Showing papers on "Line segment published in 1991"

The upper envelope of Voronoi surfaces and its applications

Abstract: Given a setS ofsources (points or segments) in ℜ211C;d, we consider the surface in ℜ211C;d+1 that is the graph of the functiond(x)=minpeSρ(x, p) for some metricρ. This surface is closely related to the Voronoi diagram, Vor(S), ofS under the metricρ. The upper envelope of a set of theseVoronoi surfaces, each defined for a different set of sources, can be used to solve the problem of finding the minimum Hausdorff distance between two sets of points or line segments under translation. We derive bounds on the number of vertices on the upper envelope of a collection of Voronoi surfaces, and provide efficient algorithms to calculate these vertices. We then discuss applications of the methods to the problems of finding the minimum Hausdorff distance under translation, between sets of points and segments.

Computing a face in an arrangement of line segments

Abstract: This paper presents a randomized incremental algorithm for computing a single face in an arrangement of n line segments in the plane that is fairly simple to implement. The expected running time of the algorithm is $O(n\alpha (n)\log n)$. The analysis of the algorithm uses a novel approach that generalizes and extends the Clarkson–Shor analysis technique [in Discrete Comput. Geom., 4 (1989), pp. 387–421]. A few extensions of the technique, obtaining efficient randomized incremental algorithms for constructing the entire arrangement of a collection of line segments and for computing a single face in an arrangement of Jordan arcs are also presented.

Efficient Processing of Spatial Queries in Line Segment Databases

Abstract: A study is performed of the issues arising in the efficient processing of spatial queries in large spatial databases. The domain is restricted to line segment databases such as those found in transportation networks and polygonal maps. Three classes of queries are identified. Those that deal with the line segments themselves, those that involve both the line segments and the space from which they are drawn (e.g., proximity queries), and those that involve attributes of the line segments. Handling the three types of queries requires that the line segments be stored implicitly using a bucketing approach on the space from which they are drawn. A number of bucketing approaches are examined and the PMR quadtree is chosen as the most suitable representation. Its storage and execution time requirements are evaluated in the context of finding the nearest line segment to a given point. This operation is shown to take time proportional to the splitting threshold (similar to the bucket capacity) and is independent of the density of the data. The evaluation uses the road networks in the data of the U.S. Bureau of the Census.

Approximating Polygons and Subdivisions with Minimum Link Paths

Abstract: We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We also discuss additional topological constraints such as simplicity.

Low-level Grouping of Straight Line Segments

Abstract: In this paper we present a formalism for the formation of self consistent, hierarchical, “Low-Level” groupings of pairs of straight line segments from which all higher level groupings may be derived. Additionally, each low-level grouping is associated with a “Quality” factor, based on evidential reasoning, which reflects how much the groupings differ from mathematically perfect ones. This formalism has been incorporated into algorithms within the “LPEG” software package produced at the University of Surrey. LPEG was developed as part of the Vision As Process [Crowley et al., 1989] project. We present results of the application of these algorithms to sets of line segments extracted from a test image.

A metric for line segments

Abstract: This correspondence presents a metric for describing line segments. This metric measures how well two line segments can be replaced by a single longer one. This depends for example on collinearity and nearness of the line segments. The metric is constructed using a new technique using so-called neighborhood functions. The behavior of the metric depends on the neighborhood function chosen. In this correspondence, an appropriate choice for the case of line segments is presented. The quality of the metric is verified by using it in a simple clustering algorithm that groups line segments found by an edge detection algorithm in an image. The fact that the clustering algorithm can detect long linear structures in an image shows that the metric is a good measure for the groupability of line segments. >

On vertical visibility in arrangements of segments and the queue size in the Bentley-Ottmann line sweeping algorithm

Abstract: Let $S = \{ e_1 , \cdots ,e_n \} $ be a collection of n (intersecting) line segments in the plane. Suppose that all segments have their right endpoints lying on the same vertical line, and that one wishes to bound the number of pairs of nonintersecting vertically visible segments that will intersect when extended to the right ($e_i$, $e_j$ are vertically visible if there exists a vertical line segment connecting a point on ei to a point on $e_i$ and not meeting any other segment). It is shown that there are at most $O(n\log ^2 n)$ such pairs, and only $O(n\log n)$ in the case of full rays, where the latter bound can be attained in the worst case. These results are applied to obtain similar upper and lower bounds on the maximum size of the queue in the original implementation of the Bentley–Ottmann algorithm for reporting all intersections between the segments in S, i.e., the implementation where future events are not deleted from the queue. It is also shown that, without the extra conditions on the segmen...

