Disjoint Line Research Articles

A Line Segment Detector for Space Target Images Robust to Complex Illumination

The relative pose estimation of the space target is indispensable for on-orbit autonomous service missions. Line segment detection is an important step in pose estimation. The traditional line segment detectors show impressive performance under sufficient illumination, while it is easy to fail under complex illumination conditions where the illumination is too bright or too dark. We propose a robust line segment detector for space applications considering the complex illumination in space environments. An improved two-dimensional histogram construction strategy is used to optimize the Otsu method to improve the accuracy of anchor map extraction. To further improve line segment detection’s effect, we introduce an aggregation method that uses the angle difference between segments, the distance between endpoints, and the overlap degree of segments to filter the aggregation candidate segments and connect disjoint line segments that probably came from the same segment. We demonstrate the performance of the proposed line segment detector using a variety of images collected on a semiphysical simulation platform. The results show that our method has better performance than traditional line segment detectors including LSD, Linelet, etc., in terms of line detection precision.

Rectangular partitions of a rectilinear polygon

We investigate the problem of partitioning a rectilinear polygon P with n vertices and no holes into rectangles using disjoint line segments drawn inside P under two optimality criteria. In the minimum ink partition, the total length of the line segments drawn inside P is minimized. We present an O(n3)-time algorithm using O(n2) space that returns a minimum ink partition of P. In the thick partition, the minimum side length over all resulting rectangles is maximized. We present an O(n3log2⁡n)-time algorithm using O(n3) space that returns a thick partition using line segments incident to vertices of P, and an O(n6log2⁡n)-time algorithm using O(n6) space that returns a thick partition using line segments incident to the boundary of P. We also show that if the input rectilinear polygon has holes, the corresponding decision problem for the thick partition problem using line segments incident to vertices of the polygon is NP-complete. We also present an O(m3)-time 3-approximation algorithm for the minimum ink partition for a rectangle containing m point holes.

Algorithms and a Library for the Exact Computation of the Cumulative Distribution Function of the Euclidean Distance Between a Point and a Random Variable Uniformly Distributed in Disks, Balls, or Polygones and Application to Probabilistic Seismic Hazard Analysis

We consider a random variable expressed as the Euclidean distance between an arbitrary point and a random variable uniformly distributed in a closed and bounded set of a three-dimensional Euclidean space. Four cases are considered for this set: a union of disjoint disks, a union of disjoint balls, a union of disjoint line segments, and the boundary of a polyhedron. In the first three cases, we provide closed-form expressions of the cumulative distribution function and the density. In the last case, we propose two algorithms with complexity [Formula: see text], [Formula: see text] being the number of edges of the polyhedron, that computes exactly the cumulative distribution function. An application of these results to probabilistic seismic hazard analysis and extensions are discussed. Finally, we present an open source library, available at https://github.com/vguigues/Areas_Library , that implements the algorithms presented in this paper.

Subgroups and Orbits by Companion Matrix in Three Dimensional Projective Space

The aim of this paper is to construct cyclic subgroups of the projective general linear group over from the companion matrix, and then form caps of various degrees in . Geometric properties of these caps as secant distributions and index distributions are given and determined if they are complete. Also, partitioned of into disjoint lines is discussed.

Circumscribing Polygons and Polygonizations for Disjoint Line Segments

Given a planar straight-line graph $$G=(V,E)$$ in $$\mathbb {R}^2$$ , a circumscribing polygon of G is a simple polygon P whose vertex set is V, and every edge in E is either an edge or an internal diagonal of P. A circumscribing polygon is a polygonization for G if every edge in E is an edge of P. We prove that every arrangement of n disjoint line segments in the plane has a subset of size $$\Omega (\sqrt{n})$$ that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to $$\mathbb {R}^3$$ . We show that it is NP-complete to decide whether a given graph G admits a circumscribing polygon, even if G is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization.

On cutting blocking sets and their codes

AbstractLetPG⁡(r,q){\operatorname{PG}(r,q)}be ther-dimensional projective space over the finite fieldGF⁡(q){\operatorname{GF}(q)}. A set𝒳{\mathcal{X}}of points ofPG⁡(r,q){\operatorname{PG}(r,q)}is a cutting blocking set if for each hyperplane Π ofPG⁡(r,q){\operatorname{PG}(r,q)}the setΠ∩𝒳{\Pi\cap\mathcal{X}}spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set ofPG⁡(3,q3){\operatorname{PG}(3,q^{3})}of size3⁢(q+1)⁢(q2+1){3(q+1)(q^{2}+1)}as a union of three pairwise disjointq-order subgeometries, and a cutting blocking set ofPG⁡(5,q){\operatorname{PG}(5,q)}of size7⁢(q+1){7(q+1)}from seven lines of a Desarguesian line spread ofPG⁡(5,q){\operatorname{PG}(5,q)}. In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimalq-ary linear code having dimension 4 and 6.

