Line Segment Intersection Research Articles

Per Segment Plane Sweep Line Segment Intersection on the GPU

Polygon overlay operations are used for various purposes such as GIS searches and queries, VLSI and basic geometric operations of intersection, union and difference. There have been recent research articles presenting algorithms using the GPU to perform line segment intersection for geometric operations. We present two parallel algorithms implemented on the GPU that focus on the active list portion of the traditional serial plane sweep algorithm. The first algorithm uses a single block of threads to simulate the active list data structure in hardware; this algorithm is slow due to GPU thread block size limitations and synchronization points, but demonstrates favorable time complexity. The second algorithm uses dynamic parallelism to remove synchronization and scales to utilize available GPU hardware (single GPU). We perform experiments on both synthetic and real world data sets. The presented results show improvement in execution time with respect to recent algorithms, and low memory usage compared to recent algorithms. We achieve speedups of up to 38.8 over the serial sweep line algorithm on real world data.

Privacy-preserving two-party computation of line segment intersection

Privacy-preserving two-party computation of line segment intersection

Hopcroft’s Problem, Log* Shaving, Two-dimensional Fractional Cascading, and Decision Trees

We revisit Hopcroft’s problem and related fundamental problems about geometric range searching. Given n points and n lines in the plane, we show how to count the number of point-line incidence pairs or the number of point-above-line pairs in O ( n 4/3 ) time, which matches the conjectured lower bound and improves the best previous time bound of \(n^{4/3}2^{O(\log ^*n)}\) obtained almost 30 years ago by Matoušek [ 58 ]. We describe two interesting and different ways to achieve the result: The first is randomized and uses a new two-dimensional version of fractional cascading for arrangements of lines; the second is deterministic and uses decision trees in a manner inspired by the sorting technique of Fredman [42]. The second approach extends to any constant dimension. Many consequences follow from these new ideas: For example, we obtain an O ( n 4/3 )-time algorithm for line segment intersection counting in the plane, O ( n 4/3 )-time randomized algorithms for distance selection in the plane and bichromatic closest pair and Euclidean minimum spanning tree in three or four dimensions, and a randomized data structure for halfplane range counting in the plane with O ( n 4/3 ) preprocessing time and space and \(O(n^{1/3})\) query time.

Hierarchical Motion Planning for Autonomous Vehicles in Unstructured Dynamic Environments

This letter presents a hierarchical motion planner for generating smooth and feasible trajectories for autonomous vehicles in unstructured environments with static and moving obstacles. The framework enables real-time computation by progressively shrinking the solution space. First, a graph searcher based on combined heuristic and partial motion planning is proposed for finding coarse trajectories in spatiotemporal space. To enable fast online planning, a time interval-based algorithm that considers obstacle prediction trajectories is proposed, which uses line segment intersection detection to check for collisions. Second, to practically smooth the coarse trajectory, a continuous optimizer is implemented in three layers, corresponding to the whole path, the near-future path and the speed profile. We use discrete points to represent the far-future path and parametric curves to represent the near-future path and the whole speed profile. The approach is validated in both simulations and real-world off-road environments based on representative scenarios, including the “wait and go” scenario. The experimental results show that the proposed method improves the success rate and travel efficiency while actively avoiding static and moving obstacles.

Nonstandard approximations to Besicovitch sets

A nonstandard approximation to a Besicovitch set in R2 is constructed. It is shown that this approximation easily yields the correct upper Minkowski dimension in the two-dimensional case through a direct computation. By constructing an approximation to a Besicovitch set in three dimensions, we use a theorem of Guth and Katz to obtain an upper bound on the number of intersections of line segments in nonstandard terms.

String indexing for top-k close consecutive occurrences

The classic string indexing problem is to preprocess a string S into a compact data structure that supports efficient subsequent pattern matching queries, that is, given a pattern string P, report all occurrences of P within S. In this paper, we study a basic and natural extension of string indexing called the string indexing for top-k close consecutive occurrences problem (Sitcco). Here, a consecutive occurrence is a pair (i,j), i<j, such that P occurs at positions i and j in S and there is no occurrence of P between i and j, and their distance is defined as j−i. Given a pattern P and a parameter k, the goal is to report the top-k consecutive occurrences of P in S of minimal distance. The challenge is to compactly represent S while supporting queries in time close to the length of P and k. We give three time-space trade-offs for the problem. Let n be the length of S, m the length of P, and ϵ∈(0,1]. Our first result achieves O(nlog⁡n) space and optimal query time of O(m+k). Our second and third results achieve linear space and query times either O(m+k1+ϵ) or O(m+klog1+ϵ⁡n). Along the way, we develop several techniques of independent interest, including a new translation of the problem into a line segment intersection problem and a new recursive clustering technique for trees.