Intersecting line segments in parallel with an output-sensitive number of processors

Abstract: An efficient parallel algorithm is given for constructing the arrangement of n line segments in the plane, i.e., the planar graph determined by the segment endpoints and intersections. This algorithm is efficient relative to three efficiency measures—it is an NC algorithm, it has a small time-processor product, and it is output-size sensitive. In particular, it runs in $O(\log n)$ time using $O(n\log n + k)$ processors, where k is the size of the output (which is $\Omega (n^2 )$ in the worst case). The algorithm does not receive the value of k as input, it determines it on-line. A method or solving an important special case of the segment arrangement problem is also shown, namely, when each input segment is parallel to one of the coordinate axes (i.e., iso-oriented). The algorithm for this problem runs in $O(\log n)$ time using an optimal $O(n + k / \log n)$ processors. The model of computation is the CREW PRAM model, where processor allocation must be explicit and global.

Matching general polygonal arcs

Abstract: In this paper we present an algorithm for matching two polygonal arcs. We first show how to match two arcs of the same length by decomposition into pairs of equal length line segments. The distance measure used is the sum of the squared distances between corresponding points on the two arcs. We find the best match by calculating analytically the relative position and orientation of the arcs that minimize the distance measure. After analyzing the case of arcs having the same length, we show how to use this result in the algorithm for arc matching, i.e., finding the piece of a long arc that best matches with a short arc. We illustrate its application with an example and suggest how to implement it efficiently. We also discuss simplifications of the algorithm in the case of digital images.

Speeding up Bresenham's algorithm

Abstract: The line segment is the basic entity in virtually all computer graphics systems. J.E. Bresenham's algorithm (1965) efficiently scan converts line segments because it requires only an integer addition and a sign test for each pixel generated. It is the standard for scan converting a line segment. A version based on the properties of linear Diophantine equations that can speed scan conversion by a factor of almost five is presented. Two approaches are used to achieve speedup. One is to parallelize the line generation process. The other is to take advantage of the repeated patterns that the algorithm generates. >

Technique for representing sampled images

Abstract: A technique for developing an analytic model of an image by operating directly upon the pixel map representation output by an optical sampling device, such as a scanner The resulting analytic representation is hierarchical, below a certain level, with each element an analytic expression representing a portion of the image Above a certain level, the model is a simple array, to facilitate certain types of graphical manipulations The model is developed by arranging pixel values into a number of groups called tiles A number of operations are then performed on each tile, to determine how it can best be represented as an analytic expression For example, if all pixel values in a tile are approximately continuously varying, they be represented as a three-dimensional linearly sloped surface However, if one or more intensity boundaries run through the tile, analytic expressions, such as a straight line or a parabolic line segment, are derived from the tile's pixel values to describe the edges

Method and apparatus for approximating polygonal line to curve

Abstract: A computer graphic apparatus uses a method and apparatus to approximate a polygonal line of a plurality of line segments (L 1 , L 2 . . . ) to a curve (X). The curve is divided into two curve portions (P 10 P 11 , P 11 P 02 ) at a division point (P 11 ). An error (e) between a line segment (P 01 P 02 ) linking the start and end points (P 01 , P 02 ) of the curve and the division point is compared with a predetermined value (V th ). When the error is small, the line segment is treated as one part of the polygonal line, but in another case, the above-mentioned process is repeated.

