Fast interference detection between geometric models

We present efficient algorithms for interference detection between geometric models described by linear or curved boundaries and undergoing rigid motion. The set of models include surfaces described by rational spline patches or piecewise algebraic functions. In contrast to previous approaches, we first describe an efficient algorithm for interference detection between convex polytopes using coherence and local features. Then an extension using hierarchical representation to concave polytopes is presented. We apply these algorithms along with properties of input models, local and global algebraic methods for solving polynomial equations, and the geometric formulation of the problem to devise efficient algorithms for convex and nonconvex curved objects. Finally, a scheduling scheme to reduce the frequency of interference detection in large environments is described. These algorithms have been successfully implemented and we discuss their performance in various environments.     

Drawing polytopal graphs with polymake

This note wants to explain how to obtain meaningful pictures of (possibly high-dimensional) convex polytopes, triangulated manifolds, and other objects from the realm of geometric combinatorics such as tight spans of finite metric spaces and tropical polytopes. In all our cases we arrive at specific, geometrically motivated, graph drawing problems. The methods displayed are implemented in the software system polymake.     

Delaunay partitions in Rn applied to non-convex programs and vertex/facet enumeration problems

Using a pair of theorems linking Delaunay partitions and linear programming, we develop a method to generate all simplices in a Delaunay partition of a set of points, and suggest an application to a piecewise linear non-convex optimization problem. The same method is shown to enumerate all facets of a polytope given as the convex hull of a finite set of points. The dual problem of enumerating all vertices of a polytope P defined as the intersection of a finite number of half-spaces is also addressed and solved by sequentially enumerating vertices of expanding polytopes defined within P. None of our algorithms are affected by degeneracy. Examples and computational results are given.     

Space sweep solves intersection of convex polyhedra

Summary Plane-sweep algorithms form a fairly general approach to two-dimensional problems of computational geometry. No corresponding general space-sweep algorithms for geometric problems in 3- space are known. We derive concepts for such space-sweep algorithms that yield an efficient solution to the problem of solving any set operation (union, intersection, ...) of two convex polyhedra. Our solution matches the best known time bound of O(n log n), where n is the combined number of vertices of the two polyhedra.     

Alternative branching rules for some nonconvex problems

Following Linderoth (A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs, Math. Program. 103 (2005), pp. 251–282), in this paper we show how a preliminary theoretical analysis about the quality of convex and concave envelopes over different regions may suggest branching rules alternative to the traditional ones based on boxes. For two nonconvex problems (minimization of the sum of products of affine functions and maximization of the sum of ratios of affine functions, in both cases over polytopes), we show how the indications of the theoretical analysis lead to computational results which confirm the superiority of the alternative branching rules with respect to those based on boxes.     

Learning Boxes in High Dimension

We present exact learning algorithms that learn several classes of (discrete) boxes in {0,...,l-1} ⁿ . In particular we learn: (1) The class of unions of O(log n) boxes in time poly(n,log l) (solving an open problem of [16] and [12]; in [3] this class is shown to be learnable in time poly(n,l) ). (2) The class of unions of disjoint boxes in time poly(n,t,log l) , where t is the number of boxes. (Previously this was known only in the case where all boxes are disjoint in one of the dimensions; in [3] this class is shown to be learnable in time poly(n,t,l) .) In particular our algorithm learns the class of decision trees over n variables, that take values in {0,...,l-1} , with comparison nodes in time poly(n,t,log l) , where t is the number of leaves (this was an open problem in [9] which was shown in [4] to be learnable in time poly(n,t,l) ). (3) The class of unions of O(1) -degenerate boxes (that is, boxes that depend only on O(1) variables) in time poly(n,t, log l) (generalizing the learnability of O(1) -DNF and of boxes in O(1) dimensions). The algorithm for this class uses only equivalence queries and it can also be used to learn the class of unions of O(1) boxes (from equivalence queries only). Received January 19, 1997; revised June 4, 1997.     

