Set Of Straight Line Segments Research Articles

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.

Accurate Outline Extraction of Individual Building From Very High-Resolution Optical Images

This letter presents a novel approach for extracting accurate outlines of individual buildings from very high-resolution (0.1–0.4 m) optical images. Building outlines are defined as polygons here. Our approach operates on a set of straight line segments that are detected by a line detector. It groups a subset of detected line segments and connects them to form a closed polygon. Particularly, a new grouping cost is defined first. Second, a weighted undirected graph $\textit {G(V,E)}$ is constructed based on the endpoints of those extracted line segments. The building outline extraction is then formulated as a problem of searching for a graph cycle with the minimal grouping cost. To solve the graph cycle searching problem, the bidirectional shortest path method is utilized. Our method is validated on a newly created data set that contains 123 images of various building roofs with different shapes, sizes, and intensities. The experimental results with an average intersection-over-union of 90.56% and an average alignment error of 6.56 pixels demonstrate that our approach is robust to different shapes of building roofs and outperforms the state-of-the-art method.

Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems

Let R denote a connected region inside a simple polygon, P. By building barriers (typically straight-line segments) in P \ R , we want to separate from R part(s) of P of maximum area. All edges of the boundary of P are assumed to be already constructed or natural barriers. In this paper we introduce two versions of this problem. In the budget fence version the region R is static, and there is an upper bound on the total length of barriers we may build. In the basic geometric firefighter version we assume that R represents a fire that is spreading over P at constant speed (varying speed can also be handled). Building a barrier takes time proportional to its length, and each barrier must be completed before the fire arrives. In this paper we are assuming that barriers are chosen from a given set B that satisfies certain conditions. Even for simple cases (e.g., P is a convex polygon and B the set of all diagonals), both problems are shown to be NP-hard. Our main result is an efficient ≈11.65 approximation algorithm for the firefighter problem, where the set B of allowed barriers is any set of straight-line segments with all endpoints on the boundary of P and pairwise disjoint interiors. Since this algorithm solves a much more general problem—a hybrid of scheduling and maximum coverage—it may find wider applications. We also provide a polynomial-time approximation scheme for the budget fence problem, for the case where barriers chosen from a set of straight-line cuts of the polygon must not cross.

Minimum Cell Connection in Line Segment Arrangements

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points [Formula: see text] and [Formula: see text] in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting [Formula: see text] to [Formula: see text] that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting [Formula: see text] to [Formula: see text] must stay inside a given polygon [Formula: see text] with a constant number of holes, the segments are contained in [Formula: see text], and the endpoints of the segments are on the boundary of [Formula: see text]. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution.

Approximability issues of guarding a set of segments

In this paper we consider the following problem: given a finite set of straight-line segments S in ℝ2, find minimum in size set V of points on the segments, such that each segment of S contains at least one point in V. We call this problem guarding a set of segments (GSS). GSS is a special case of the set cover problem where the given family of subsets can be taken as a set of intersections of the straight-line segments in S. Requiring that the given subsets can be interpreted geometrically this way is a major restriction on the input, yet it has been shown that the problem is still strongly NP-complete [V.E. Brimkov, A. Leach, M. Mastroianni, and J. Wu, Guarding a set of line segments in the plane, Theor. Comput. Sci. 412 (2011), pp. 1313–1324]. In light of this result, in Brimkov et al. [Experimental studies on approximation algorithms for guarding sets of line segments, in Advances in Visual Computing, G. Bebis, R. Boyle, B. Parvin, D. Koracin, R. Chung, R. Hammoud, M. Hussain, T. Kar-Han, R. Crawfis, D. Thalmann, D. Kao, and L. Avila, eds., ISVC 2010, Part I, Lecture Notes in Computer Science, Vol. 6453, Springer, Berlin, 2010, pp. 592–601; V.E. Brimkov, A. Leach, M. Mastroianni, and J. Wu, Approximation algorithms for a geometric set cover problem, Discrete Appl. Math. 160 (2012), pp. 1039–1052] the GSS approximability was studied both theoretically and experimentally. Here we continue these investigations. In particular, we obtain conditions under which GSS admits good approximation.