Development of a Geometric Cutting Aided System for Tin-sprayed Steel Mesh

The design of tin-sprayed steel mesh is particularly important in the production of tin-sprayed steel mesh, which is the key to ensuring the function and quality of the tin-sprayed steel mesh. At present, the tin-sprayed steel mesh is still manually designed, and the design process is error-prone and inefficient. Therefore, a geometric cutting assistance system for tin-sprayed steel mesh of the combiner based on AutoCAD and ObjectARX was developed. The system adopts a parameterized design to realize the automatic operation of the steel mesh cutting process. First introduced the system development environment and composition architecture, and briefly explained the system interaction interface. Then it explains two key methods in the software system design and development process, including cutting quality control and finding the intersection of line segments. Finally, the actual case proves that the system greatly improves the efficiency and quality of steel mesh cutting.

Further results on generalized Holmgren's principle to the Lamé operator and applications

In our earlier paper [14], it is proved that a homogeneous rigid, traction or impedance condition on one or two intersecting line segments together with a certain zero point-value condition implies that the solution to the Lamé system must be identically zero, which is referred to as the generalized Holmgren principle (GHP). The GHP enables us to solve a longstanding inverse scattering problem of determining a polygonal elastic obstacle of general impedance type by at most a few far-field measurements. In this paper, we include all the possible physical boundary conditions from linear elasticity into the GHP study with the soft-clamped, simply-supported as well as the associated impedance-type conditions. We derive a comprehensive and complete characterization of the GHP associated with all of the aforementioned physical conditions. As significant applications, we establish novel unique identifiability results by a few scattering measurements not only for the inverse elastic obstacle problem but also for the inverse elastic diffraction grating problem within polygonal geometry in the most general physical scenario. We follow the general strategy from [14] in establishing the results. However, we develop technically new ingredients to tackle the more general and challenging physical and mathematical setups. It is particularly worth noting that in [14], the impedance parameters were assumed to be constant whereas in this work they can be variable functions.

Block cutting and searching method of fractured rock mass based on topological matrix

Aiming at the problem of block cutting and searching in fractured rock mass, a method of block cutting and searching in fractured rock mass based on topological matrix is proposed. According to the structural characteristics of fractured rock mass, the data structure of the model is established, which only contains node coordinates (vectors) and a numbering system; the line segments are divided into line segment units, which only intersect at the boundary by the algorithm of line segment intersection and discretization. The number matrix describing the topological relationship of points and lines system is constructed with the intersection and line segment as the core. According to the number matrix and geometric characteristics of spatial line segments, the number matrix for block search is obtained by cutting a non-closed polygon; without any geometric operation, only the index of the number in the matrix is used to search the blocks and form a seamless rock mass model; according to the spatial relationship between the clipped line segment and the block, the block reflecting the internal fracture is generated to form the final fractured rock mass model. Results indicate that this method can identify the distribution of complex blocks and their internal joints and fissures. It also has a great application prospect in the analysis of fractured rock mass.

Alignment Percolation

The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in $d \geqslant 2$ dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on $\mathbb {Z}^{d}$ with parameter p ∈ (0,1]. For each occupied site v, and for each of the 2d possible coordinate directions, declare the entire line segment from v to the next occupied site in the given direction to be either blue or not blue according to a given stochastic rule. In the ‘one-choice model’, each occupied site declares one of its 2d incident segments to be blue. In the ‘independent model’, the states of different line segments are independent.

IC Microscopic Image Stitching Based on Improved Line Matching

Integrated Circuit(IC) microscopic image stitching is an important process in IC reverse engineering. However, IC is highly integrated and miniaturized, with many similar structures and dense distribution, and the overlapping region of some IC microimages may be the low-texture region. As a result, the traditional image stitching method is prone to mismatch, and the stitching effect is poor. This paper proposes an IC microscopic image stitching method based on improved line matching. The method first extract line segments from two IC microscopic images to be stitched by using Edline algorithm. Secondly, the intersection of line segments is generated by using the position relation of line segments in a certain strategy. Then, combined with the 2D capture method of the IC microscopic image, we limit the matching region. Finally, image registration is completed by combining points and lines, and a transformation matrix is obtained to stitch IC microimages. Experimental results demonstrate that the proposed method reduces the possibility of mismatching in the stitching process while using the stable line features and the position constraints between line segments in IC microscopic images. Moreover, the image stitching guided by line features can also achieve good stitching results in low-texture IC micrograph.