Jacobi and Gauss-Seidel Iterations for Polytopic Systems: Convergence via Convex M-Matrices

A natural generalization of the Jacobi and Gauss-Seidel iterations for interval systems is to allow the matrices to reside in convex polytopes. In order to apply the standard convergence criteria involving M-matrices to iterations for polytopic systems, we derive conditions for a convex polytope of matrices to be a polytope of M-matrices in terms of its vertices. We show the conditions are used in the convergence analysis of iterations for block and nonlinear polytopic systems.     

Covering a convex 3D polytope by a minimal number of congruent spheres

The problem of covering a convex 3D polytope by the minimal number of congruent spheres is reduced to a sequence of problems of minimising sphere radius when fixing the number of the spheres. We form a mathematical model of the problem using the Voronoi polytopes. Characteristics of the model are investigated. Extrema are attained at the vertices of the Voronoi polytopes constructed for sphere centres. To search for local minima, a modification of the Zoutendijk feasible directions method in a combination with random search is developed. Some numerical results for a cube and a non-regular octahedron are obtained.     

An Improved Geometric Approach for Palette‐based Image Decomposition and Recoloring

Palette‐based image decomposition has attracted increasing attention in recent years. A specific class of approaches have been proposed basing on the RGB‐space geometry, which manage to construct convex hulls whose vertices act as palette colors. However, such palettes do not guarantee to have the representative colors which actually appear in the image, thus making it less intuitive and less predictable when editing palette colors to perform recoloring. Hence, we proposed an improved geometric approach to address this issue. We use a polyhedron, but not necessarily a convex hull, in the RGB space to represent the color palette. We then formulate the task of palette extraction as an optimization problem which could be solved in a few seconds. Our palette has a higher degree of representativeness and maintains a relatively similar level of accuracy compared with previous methods. For layer decomposition, we compute layer opacities via simple mean value coordinates, which could achieve instant feedbacks without precomputations. We have demonstrated our method for image recoloring on a variety of examples. In comparison with state‐of‐the‐art works, our approach is generally more intuitive and efficient with fewer artifacts.     

Maximum Entropy Coordinates for Arbitrary Polytopes

Barycentric coordinates can be used to express any point inside a triangle as a unique convex combination of the triangle's vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices of a triangle. In recent years, the ideas of barycentric coordinates and barycentric interpolation have been extended to arbitrary polygons in the plane and general polytopes in higher dimensions, which in turn has led to novel solutions in applications like mesh parameterization, image warping, and mesh deformation. In this paper we introduce a new generalization of barycentric coordinates that stems from the maximum entropy principle. The coordinates are guaranteed to be positive inside any planar polygon, can be evaluated efficiently by solving a convex optimization problem with Newton's method, and experimental evidence indicates that they are smooth inside the domain. Moreover, the construction of these coordinates can be extended to arbitrary polyhedra and higher‐dimensional polytopes.     

Time-Optimal Nearest-Neighbor Computations on Enhanced Meshes

The All Nearest Neighbor problem (ANN, for short) is stated as follows: given a setSof points in the plane, determine for every point inS, a point that lies closest to it. The ANN problem is central to VLSI design, computer graphics, pattern recognition, and image processing, among others. In this paper we propose time-optimal algorithms to solve the ANN problem for an arbitrary set of points in the plane and also for the special case where the points are vertices of a convex polygon. Both our algorithms run on meshes with multiple broadcasting. Our first main contribution is to establish an Ω(logn) time lower bound for the task of solving an arbitraryn-point instance of the ANN problem, even if the points are the vertices of a convex polygon. We obtain our time lower bound results for the CREW-PRAM by using a novel technique involving geometric constructions. These constructions allow us to reduce the well-known OR problem to each of the geometric problems of interest. We then port these time lower bounds to the mesh with multiple broadcasting using simulation results. Our second main contribution is to show that the time lower bound obtained is tight, by exhibiting algorithms solving the problem inO(logn) time on a mesh with multiple broadcasting of sizen×n.     

