Range searching in a set of line segments

Given a collection of line segments in the plane we would like to connect the segments by their endpoints to construct a simple circuit. (A simple circuit is the boundary of a simple polygon). However, there are collections of line segments where this cannot be done. In this note it is proved that deciding whether a set of line segments admits a simple circuit is NP-complete. Deciding whether a set of horizontal line segments can be connected with horizontal and vertical line segments to construct an orthogonal simple circuit is also shown to be NP-complete.

The convex hull of a planer set of points can be defined as the set of vertices of the smallest convex polygon containing all the points. If S is a planer set of points then convex layers of S can be derived by iteratively computing the convex hull of S and then removing it from S. Computation of the convex layers has widely been studied in the static environment where the set of points is fixed. The theoretical lower bound of computing the convex layers of a fixed set of points is O (n log n). The static convex layers algorithms with the optimal time complexity have already been proposed in the literature. A set of points where the points can be inserted or deleted is called a dynamic set of points. The set of convex layers should be reconstructed from the scratch at each insertion or deletion if a static convex layers algorithm was used to handle a dynamic set of points. Therefore, it takes O (n log n) time to handle an insertion or a deletion even for an optimal static convex layers algorithm. A dynamic convex layers algorithm has been proposed recently that can perform an insertion or a deletion by doing a slight modification to the existing set of convex layers. It takes O (n) time for an insertion or a deletion. It assumes that the set of points does not contain collinear points and that assumption is not valid in practical applications. Furthermore, the notion of tangent used in that approach restricts the extension of the algorithm into higher dimensions. This paper proposes a novel dynamic convex layers algorithm to eliminate the drawbacks in the existing algorithm. Salient feature of this algorithm is that it represents each layer as a set of line segments. The layers are modified upon an insertion or a deletion of a point by adding some new line segments and deleting some existing line segments. The proposed algorithm takes O (n 3 /k 2 ) time for an insertion or a deletion where k is the number of convex layers. A computer implementation is also presented. The proposed algorithm can work with set of points with collinear points and coincident points. The notion used in the algorithm can easily be extended to higher dimensions. Suppose the set of convex layers have already been found for a given set of points. Further, suppose that the layers are close to each other and new points are expected to fall within the region bounded by the outermost layer. This kind of situations widely occurs in practice and then the proposed algorithm takes O (n) time for an inclusion or a deletion.

Based on the unique geometric features of the ruled surface, a new efficient algorithm for ruled surface/plane intersection is proposed. The ruled surface is firstly dispersed into a set of line segments, and the ruled surface/plane intersection is transformed to the intersection of a group line segments with a plane. Then a set of ordered intersection points can be obtained by the proposed line/plane intersection algorithm. According to the serial number of every intersection point, all the intersection points are grouped and reorganized. Each group point corresponds to an intersection curve. All the intersection curves can be reconstructed by curve interpolation. Compared with the traditional tracing method, the proposed algorithm can avoid complex calculation including initial points searching and intersection points sorting, which is more efficient and stable.

We propose a compact optical system made up of a thin collimator and a free-form lens array to provide the uniform illumination of various-shape regions. Analytical formulas to design the lens array are derived. The simulation results for the designed optical elements demonstrate the high-level illumination uniformity of various regions in the form of a rectangle, a cross, a flag-shaped region, a line segment, and a set of line segments.

Event-based temporal contrast vision sensors such as the Dynamic Vison Sensor (DVS) have advantages such as high dynamic range, low latency, and low power consumption. Instead of frames, these sensors produce a stream of events that encode discrete amounts of temporal contrast. Surfaces and objects with sufficient spatial contrast trigger events if they are moving relative to the sensor, which thus performs inherent edge detection. These sensors are well-suited for motion capture, but so far suitable event-based, low-level features that allow assigning events to spatial structures have been lacking. A general solution of the so-called event correspondence problem, i.e. inferring which events are caused by the motion of the same spatial feature, would allow applying these sensors in a multitude of tasks such as visual odometry or structure from motion. The proposed Event-based Line Segment Detector (ELiSeD) is a step towards solving this problem by parameterizing the event stream as a set of line segments. The event stream which is used to update these low-level features is continuous in time and has a high temporal resolution; this allows capturing even fast motions without the requirement to solve the conventional frame-to-frame motion correspondence problem. The ELiSeD feature detector and tracker runs in real-time on a laptop computer at image speeds of up to 1300 pix/s and can continuously track rotations of up to 720 deg/s. The algorithm is open-sourced in the jAER project.