Dispersing and grouping points on planar segments

Motivated by (continuous) facility location, we study the problem of dispersing and grouping points on a set of segments (of streets) in the plane. In the former problem, given a set of n disjoint line segments in the plane, we investigate the problem of computing a point on each of the n segments such that the minimum Euclidean distance between any two of these points is maximized. We prove that this 2D dispersion problem is NP-hard, in fact, it is NP-hard even if all the segments are parallel and are of unit length. This is in contrast to the polynomial solvability of the corresponding 1D problem by Li and Wang (2016), where the intervals are in 1D and are all disjoint. With this result, we also show that the Independent Set problem on Colored Linear Unit Disk Graph (meaning the convex hulls of points with the same color form disjoint line segments) remains NP-hard, and the parameterized version of it is in W[2]. In the latter problem, given a set of n disjoint line segments in the plane we study the problem of computing a point on each of the n segments such that the maximum Euclidean distance between any two of these points is minimized. We present a factor-1.1547 approximation algorithm which runs in O(nlog⁡n) time. Our results can be generalized to the Manhattan distance.

On the Hartshorne–Hirschowitz theorem

The Hartshorne–Hirschowitz theorem says that a generic union of lines in P n , ( n ≥ 3 ) , has good postulation. The proof of Hartshorne and Hirschowitz in the initial case P 3 is handled by a method of specialization via a smooth quadric surface with the property of having two rulings of skew lines. We provide a proof in the case P 3 based on a new degeneration of disjoint lines via a plane H ≅ P 2 , which we call ( 2 , s ) -cone configuration , that is a schematic union of s intersecting lines passing through a single point P together with the trace of an s -multiple point supported at P on the double plane 2 H . In the first part of this paper, we discuss our degeneration inductive approach. We prove that a ( 2 , s ) -cone configuration is a degeneration of s disjoint lines in P 3 , or more generally in P n . In the second part of the paper, we use this degeneration in an effective method to show that a generic union of lines in P 3 imposes independent conditions on the linear system | O P 3 ( d ) | of surfaces of given degree d . The basic motivation behind our degeneration approach is that it looks more systematic that gives some hope of extensions to the analogous problem in higher dimensional spaces, that is the postulation problem for m -dimensional planes in P 2 m + 1 .

On partial geometries arising from maximal arcs

The subject of this paper are partial geometries \(pg(s,t,\alpha )\) with parameters \(s=d(d'-1), \ t=d'(d-1), \ \alpha =(d-1)(d'-1)\), \(d, d' \ge 2\). In all known examples, \(q=dd'\) is a power of 2 and the partial geometry arises from a maximal arc of degree d or \(d'\) in a projective plane of order q via a known construction due to Thas [28] and Wallis [34], with a single known exception of a partial geometry pg(4, 6, 3) found by Mathon [22] that is not associated with a maximal arc in the projective plane of order 8. A parallel class of lines is a set of pairwise disjoint lines that covers the point set. Two parallel classes are called orthogonal if they share exactly one line. An upper bound on the maximum number of pairwise orthogonal parallel classes in a partial geometry G with parameters \(pg(d(d'-1),d'(d-1),(d-1)(d'-1))\) is proved, and it is shown that a necessary and sufficient condition for G to arise from a maximal arc of degree d or \(d'\) in a projective plane of order \(q=dd'\) is that both G and its dual geometry contain sets of pairwise orthogonal parallel classes that meet the upper bound. An alternative construction of Mathon’s partial geometry is presented, and the new necessary condition is used to demonstrate why this partial geometry is not associated with any maximal arc in the projective plane of order 8. The partial geometries associated with all known maximal arcs in projective planes of order 16 are classified up to isomorphism, and their parallel classes of lines and the 2-rank of their incidence matrices are computed. Based on these results, some open problems and conjectures are formulated.