Direct computation of qualitative 3-D shape and motion invariants

Abstract: Structure from motion often refers to the computation of three-dimensional structure from a matched sequence of images. However, a depth map of a surface is difficult to compute and may not be a good representation for storage and recognition. Given matched images it is shown that the sign of the normal curvature in a given direction at a given point in the image can be computed from a simple difference of slopes of line segments in one image. Using this result, local surface patches can be classified as convex, concave, cylindrical, hyperbolic (saddle point), or planar. At the same time, the translational component of the optical flow, from which the focus of expansion can be computed, is obtained. >

Angular initialization method for orientating a cutting edge portion in a cutting plotter

Abstract: The present invention relates to an angular initialization method for orientating a direction of a cutting edge portion in a cutting plotter which cuts out a cutting medium into a desired shape sheet, in which a cutter member is continuously shifted to cut out the cutting medium along two line segments from a first line segment to a second line segment which intersect with each other at a predetermined angle, and it is possible, when the cutter member is shifting on the second line segment, to change the direction of the cutting edge portion and to securely locate the direction of the cutting edge portion to coincide with the second line segment shortly before a cutting operation is commenced, even if the cutter member is not turned to a desired cutting direction while the cutter member has been shifting on the first line segment.

A new approach to plane-sweep overlay: topological structuring and line-segment classification

Abstract: An integrated approach to spatial overlay was developed with the objective of creating a single function that can perform most of the tasks now assigned to discrete functions in current systems. Two important components of this system are a unique method for topological structuring, and a method for attribute propagation and line-segment classification.

Towards structure from motion for linear features through reference points

Abstract: The 3D reconstruction from projections and feature correspondence between projections are two major problems in structure from motion. The authors show a geometric method on how a 3D affine shape is obtained from two parallel projections through general 4 reference points, or from two central projections when additional vanishing points are available. And they show also that correspondences can be established through reference points; the reference points configuration for central projections should contain at least a set of 4 coplanar points. The method is developed both for point and line segment features. Apart from that the correspondence of the necessary reference points should be known in advance, one needs neither to calibrate cameras nor to have any knowledge of its position and displacement. Projective geometry and affine geometry invariants are manipulated to perform both reconstruction and correspondence. >

Method for hidden line and surface removal in a three dimensional display

Abstract: A method for displaying three dimensional images starts by initially forming a "stripped" partition of a view window by passing a contractible, vertical attachment through each scene vertex in the view window. Faces from the scene are chosen and added to a planar partition by selecting a vertex of the chosen face; locating its corresponding vertical attachment and traveling along the line segment which is initiated at that vertex. As the line segment traverses each region and makes region to region transitions, the partitions are updated. Concurrently, each new region in the partition has a visibility pointer assigned to it. When an entire scene face has been added, a set of visibility pointers associated with regions interior to the added scene face enable depth comparisons to be carried out.

Generalization effects of k-neighbor interpolation training

Abstract: This paper describes a new training method for a continuous mapping and/or pattern classification neural network that performs local sample-density smoothing. A conventional training method uses point-to-point mapping from an input space to an output space. Even though the mapping may be precise at two given training sample points, there are no guarantees of mapping accuracy at points on a line segment connecting the sample points. This paper first discusses a theory for formulating line-to-line mapping. The theory is called interpolation training. This paper then expands the theory to k-nearest neighbor interpolation. The k-neighbor interpolation training (KNIT) method connects an input sample training point to its k-neighbor points via k line segments. Then, the method maps these k line segments in the input space for each training sample to linear line segments in the output space that interpolate between training output values. Thus, a web structure made by connecting input samples is mapped into the same structure in an output space. The KNIT method reduces the over learning problem caused by point-to-point training by smoothing input/output functions. Simulation tasks show that KNIT improves vowel recognition on a small speech database.

Digital communication channel interface

Abstract: A differential protective relay system includes protective relays 19A and 19B at the terminals of a protected transmission line segment 21. Analog signals QL1 and QL2 are sampled at the respective ends of the protected line segment 21, digitally encoded using delta modulation and biphase-mark encoding, multiplexed and transmitted over a 1.544 Mbps T1 communication channel 20 to the other end of the line where they are decoded and compared with the local signal at that end. Power is removed from the protected line segment if the difference between the local and remote signals exceeds a predefined value. Biphase-mark encoding is employed so that a clock signal may be recovered from the decoded signals.