Given a simple polygon P on n vertices, two points x, y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum-size set S such that every point in P is visible from a point in S. The set S is referred to as guards. Assuming integer coordinates and a specific general position on the vertices of P, we present the first O(log OPT)-approximation algorithm for the point guard problem. This algorithm combines ideas in papers of Efrat and Har-Peled and Deshpande et al. We also point out a mistake in the latter.

Exploration of the Most Effective Dural Incision Design in a Decompressive Craniectomy.

We study three covering problems in the plane. Our original motivation for these problems come from trajectory analysis. The first is to decide whether a given set of line segments can be covered by up to four unit-sized, axis-parallel squares. The second is to build a data structure on a trajectory to efficiently answer whether any query subtrajectory is coverable by up to three unit-sized axis-parallel squares. The third problem is to compute a longest subtrajectory of a given trajectory that can be covered by up to two unit-sized axis-parallel squares.

This paper describes an approach to constructing a 3D paper-made object from hand-written sketches. This approach consists of two phases. One is the phase that constitutes a crease pattern based on a sketch. Another is the phase that constructs a virtual origami model from the obtained crease pattern. The crease pattern is a set of line segments in an unfolded sheet of paper and often designed by origami design methods. When an origami model is folded from a crease pattern, faces in the crease pattern may be transformed into the same plane and some inconsistent objects may be represented. In order to construct feasible (fold-able) objects, we need to dispose the faces on the same plane consistently. Therefore, in this paper, a method for analyzing overlap order of faces based on simulated annealing is proposed. Furthermore, we show some examples of 3D objects constructed by our method. The proposed method is useful for packaging and architectural modeling

This paper we are proposing to detect the localization and recognition of an indoor stairway. This is a fundamental step in the implementing of autonomous stair climbing navigation, as well as the implementing of passive alarm systems intended for the blind and visually impaired. This method analyzes the edges of a stairway based on planar motion tracking and directional filters. The horizontal edges of the stairs are extracted by using the Gabor Filter. Then, the vanishing point is extracted from the specified set of line segments in the aim of facilitating the reconstruction of the stair treads. After this stage, we extract a hypothetical set of targets by using the correlation method. Finally, we employ the discrimination method to find the ground plane, using the behavioral distance measurement. Consequently, the remaining information is considered as an indoor stairway candidate region. As a result, testing is able to prove its effectiveness.

In this paper we give a parallel algorithm for constructing the Voronoi diagram of a polygonal scene, i.e., a set of line segments in the plane such that no two segments intersect except possibly at their endpoints. Our algorithm runs inO(log2n) time usingO(n) processors in the CREW PRAM model.

Controlled Perturbation of sets of line segments in [formula omitted] with smart processing order

Chapter 34 - The Construction of Perceptual and Semantic Features During Category Learning

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya’s problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal Θ(nlogn)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.

This paper describes an algorithm that robustly locates salient convex collections of line segments in an image. The algorithm is guaranteed to find all convex sets of line segments in which the length of the gaps between segments is smaller than some fixed proportion of the total length of the lines. This enables the algorithm to find convex groups whose contours are partially occluded or missing due to noise. We give an expected case analysis of the algorithm performance. This demonstrates that salient convexity is unlikely to occur at random, and hence is a strong clue that grouped line segments reflect underlying structure in the scene. We also show that our algorithm run time is O(n/sup 2/log(n)+nm), when we wish to find the m most salient groups in an image with n line segments. We support this analysis with experiments on real data, and demonstrate the grouping system as part of a complete recognition system.