Showing papers on "Line segment published in 1988"

Parallel computational geometry

Abstract: We present efficient parallel algorithms for several basic problems in computational geometry: convex hulls, Voronoi diagrams, detecting line segment intersections, triangulating simple polygons, minimizing a circumscribing triangle, and recursive data-structures for three-dimensional queries.

New methods for computing visibility graphs

Abstract: Let S be a set of n non-intersecting line segments in the plane The visibility graph GS of S is the graph that has the endpoints of the segments in S as nodes and in which two nodes are adjacent whenever they can “see” each other (ie, the open line segment joining them is disjoint from all segments or is contained in a segment) Two new methods are presented to construct GS Both methods are very simple to implement The first method is based on a new solution to the following problem: given a set of points, for each point sort the other points around it by angle It runs in time O(n2) The second method uses the fact that visibility graphs often are sparse and runs in time O(m log n) where m is the number of edges in GS Both methods use only Ogr;(n) storage

An optimal algorithm for intersecting line segments in the plane

Abstract: The authors present the first optimal algorithm for the following problem: given n line segments in the plane, compute all k pairwise intersections in O(n log n+k) time. Within the same asymptotic cost the algorithm will also compute the adjacencies of the planar subdivision induced by the segments, which is a useful data structure for contour-filling on raster devices. >

Measuring Image Flow By Tracking Edge-lines

Abstract: This paper describes a technique for measuring the movement of edge-lines in a sequence of images by maintalning an image plane "flow model". Edge-lines are expressed as a set of parameter vectors representing the center-point, orientation and length of a segment. Each parameter vector is composed of an estimate, a temporal derivative, and their covariance matrix. Line segment parameters in the flow model are updated using a Kalman filter. The eorrespondance of observed edge-lines segments to segments predicted from the flow model is determined by a linear complexity algorithm using distance normalized by covariance. The existence of segments in the flow model is controlled using a confidence factor. This technique is in everyday use as part of a larger system for building 3-D scene descriptions using a camera mounted on a robot arm. A near video-rate hardware implementation is currently under development

Constraints on length and angle

Abstract: Given a perspective projection of line segments on the image plane, the constraints on their 3D positions and orientations are derived on the assumption that their true lengths or the true angles they make are known. The approach here is first to transform images of line segments to the center of the image plane as if the camera were rotated to aim at them. The 3D information extracted in this canonical position is then transformed back to the original configuration. Examples are given, by using real images, for analytical 3D recovery of a rectangular corner and a corner with two right angles.

Method and apparatus for drawing high quality lines on color matrix displays

Abstract: A method and apparatus for drawing high quality lines on color matrix displays wherein a line segment is created by activating a series of linear elements substantially centered about the predetermined line segment position and providing for various intensities for each element, the notion of a pixel group is completely discarded and each individual display element is individually addressed and individually assigned an intensity depending upon the predetermined line segment to be displayed and the orientation of that line segment wherein the method comprises generating element intensity, position and line slope information for a given line segment; inverting and registering the element intensity information; centering an array of elements around the element position information; determining the color of elements in the array of elements; determining the proper intensity for each element in the array of elements in order to produce the predetermined position of the line; and providing the proper intensity for each element and the array of elements in order to provide the proper line color.

Intersecting line segments, ray shooting, and other applications of geometric partitioning techniques

Abstract: We present a variety of applications of certain techniques, based on partition trees, that were originally developed for range searching problems. Our results are obtained by enhancing and extending these techniques, and include: (i) An O(n4/3+δ+k)-time (for any δ>0), O(n)-space randomized algorithm for finding all k intersections of n line segments in the plane (we can count the number of these intersections in O(n4/3+δ) time and linear space). (ii) Preprocessing a collection of n (possibly intersecting) segments in the plane so that, given any query ray, we can find quickly the first segment it hits. Other applications concern “implicit” point location, hidden surface removal in three dimensions, polygon placement queries, and problems involving overlapping planar maps. We also present several efficient algorithms involving the analysis of the connectivity and other useful properties of arrangements of line segments.