X-LineNet: Detecting Aircraft in Remote Sensing Images by a Pair of Intersecting Line Segments

Motivated by the development of deep convolution neural networks (DCNNs), aircraft detection has gained tremendous progress. State-of-the-art DCNN-based detectors mainly belong to top-down approaches, which enumerate massive potential locations of aircraft with the form of rectangular regions, and then identify whether they are objects or not. Compared with these top-down detectors, this article shows that aircraft detection via a type of bottom-up method can have better performances in the era of deep learning. In this article, we propose a novel bottom-up detector named X-LineNet. It formulates the aircraft detection task as prediction and clustering of paired intersecting line segments inside each target. Aircraft detection is then a purely appearance-based line segments estimation problem, without any rectangular regions classification or implicit features learning. With simple postprocessing, X-LineNet can simultaneously provide multiple representation forms of the detection result: the horizontal bounding box, the oriented bounding box, and the pentagonal mask. The pentagonal mask is a more accurate representation form of aircraft which has less redundancy than that of a rectangular box. Experiments show that X-LineNet outperforms prevalent top-down and region-based detectors on UCAS-AOD, NWPU VHR-10, and DIOR public data sets in the field of aircraft detection.

A Model-Driven Method for Pylon Reconstruction from Oblique UAV Images.

Pylons play an important role in the safe operation of power transmission grids. Directly reconstructing pylons from UAV images is still a great challenge due to problems of weak texture, hollow-carved structure, and self-occlusion. This paper presents an automatic model-driven method for pylon reconstruction from oblique UAV images. The pylons are reconstructed with the aid of the 3D parametric model library, which is represented by connected key points based on symmetry and coplanarity. First, an efficient pylon detection method is applied to detect the pylons in the proposed region, which are obtained by clustering the line segment intersection points. Second, the pylon model library is designed to assist in pylon reconstruction. In the predefined pylon model library, a pylon is divided into two parts: pylon body and pylon head. Before pylon reconstruction, the pylon type is identified by the inner distance shape context (IDSC) algorithm, which matches the shape contours of pylon extracted from UAV images and the projected pylon model. With the a priori shape and coplanar constraint, the line segments on pylon body are matched and the pylon body is modeled by fitting four principle legs and four side planes. Then a Markov Chain Monte Carlo (MCMC) sampler is used to estimate the parameters of the pylon head by computing the maximum probability between the projected model and the extracted line segments in images. Experimental results on several UAV image datasets show that the proposed method is a feasible way of automatically reconstructing the pylon.

A Corner Detection Method for Conventional Light Field Camera by Jointly Using Line-Features

Micro-lens array (MLA) based light field cameras have been commercialized and had a wide range of applications in various fields in recent years. To promote these applications, especially for 3D reconstruction, it is crucial to calibrate light field cameras. Among the calibration steps, it is of great importance to detect corners in checkerboard images and establish the point-to-point correspondence between the checkerboard and its images. However, almost all existing algorithms detect corners in the sub-aperture images instead of raw images for conventional light field cameras. In this paper, a corner detection algorithm that can recognize corners in the raw images of conventional light field cameras was proposed. Firstly, template matching was used to detect the 3D lines in the image space of the main lens. Then, the locations of the corners were obtained by calculating the intersection of 3D line segments and reprojecting it to the raw image again. Experimental results have demonstrated that the algorithm can still achieve good performance in actual datasets with blurred and low resolution micro lens image.

Cable routing optimization for offshore wind power plants via wind scenarios considering power loss cost model

Offshore wind power plants have been considered as one of the fastest-growing types of renewable energy technologies that is superior to the onshore wind farms with low impacts on habitat, better wind condition, higher energy efficiency, etc. The cost of submarine cables takes a significant proportion of the overall capital cost for a large-scale offshore wind farm, rendering the task of optimization of electrical infrastructure a critical role in modern wind farm design. With the increasing capacity and offshore distance, the impact of power losses in the cables on the economic performance of the wind farm becomes significant. Therefore, both the investment on the cables and the cost from the associated energy loss need to be considered in the optimization model. In this work, a detailed power loss cost model accounting for the wake effect’s impact on the wind turbine output is proposed. The cable cost and the associated power losses cost are considered in the objective function. The offshore substation location, cable connection layout, and cable sectional area are optimized simultaneously while ensuring an uncrossed cable connection layout via a line segment intersection detection algorithm. Due to the non-convexity of the optimization model, an adaptive particle swarm optimization algorithm is adopted. The proposed method was validated through a case of a real offshore wind farm, where the simulation results show that the cable connection layout formulation and sectional area selection varies significantly when different power loss model is applied. A 3.14% total cost reduction can be achieved by using the proposed method compared with the case without the power loss model.