Approximation algorithms for a geometric set cover problem

Given a finite set of straight line segments S in R2 and some k∈N, is there a subset V of points on segments in S with |V|≤k such that each segment of S contains at least one point in V? This is a special case of the set covering problem where the family of subsets given can be taken as a set of intersections of the straight line segments in S. Requiring that the given subsets can be interpreted geometrically this way is a major restriction on the input, yet we have shown that the problem is still strongly NP-complete [5]. In light of this result, we studied the accuracy of two polynomial-time approximation algorithms along with a third heuristic based algorithm which return segment coverings. We obtain certain theoretical results, and in particular we show that the performance ratio for each of these algorithms is unbounded, in general. Despite this, our experimental results suggest that the cases where this ratio exceeds 2 are rare for “reasonable” methods of placing segments. This is evidence that these polynomial-time algorithms solve the problem accurately in practice.

Guarding a set of line segments in the plane

We consider the following problem: Given a finite set of straight line segments in the plane, find a set of points of minimum size, so that every segment contains at least one point in the set. This problem can be interpreted as looking for a minimum number of locations of policemen, guards, cameras or other sensors, that can observe a network of streets, corridors, tunnels, tubes, etc. We show that the problem is strongly NP-complete even for a set of segments with a cubic graph structure, but in P for tree structures.

Edge-Removal and Non-Crossing Configurations in Geometric Graphs

Graphs and Algorithms A geometric graph is a graph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges.

Distortion of orientation data introduced by digitizing procedures

SUMMARYFor a given set of lines or outlines, the orientation distribution function, h(α), i.e. the line length per angle of orientation, is a useful measure of anisotropy. In order to obtain this function, it is most convenient to digitize the original, continuously curved lines, that is to replace them by an approximate set of straight line segments. h(α) is then given by the total length of line segment per angular interval. It is generally assumed that the orientation distribution function of the digitized lines approaches that of the original lines if the line segments are sufficiently small. However, in as much as the point density of a square lattice is not constant for all directions, the digitized lines, whose end points are lattice points of the digitizing grid, cannot represent all directions equally well. In this paper, the nature of the distortions introduced by the digitizing procedure and their dependence on sampling length and angular sampling interval are explored, and conditions for minimal distortions are discussed. For computer applications, it is recommended to perform a two‐dimensional smoothing spline operation on the digitized input. In doing so the distortion problem is altogether avoided.

An application of the c-varieties clustering algorithms to polygonal curve fitting

An algorithm is described that fits boundary data of planar shapes in either rectangular coordinate or chain-code format with a set of straight line segments. The algorithm combines a new vertex detection method, which locates initial vertices and segments in the data, with the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i> -elliptotype clustering algorithm, which iteratively adjusts the location of these initial segments, thereby obtaining a best polygonal fit for the data in the mean-squared error sense. Several numerical examples are given to exemplify the implementation and utility of this new approach.

Analytic evaluation of depths of dose calculation points for external beam radiation therapy treatment planning.

A method has been developed to calculate the depth of a point in a patient contour analytically. In this method, the patient contour is approximated by a Fourier series comprising 5-10 terms. The depth of a calculation point is the distance between the calculation point and the intersection of the ray line with the contour line. Depth calculations using this method are compared with calculations using a more conventional method in which the intersection is found by representing the contour as a set of straight-line segments between pairs of contour points. This method gives depth values comparable in accuracy to the more conventional method and computed approximately twice as fast.

Computer-aided hierarchical diagrams

The user interface of a diagram-drawing program, DIAG2, is described. The user is provided with three basic constructs: the straight line, the segment of a circle and the block of text. These constructs may be assembled into larger groups. Thus a diagram can be built up in a hierarchical manner by the user. DIAG2 performs the unwrapping operations necessary to decompose the diagram into a set of straight line segments, which can be output to the appropriate display peripheral. No intelligence is required in the graphics device itself.

A Programmed Variable-Rate Counter for Generating the Sine Function

The sine curve is approximated by a set of straight line segments whose slopes are chosen to be integral multiples of a binary fraction. A programmed counter counts up or down at a rate proportional to the slope, thus generating an approximate sine function. Using (360/256)?1.4 degree intervals and four integral slopes, ±0(1/128), ±1(1/128), ±2(1/128), ±3(1/128), the maximum difference between the true and the generated value is 0.014 and occurs at 36.6 degrees. The extension of this method to higher accuracy and to other functions is indicated.