A Duality-Based Unified Approach to Bayesian Mechanism Design

We provide a unified view of many recent developments in Bayesian mechanism design, including the black-box reductions of Cai, Daskalakis, and Weinberg [in Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, 2013], simple auctions for additive buyers [S. Hart and N. Nisan, in Proceedings of the 13th ACM Conference on Electronic Commerce, 2012], and posted-price mechanisms for unit-demand buyers [S. Chawla, J. D. Hartline, and R. D. Kleinberg, in Proceedings of the 8th ACM Conference on Electronic Commerce, 2007, pp. 243--251]. Additionally, we show that viewing these three previously disjoint lines of work through the same lens leads to new developments as well. First, we provide a duality framework for Bayesian mechanism design, which naturally accommodates multiple agents and arbitrary objectives/feasibility constraints. Using this, we prove that either a posted-price mechanism or the Vickrey--Clarke--Groves auction with per-bidder entry fees achieves a constant factor of the optimal revenue achievable by a Bayesian Incentive Compatible mechanism whenever buyers are unit-demand or additive, unifying previous breakthroughs of Chawla et al. [in Proceedings of the 42nd ACM Symposium on Theory of Computing, 2010] and Yao [in Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 2015, pp. 92--109], and improving both approximation ratios (from 30 to 24 and 69 to 8, respectively). Finally, we show that this view also leads to improved structural characterizations in the framework of Cai, Daskalakis, and Weinberg.

On families of invariant lines of a Brouwer homeomorphism

ABSTRACT We present properties of equivalence classes of the codivergency relation defined for a Brouwer homeomorphism for which there exists a family of invariant pairwise disjoint lines covering the plane. In particular, using the codivergency relation we describe the sets of regular and irregular points for such Brouwer homeomorphisms. Moreover, we show that under this assumption the interior of each equivalence class of this relation is invariant and simply connected.

The maximal number of skew lines on Schur’s quartic

Since 1882 it is known that the so-called Schur’s quartic contains exactly 64 lines.However, it has not yet been establishedwhat is themaximumnumber of pairwise disjoint lines that it can have. The aim of our work is to show in an elementary and self-contained way that the maximum number of pairwise disjoint lines in Schur’s quartic is 16 (without using Nikulins’s theorem or Miyaoka’s upper bound).

Topology of real Schläfli six-line configurations on cubic surfaces and in ℝℙ³

A famous configuration of 27 lines on a non-singular cubic surface in P 3 \mathbb {P}^3 contains remarkable subconfigurations, and in particular the ones formed by six pairwise disjoint lines. We study such six-line configurations in the case of real cubic surfaces from a topological viewpoint, as configurations of six disjoint lines in the real projective 3-space, and show that the condition that they lie on a cubic surface implies a very special property of homogeneity. This property distinguishes them in the list of 11 deformation types of configurations formed by six disjoint lines in R P 3 \mathbb {RP}^3 .

Quasi-Coordinate Search for a Randomly Moving Target

In this paper, we study the quasi-coordinated search technique for a lost target assumed to move randomly on one of two disjoint lines according to a random walk motion, where there are two searchers beginning their search from the origin on the first line and other two searchers begin their search from the origin on the second line. But the motion of the two searchers on the first line is independent from the motion of the other two searchers on the second line. Here we introduce a model of search plan and investigate the expected value of the first meeting time between one of the searchers and the lost target. Also, we prove the existence of a search plan which minimizes the expected value of the first meeting time between one of the searchers and the target.

Twistor lines on algebraic surfaces

We give quantitative and qualitative results on the family of surfaces in $\mathbb{CP}^3$ containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines $E$. We prove that its general element is a smooth surface containing $E$ and no other line. Afterwards we prove that twistor lines are Zariski dense in the Grassmannian $Gr(2,4)$. Then, for any degree $d\ge 4$, we give lower bounds on the maximum number of twistor lines contained in a degree $d$ surface. The smooth and singular cases are studied as well as the $j$-invariant one.

Extactic Divisors for Webs and Lines on Projective Surfaces

Given a web (multifoliation) and a linear system on a projective surface, we construct divisors cutting out the locus where some element of the linear system has abnormal contact with the leaf of the web. We apply these ideas to reobtain a classical result by Salmon on the number of lines on a projective surface. In a different vein, we investigate the numbers of lines and disjoint lines contained in a projective surface and tangent to a contact distribution.

Randomized approximation algorithms for planar visibility counting problem

Given a set S of n disjoint line segments in R2, the visibility counting problem (VCP) is to preprocess S such that the number of segments in S visible from any query point p can be computed quickly. This problem can be solved trivially in O(log⁡n) query time using O(n4log⁡n) preprocessing time and O(n4) space.Gudmundsson and Morin (2010) [10] proposed a 2-approximation algorithm for this problem with a tradeoff between the space and the query time. For any constant 0≤α≤1, their algorithm answers any query in Oϵ(m(1−α)/2) time with Oϵ(m1+α) of preprocessing time and space, where ϵ>0 is a constant that can be made arbitrarily small and Oϵ(f(n))=O(f(n)nϵ) and m=O(n2) is a number that depends on the configuration of the segments.In this paper, we propose two randomized approximation algorithms for VCP. The first algorithm depends on two constants 0≤β≤23 and 0<δ≤1, and the expected preprocessing time, the expected space, and the expected query time are O(m2−3β/2log⁡m), O(m2−3β/2), and O(1δ2mβ/2log⁡m), respectively. The algorithm, in the preprocessing phase, selects a sequence of random samples, whose size and number depend on the tradeoff parameters. When a query point p is given by an adversary unaware of the random sample of our algorithm, it computes the number of visible segments from p, denoted by mp, exactly, if mp≤3δ2mβ/2log⁡(2m). Otherwise, it computes an approximated value, mp′, such that with the probability of at least 1−1m, we have (1−δ)mp≤mp′≤(2+2δ)mp. The preprocessing time and space of the second algorithm are O(n2log⁡n) and O(n2), respectively. This algorithm computes the exact value of mp if mp≤1δ2nlog⁡n, otherwise it returns an approximated value mp″ in expected O(1δ2nlog⁡n) time, such that with the probability at least 1−1log⁡n, we have (1−3δ)mp≤mp″≤(1.5+3δ)mp.