Method of recognizing a circular arc segment for an image processing apparatus

Abstract: A method of recognizing an arcuate or circular arc segment is used with an image processing apparatus. In a first step, two arbitray image structuring points are selected as segmenting end points from the dot-line image data. next, an image structuring point among the image structuring points present within a segment divided by the segment end points that is located furthest from and in a direction perpendicular to a straight line connecting the given structuring points and within a distance which is equal to or larger than a constant distance is extracted. The, three consecutive, adjacent polygonal-line structuring points are recognized as an arcuate or circular arc segment when a distance between a virtual circle passing through the three adjacent points and a median point of two line segments formed by the three consecutive points is equal to or less than the constant distance.

Polyhedral line transversals in space

Abstract: Algorithms are developed for determining if a set of polyhedral objects inR3 can be intersected by a common transversal (stabbing) line. It can be determined inO(n logn) time if a set ofn line segments in space has a line transversal, and such a transversal can be found in the same time bound. For a set of polyhedra with a total ofn vertices, we give anO(n4 logn) algorithm for determining the existence of, and computing, a line transversal. Helly-type theorems for lines and segments are also given. In particular, it is shown that if every six of a set of lines in space are intersected by a common transversal, then the entire set has a common transversal.

First passage time problems in time-dependent fields

Abstract: This paper discusses the simplest first passage time problems for random walks and diffusion processes on a line segment When a diffusing particle moves in a time-varying field, use of the adjoint equation does not lead to any simplification in the calculation of moments of the first passage time as is the case for diffusion in a time-invariant field We show that for a discrete random walk in the presence of a sinusoidally varying field there is a resonant frequency ϖ* for which the mean residence time on the line segment is a minimum It is shown that for a random walk on a line segment of lengthL the mean residence time goes likeL2 for largeL when ϖ≠ϖ*, but when ϖ=ϖ* the dependence is proportional toL The results of our simulation are numerical, but can be regarded as exact Qualitatively similar results are shown to hold for diffusion processes by a perturbation expansion in powers of a dimensionless velocity These results are extended to higher values of this parameter by a numerical solution of the forward equation

Graph-based vectorization method for line patterns

Abstract: A vectorization method for line patterns is proposed which converts digital binary images into line segment vectors. The vector data is more compact and more natural than that obtained by conventional methods using thinning operations. The proposed method consisted of four steps. First, thinning of an input binary image is performed. Then a medical line image obtained by the thinning operation is transformed into a graph, in which pixels on the medial line correspond to nodes and neighboring nodes are connected by edges. Next, extra edges unnecessary for preserving the topology of the medial line image are deleted. The deletion can be implemented as an iterative parallel operation. Finally, the graph is simplified by line approximation. Every step except the line approximation is suitable for parallel processing. The experimental results of applying the proposed method to geographical maps show that the method reduces data volume by about 30%, as compared with conventional methods. >

On The Congruence Of Noisy Images To Line Segment Models

Abstract: An algorithm is described for matching two-dimensional images that does not depend on the extraction of features, a step which is often difficult or impractical. It combines computational efficiency with robustness against noisy data. It has been tested with two kinds of data. The algorithm matches a two-dimensional image, in the form of a set of points, to a line segment model. It estimates a positive congruence (i.e., a translation plus a rotation) that moves the image onto the model. The algorithm is most appropriate for matching when the required congruence is small, and it seems very well suited to motion-tracking. Experimental results have been obtained for two kinds of images: (1) points obtained from an optical range finder, and (2) edge points extracted from an intensity image. Only the former results are presented.

Egomotion And The Stabilized World

Abstract: This paper formulates the tasks of moving-object detection and motion-based depth recovery as a problem of sensor fusion in the presence of uncertainty. We utilize two sensor systems, one providing information about the local image velocity (spsecifying a point in image-velocity space), and the othter providing information about the camera motion (specifying a line segment in image-velocity space). 1Ne utilize Mahalanobis distance as a threshold rule for determining consistency between the measurements from the two sensor systems. We also suggest a framework for using the resulting segmented flow field to update estimates of the egomotion parameters.

Probabilistic Analysis of a Greedy Heuristic for Euclidean Matching

Abstract: Given a collection of n points in the plane, the Euclidean matching problem is the task of decomposing the collection into matched pairs connected by line segments in such a way as to minimize the sum of all the segment lengths. The greedy heuristic provides an approximate solution to the Euclidean matching problem by successively matching the two closest unmatched points. We study the behavior of Gn, the sum of the lengths of the segments produced by the greedy heuristic.