Stock market prediction with multiple classifiers

Stock market prediction is attractive and challenging. According to the efficient market hypothesis, stock prices should follow a random walk pattern and thus should not be predictable with more than about 50 percent accuracy. In this paper, we investigated the predictability of the Dow Jones Industrial Average index to show that not all periods are equally random. We used the Hurst exponent to select a period with great predictability. Parameters for generating training patterns were determined heuristically by auto-mutual information and false nearest neighbor methods. Some inductive machine-learning classifiers—artificial neural network, decision tree, and k-nearest neighbor were then trained with these generated patterns. Through appropriate collaboration of these models, we achieved prediction accuracy up to 65 percent.     

基于启发式搜索分离向量的凸多面体碰撞检测

碰撞检测是计算机模拟物理过程的基础，在计算机图形学、CAD／CAM、虚拟现实和机器人等领域有着广泛的应用．该文给出了一个新的用于凸多面体碰撞检测的算法——HP-jump．HP-jump建立了一个有效的碰撞检测模型用于报告物体的碰撞，同时提供了一个快速的启发式的策略用于搜索两个凸多面体的分离向量．该算法是利用凸多面体的层次表示来搜索支撑顶点对，用平衡二叉树来记录球面凸多边形的顶点，同时还利用了时间、空间相关性，这些都加速了算法的执行．该文的最后给出了HP-jump与GJK，I-COLLIDE算法的比较．     

Algorithm to calculate the Minkowski sums of 3-polytopes based on normal fans

Prompted by the development of algorithms for analysing geometric tolerancing, this article describes a method to determine the Minkowski sum for 3-dimensional polytopes. This method is based exclusively on intersection operations on normal cones, using the properties of the normal fan of a Minkowski sum obtained by common refinement of the normal fans of the operands. It can be used to determine from which vertices of the operands the vertices of the Minkowski sum derive. It is also possible to determine to which facets of the operands each facet of the Minkowski sum is oriented. The basic properties of the algorithms can be applied to n-polytopes.First, the main properties of the duality of normal cones and primal cones associated with the vertices of a polytope are described. Next, the properties of normal fans are applied to define the vertices and facets of the Minkowski sum of two polytopes. An algorithm is proposed, which generalises the method. Lastly, there is a discussion of the features of this algorithm, developed using the OpenCascade environment.     

New applications of clique separator decomposition for the Maximum Weight Stable Set problem

Graph decompositions such as decomposition by clique separators and modular decomposition are of crucial importance for designing efficient graph algorithms. Clique separators in graphs were used by Tarjan as a divide-and-conquer approach for solving various problems such as the Maximum Weight Stable Set (MWS) problem, Colouring and Minimum Fill-in. The basic tool is a decomposition tree of the graph whose leaves have no clique separator (so-called atoms), and the problem can be solved efficiently on the graph if it is efficiently solvable on its atoms. We give new examples where the clique separator decomposition works well for the MWS problem; our results improve and extend various recently published results. In particular, we describe the atom structure for some new classes of graphs whose atoms are P₅-free (the P₅ is the induced path with five vertices) and obtain new polynomial time results for the MWS problem. The complexity of this problem on the class of P₅-free graphs is still unknown.     

Polytope-based computation of polynomial ranges

Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate a polytope defined as the convex hull of the coefficients with respect to the tensorial Bernstein basis, and we formulate several polytopes based on the Bernstein polynomials of the domain. These Bernstein polytopes can be defined by a polynomial number of halfspaces. We give the number of vertices, the number of hyperfaces, and the volume of each polytope for n=1,2,3,4, and we compare the computed range widths for random n-variate polynomials for n?10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.     

Some problems in distributed computational geometry

In a planar geometric network vertices are located in the plane, and edges are straight line segments connecting pairs of vertices, such that no two of them intersect. In this paper we study distributed computing in asynchronous, failure-free planar geometric networks, where each vertex is associated to a processor, and each edge to a bidirectional message communication link. Processors are aware of their locations in the plane.We consider fundamental computational geometry problems from the distributed computing point of view, such as finding the convex hull of a geometric network and identification of the external face. We also study the classic distributed computing problem of leader election, to understand the impact that geometric information has on the message complexity of solving it.We obtain an O(nlog²n) message complexity algorithm to find the convex hull, and an O(nlogn) message complexity algorithm to identify the external face of a geometric network of n processors. We present a matching lower bound for the external face problem. We prove that the message complexity of leader election in a geometric ring is Ω(nlogn), hence geometric information does not help in reducing the message complexity of this problem.     