Construction of line densities for parallel coordinate plots

Abstract: John J. MillerEdward J. WegmanCenter for Computational StatisticsGeorge Mason UniversityFairfax, VA 22030ABSTRACTThe graphical representation of high dimensional data may be accomplished by usingthe parallel coordinate plotting system. In such a representation points in Euclidian n-space aremapped into line segments in Euclidian 2-space by a projective transformation. Thisrepresentation preserves much of the geometric structure found in hyperspace but not easilyrepresentable by other methods. In large data set applications, the graphical displays may beheavily overplotted like ordinary scatterplots yielding uninterpretable displays. In this paper,we suggest replacing the raw data display with a density plot. In order to define densities forlines sensibly, we introduce the notion of line densities and develop their basic construction.We illustrate with theoretical parallel coordinate density plots for the normal and the uniformcases. Finally we illustrate sample density plots with 4-dimensional spheres.1. Introduction.The graphical display of high dimensional data is a challenging and difficult task. Anumber of interesting techniques including rotating three-dimensional scatterplots, thescatterplot matrix with brushing, the grand tour and other related graphical and computationalmethodologies have been invented with the ambition of seeing structural relationships amongrandom variables in three, four and more dimensions. One such technique is the parallelcoordinate display. This has been exposited by Inselberg (1985) as a tool for computationalgeometry and by Bolorforoush and Wegman (1988) and Wegman (1990) as a tool forexploratory data analysis. Recognizing the basic weakness of Cartesian plots as an inability tohave more than three orthogonal coordinate axes, the parallel coordinate idea is to give uporthogonality in favor of a system of parallel coordinate axes drawn in the 2-plane. Of course,as many parallel axes as one likes can be drawn in the 2-plane so that there is no inherentdimensionality limit. The idea then is to label each axis with an appropriate variable name andto connect points on adjacent axes with straight line segments. Figure 1.1 illustrates the basicidea.

A linear time algorithm for computing the shortest line segment from which a polygon is weakly externally visible

Abstract: A simple polygon P is said to be weakly externally visible from a line segment if the line segment is outside P and if for every point x on the boundary of P there is a point y on the line segment such that the interior of the line segment xy does not intersect the interior of P. In this paper a linear time algorithm is proposed for computing the shortest line segment from which a simple polygon is weakly externally visible. This is done by a suitable generalization of a linear time algorithm which solves the same problem for a convex polygon.

Geometric searching and link distance

Abstract: Given n orthogonal line segments on the plane, their intersection graph is defined such that each vertex corresponds to a segment, and each edge corresponds to a pair of intersecting segments. Although this graph can have Θ(n2) edges, we show that breadth first search can be accomplished in O(nlogn) time and O(n) space. As an application, we show that the minimum link rectilinear path between two points s and t amidst rectilinear polygonal obstacles can be computed in O(nlogn) time and O(n) space, which is optimal. We mention other related results in the paper.

Computational geometry algorithms for the systolic screen

Abstract: Adigitized plane ź of sizeM is a rectangular źM × źM array of integer lattice points called pixels. A źM × źM mesh-of-processors in which each processorPij represents pixel (i,j) is a natural architecture to store and manipulate images in ź; such a parallel architecture is called asystolic screen. In this paper we consider a variety of computational-geometry problems on images in a digitized plane, and present optimal algorithms for solving these problems on a systolic screen. In particular, we presentO(źM)-time algorithms for determining all contours of an image; constructing all rectilinear convex hulls of an image (peeling); solving the parallel and perspective visibility problem forn disjoint digitized images; and constructing the Voronoi diagram ofn planar objects represented by disjoint images, for a large class of object types (e.g., points, line segments, circles, ellipses, and polygons of constant size) and distance functions (e.g., allLp metrics). These algorithms implyO(źM)-time solutions to a number of other geometric problems: e.g., rectangular visibility, separability, detection of pseudo-star-shapedness, and optical clustering. One of the proposed techniques also leads to a new parallel algorithm for determining all longest common subsequences of two words.