (p₁, p₂,…,p_n) be a monotone polygon whose vertices are specified in terms of cartesian coordinates in order. A new simple two-step procedure is presented for triangulating P, without the addition of new vertices, in O(n) time. Unlike the previous algorithm no specialized code is needed since the new approach uses well-known existing algorithms for first decomposing P into edge-visible polygons and subsequently triangulating these.     

A Differential Geometric Approach to Multiple View Geometry in Spaces of Constant Curvature

Based upon an axiomatic formulation of vision system in a general Riemannian manifold, this paper provides a unified framework for the study of multiple view geometry in three dimensional spaces of constant curvature, including Euclidean space, spherical space, and hyperbolic space. It is shown that multiple view geometry for Euclidean space can be interpreted as a limit case when (sectional) curvature of a non-Euclidean space approaches to zero. In particular, we show that epipolar constraint in the general case is exactly the same as that known for the Euclidean space but should be interpreted more generally when being applied to triangulation in non-Euclidean spaces. A special triangulation method is hence introduced using trigonometry laws from Absolute Geometry. Based on a common rank condition, we give a complete study of constraints among multiple images as well as relationships among all these constraints. This idealized geometric framework may potentially extend extant multiple view geometry to the study of astronomical imaging where the effect of space curvature is no longer negligible, e.g., the so-called gravitational lensing phenomenon, which is currently active study in astronomical physics and cosmology.     

A Multiscale Approach to Optimal Transport

In this paper, we propose an improvement of an algorithm of Aurenhammer, Hoffmann and Aronov to find a least square matching between a probability density and finite set of sites with mass constraints, in the Euclidean plane. Our algorithm exploits the multiscale nature of this optimal transport problem. We iteratively simplify the target using Lloyd's algorithm, and use the solution of the simplified problem as a rough initial solution to the more complex one. This approach allows for fast estimation of distances between measures related to optimal transport (known as Earth‐mover or Wasserstein distances). We also discuss the implementation of these algorithms, and compare the original one to its multiscale counterpart.     

A Viewpoint-Dependent Approach to Ray Trace Free-Form Surfaces

For convincing realistic scenes objects with free‐form surfaces are essential. Especially for photorealistic rendering pure polygonal models are often not sufficient. We present a new kind of algorithm to render free‐form surfaces in a rendering system based on ray tracing. We describe a triangular patch as usual by its three points and normal vectors, but base the intersection calculation as well on the viewpoint of the camera (or, in general, on the ray itself). Hence, the shape of the object depends to some extent on the sampling rays. However, the resulting differences of, for instance, the shape of the silhouette to the shape of the corresponding shadow is usually not perceived by the observer of the rendered image. Because we perform a direct computation without a tessellation process, the resulting surface, its shadows, and its reflections appear smooth independent of the distance to the camera. Furthermore, the memory consumption depends only linearly on the number of input triangles. Special features like creases, T‐vertices, and darts are also well supported. The computed uv‐coordinates provide a direct means for texture mapping whose visual appearance improves significantly compared to triangle meshes of the same resolution.     

A Projective Framework for Polyhedral Mesh Modelling

We present a novel framework for polyhedral mesh editing with face‐based projective maps that preserves planarity by definition. Such meshes are essential in the field of architectural design and rationalization. By using homogeneous coordinates to describe vertices, we can parametrize the entire shape space of planar‐preserving deformations with bilinear equations. The generality of this space allows for polyhedral geometric processing methods to be conducted with ease. We demonstrate its usefulness in planar‐quadrilateral mesh subdivision, a resulting multi‐resolution editing algorithm, and novel shape‐space exploration with prescribed transformations. Furthermore, we show that our shape space is a discretization of a continuous space of conjugate‐preserving projective transformation fields on surfaces. Our shape space directly addresses planar‐quad meshes, on which we put a focus, and we further show that our framework naturally extends to meshes with faces of more than four vertices as well.     

A Parallel Approach to Compression and Decompression of Triangle Meshes using the GPU

Most state‐of‐the‐art compression algorithms use complex connectivity traversal and prediction schemes, which are not efficient enough for online compression of large meshes. In this paper we propose a scalable massively parallel approach for compression and decompression of large triangle meshes using the GPU. Our method traverses the input mesh in a parallel breadth‐first manner and encodes the connectivity data similarly to the well known cut‐border machine. Geometry data is compressed using a local prediction strategy. In contrast to the original cut‐border machine, we can additionally handle triangle meshes with inconsistently oriented faces. Our approach is more than one order of magnitude faster than currently used methods and achieves competitive compression rates.