Computational Complexity for the Problem of Optimal Intersection of Straight Line Segments by Disks

Computational complexity and exact polynomial algorithms are reported for the problem of stabbing a set of straight line segments with a least cardinality set of disks of fixed radii r > 0, where the set of segments forms a straight line drawing G = (V,E) of a plane graph without edge crossings. Similar geometric problems arise in network security applications (Agarwal et al., 2013). We establish the strong NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel graphs, and other subgraphs (which are often used in network design) for r ∈ [dmin, ηdmax] and some constant η, where dmax and dmin are the Euclidean lengths of the longest and shortest graph edges, respectively.

Privacy‐preserving collision detection of moving objects

AbstractIn this paper, we present secure protocols for obtaining the collision points of two or more moving objects, while preserving their movements' privacy. We assume that the paths of moving objects can be presented with polynomial functions of time. We reduce the problem of obtaining collision points of two moving objects to the problem of obtaining common roots of polynomials. Then, we propose a secure protocol for obtaining the common roots of two polynomials, which is based on a Gröbner basis. Moreover, we propose another secure protocol for obtaining collision points of more than two moving objects based on a line segment intersection algorithm. We also demonstrate the security proof of our proposed protocols based on the ideal/real simulation paradigm.

An Efficient Algorithm for the 1D Total Visibility-Index Problem and Its Parallelization

Let T be a terrain and P be a set of points on its surface. An important problem in Geographic Information Science (GIS) is computing the visibility index of a point p on P , that is, the number of points in P that are visible from p . The total visibility-index problem asks for the visibility index of every point in P . We present the first subquadratic-time algorithm to solve the one-dimensional total-visibility-index problem. Our algorithm uses a geometric dualization technique to reduce the problem to a set of instances of the red--blue line segment intersection counting problem, allowing us to find the total visibility-index in O ( n log 2 n ) time. We implement a naive O ( n 2 ) approach and four variations of our algorithm: one that uses an existing red--blue line segment intersection counting algorithm and three new approaches that leverage features specific to our problem. Two of our implementations allow for parallel execution, requiring O (log 2 n ) time and O ( n log 2 n ) work in the CREW PRAM model. We present experimental results for both serial and parallel implementations on synthetic and real-world datasets using two hardware platforms. Results show that all variants of our algorithm outperform the naive approach by several orders of magnitude. Furthermore, we show that our special-case red--blue line segment intersection counting implementations out-perform the existing general-case solution by up to a factor 10. Our fastest parallel implementation is able to process a terrain of more than 100 million vertices in under 3 minutes, achieving up to 85% parallel efficiency using 16 cores.

Bend-optimal orthogonal drawings of triconnected plane graphs

A drawing of a plane graph G in which each edge is represented by a sequence of alternating horizontal and vertical line segments is called an orthogonal drawing. The points of intersection of horizontal and vertical line segments of an edge in an orthogonal drawing are called bends. The best known algorithm to find a bend-optimal orthogonal drawing of a plane graph takes time O(n1.5) where n is the number of vertices in the graph. In this paper we present a new linear time algorithm to find an orthogonal drawing of a plane 3-connected graph (with maximum degree 4) and give bounds on number of bends (in terms of number k of degree-4 vertices in the graph) in the resulting drawing with respect to the number b(G) of bends in the bend-optimal orthogonal drawing of the same graph. The bound is b(G)+3k.

New Environmental Line Feature-based Vision Navigation: Design and Analysis

Vision navigation using environmental features has been widely applied when satellite signals are not available. However, the matching performance of traditional environmental features such as keypoints degrades significantly in weakly textured areas, deteriorating navigation performance. Further, the user needs to evaluate and assure feature matching quality. In this paper, a new feature, named Line Segment Intersection Feature (LSIF), is proposed to solve the availability problem in weakly textured regions. Then a combined descriptor involving global structure and local gradient is designed for similarity comparison. To achieve reliable point-to-point matching, a coarse-to-fine matching algorithm is developed, which improves the performance of the point set matching algorithm. Finally, a framework of matching quality evaluation is proposed to assure matching performance. Through the comparison, it is demonstrated that the proposed new feature has superior overall performance especially on correctly matched numbers of keypoints and matching correctness. Also, using real image sets with weak texture, it is shown that the proposed LSIF can achieve improved navigation solutions with high continuity and accuracy.