The Induced Separation Dimension of a Graph

A linear ordering of the vertices of a graph G separates two edges of G if both the endpoints of one precede both the endpoints of the other in the order. We call two edges $$\{a,b\}$$ and $$\{c,d\}$$ of G strongly independent if the set of endpoints $$\{a,b,c,d\}$$ induces a $$2K_2$$ in G. The induced separation dimension of a graph G is the smallest cardinality of a family $$\mathcal {L}$$ of linear orders of V(G) such that every pair of strongly independent edges in G are separated in at least one of the linear orders in $$\mathcal {L}$$ . For each $$k \in \mathbb {N}$$ , the family of graphs with induced separation dimension at most k is denoted by $${\text {ISD}}(k)$$ . In this article, we initiate a study of this new dimensional parameter. The class $${\text {ISD}}(1)$$ or, equivalently, the family of graphs which can be embedded on a line so that every pair of strongly independent edges are disjoint line segments, is already an interesting case. On the positive side, we give characterizations for chordal graphs in $${\text {ISD}}(1)$$ which immediately lead to a polynomial time algorithm which determines the induced separation dimension of chordal graphs. On the negative side, we show that the recognition problem for $${\text {ISD}}(1)$$ is NP-complete for general graphs. Nevertheless, we show that the maximum induced matching problem can be solved efficiently in $${\text {ISD}}(1)$$ . We then briefly study $${\text {ISD}}(2)$$ and show that it contains many important graph classes like outerplanar graphs, chordal graphs, circular arc graphs and polygon-circle graphs. Finally, we describe two techniques to construct graphs with large induced separation dimension. The first one is used to show that the maximum induced separation dimension of a graph on n vertices is $$\Theta (\lg n)$$ and the second one is used to construct AT-free graphs with arbitrarily large induced separation dimension. The second construction is also used to show that, for every $$k \ge 2$$ , the recognition problem for $${\text {ISD}}(k)$$ is NP-complete even on AT-free graphs.

A randomized divide and conquer algorithm for higher-order abstract Voronoi diagrams

Given a set of n sites in the plane, their order-k Voronoi diagram partitions the plane into regions such that all points within one region have the same k nearest sites. The order-k abstract Voronoi diagram offers a unifying framework that represents a wide range of concrete order-k Voronoi diagrams. It is defined in terms of bisecting curves satisfying some simple combinatorial properties, rather than the geometric notions of sites and distance.In this paper we develop a randomized divide-and-conquer algorithm to compute the order-k abstract Voronoi diagram in expected O(kn1+ε) operations. For solving small sub-instances in the divide-and-conquer process, we also give two auxiliary algorithms with expected O(k2nlog⁡n) and O(n22α(n)log⁡n) time, respectively, where α(⋅) is the inverse of the Ackermann function. Our approach directly implies an O(kn1+ε)-time algorithm for several concrete order-k instances such as points in any convex distance, disjoint line segments or convex polygons of constant size in the Lp norms, and others. It also provides basic techniques that can enable the application of well-known random sampling techniques to the construction of Voronoi diagrams in the abstract setting and for non-point sites.

Contractions of del Pezzo surfaces to $\mathbb P^2$ or $\mathbb P^1\times \mathbb P^1$

In this article, we consider $r-1$ disjoint lines given in a del~Pezzo surface $S_{r}$ and study how to determine if a contraction given by the lines produces a map to $S_{1}$ (one point blow up of $\mathbb {P}^{2}$) or $\mathbb {P}^{1}\times \mathbb {P}^{1}$ by checking only the configuration of lines. Here, we show that we can determine if the disjoint lines produce a contraction to $\mathbb {P}^{1}\times \mathbb {P}^{1}$ by combining a quartic rational divisor class to them. We also study the quartic rational divisor classes along the configuration of lines in del Pezzo surfaces.