On Minimum-Area Hulls

We study some minimum-area hull problems that generalize the notion of convex hull to star-shaped and monotone hulls. Specifically, we consider the minimum-area star-shaped hull problem: Given an n -vertex simple polygon P , find a minimum-area, star-shaped polygon P ^* containing P . This problem arises in lattice packings of translates of multiple, nonidentical shapes in material layout problems (e.g., in clothing manufacture), and has been recently posed by Daniels and Milenkovic. We consider two versions of the problem: the restricted version, in which the vertices of P ^* are constrained to be vertices of P , and the unrestricted version, in which the vertices of P ^* can be anywhere in the plane. We prove that the restricted problem falls in the class of ``3sum-hard' (sometimes called ``n ² -hard') problems, which are suspected to admit no solutions in o(n ² ) time. Further, we give an O(n ² ) time algorithm, improving the previous bound of O(n ⁵ ) . We also show that the unrestricted problem can be solved in O(n ² p(n)) time, where p(n) is the time needed to find the roots of two equations in two unknowns, each a polynomial of degree O(n) . We also consider the case in which P ^* is required to be monotone, with respect to an unspecified direction; we refer to this as the minimum-area monotone hull problem. We give a matching lower and upper bound of Θ(n log n) time for computing P ^* in the restricted version, and an upper bound of O(n q(n)) time in the unrestricted version, where q(n) is the time needed to find the roots of two polynomial equations in two unknowns with degrees 2 and O(n) . Received November 1996; revised March 1997.     

LMI stability conditions for uncertain rational nonlinear systems

This paper presents LMI conditions for local, regional, and global robust asymptotic stability of rational uncertain nonlinear systems. The uncertainties are modeled as real time varying parameters with magnitude and rate of variation bounded by given polytopes and the system vector field is a rational function of the states and uncertain parameters. Sufficient LMI conditions for asymptotic stability of the origin are given through a rational Lyapunov function of the states and uncertain parameters. The case where the time derivative of the Lyapunov function is negative semidefinite is also considered and connections with the well known LaSalle's invariance conditions are established. In regional stability problems, an algorithm to maximize the estimate of the region of attraction is proposed. The algorithm consists of maximizing the estimate for a given target region of initial states. The size and shape of the target region are recursively modified in the directions where the estimate can be enlarged. The target region can be taken as a polytope (convex set) or union of polytopes (non‐convex set). The estimates of the region of attraction are robust with respect to the uncertain parameters and their rate of change. The case of global and orthant stability problems are also considered. Connections with some results found in sum of squares based methods and other related methods found in the literature are established. The LMIs in this paper are obtained by using the Finsler's Lemma and the notion of annihilators. The LMIs are characterized by affine functions of the state and uncertain parameters, and they are tested at the vertices of a polytopic region. It is also shown that, with some additional conservatism, the use of the vertices can be avoided by modifying the LMIs with the S‐Procedure. Several numerical examples found in the literature are used to compare the results and illustrate the advantages of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

Polyhedral Characteristics of Balanced and Unbalanced Bipartite Subgraph Problems

We study the polyhedral properties of three problems of constructing an optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the first problem, we consider a balanced biclique with the same number of vertices in both parts and arbitrary edge weights. In the other two problems we are dealing with unbalanced subgraphs of maximum and minimum weight with non-negative edges. All three problems are established to be NP-hard. We study the polytopes and the cone decompositions of these problems and their 1-skeletons. We describe the adjacency criterion in the 1-skeleton of the polytope of the balanced complete bipartite subgraph problem. The clique number of the 1-skeleton is estimated from below by a superpolynomial function. For both unbalanced biclique problems we establish the superpolynomial lower bounds on the clique numbers of the graphs of nonnegative cone decompositions. These values characterize the time complexity in a broad class of algorithms based on linear comparisons.