A system for recognition and description of graphics

Abstract: A system for automatic recognition and description of graphical primitives and their interconnections in a mixed text/graphics image is described. The input to the system is a high-resolution binary image obtained by scanning paper-based documents. The Hough transform is applied to the connected components of the image to locate and separate text strings of various font size and orientation. The graphics in the segmented image is processed to represent thin entities by their core lines and thick objects by their boundaries. Lines of various types are identified and segmented into straight line and curved line segments during this process. The line segments and their interconnections are analyzed to locate minimum redundancy loops which are adequate to generate a succinct description of the graphics. Various hatching and filling patterns in the image are identified and described. The algorithm has been implemented as software, and its robustness and performance have evaluated tested using a number of test images. The results for a simple test image are reported. >

An iterative edge linking algorithm with noise removal capability

Abstract: A linking algorithm to fill gaps between edge segments is presented. The gap filling operation is performed in an iterative manner rather than a single step. During each iteration, the noises are also removed gradually. The proposed algorithm starts with connecting the tip ends of two (line or curve) segments with a line segment and then tries to modify the resulting segment by straight-line fitting. The two major phases of the algorithm are detailed step by step. The advantages of using the algorithm are also discussed. >

Test pattern for use monitoring variations of critical dimensions of patterns during fabrication of semiconductor devices

Abstract: An improved test pattern for monitoring variations of critical dimensions of patterns produced during the fabrication of semiconductor devices is disclosed. The test pattern which allows the critical dimensions to be easily monitored by a microscope, has a reference pattern including a reference line corresponding to one vertical edge of the reference pattern, said reference pattern formed flatly in a given layer below the present layer, a step shaped first pattern including a plurality of vertical indication step line segments corresponding to each vertical edge of the step and being formed flatly in the present layer, each extending line of said line segments being spaced by an equal horizontal interval and one of said line segments lying in the reference line, and a step shaped second pattern including a plurality of vertical indication step line segments corresponding to vertical edges of the step of the second pattern and being formed in separation from the first pattern in the present layer, each of said line segments of the second pattern being arranged on the extending line of each line segment of the first pattern.

Polygonizations of point sets in the plane

Abstract: We examine the different ways a set ofn points in the plane can be connected to form a simple polygon. Such a connection is called apolygonization of the points. For some point sets the number of polygonizations is exponential in the number of points. For this reason we restrict our attention to star-shaped polygons whose kernels have nonempty interiors; these are callednondegenerate star-shaped polygons. We develop an algorithm and data structure for determining the nondegenerate star-shaped polygonizations of a set ofn points in the plane. We do this by first constructing an arrangement of line segments from the point set. The regions in the arrangement correspond to the kernels of the nondegenerate star-shaped polygons whose vertices are the originaln points. To obtain the data structure representing this arrangement, we show how to modify data structures for arrangements of lines in the plane. This data structure can be computed inO(n4) time and space. By visiting the regions in this data structure in a carefully chosen order, we can compute the polygon associated with each region inO(n) time, yielding a total computation time ofO(n5) to compute a complete list ofO(n4) nondegenerate star-shaped polygonizations of the set ofn points.

Geometric data structures for computer graphics: an overview

Abstract: In computational geometry many sophisticated data structures have been designed for storing geometric objects. Many of these structures might be very useful for solving a number of problems in computer graphics and, hence, it is important that people in computer graphics know these data structures. In this paper we discuss a number of such geometric data structures and indicate their advantages and uses, especially in computer graphics. The structures we consider are: The quad-tree is probably the most well-known data structure in computer graphics. Quad-trees can be used to store planar objects, both in image space and in object space. They use little storage and often allow for fast retrieval of geometric information. Range trees are a clear example of so-called multi-dimensional data structures. They allow for very fast searching to locate e.g. the points in a set lying in a given rectangle. Segment trees store intervals in an efficient way. They can be used for solving problems concerning e.g. line segments. Many geometric data structures are static, i.e., they don’t allow for insertions and deletions of objects. Some general dynamisation techniques will be discussed that can be used for turning static data structures into dynamic structures. With the description of the different data structures examples are given to show how the structures can be used for solving actual geometric problems. For example we will show how the segment tree can be used to solve the windowing problem efficiently.

Distortion of orientation data introduced by digitizing procedures

Abstract: SUMMARY For 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.

Approximate coding of digital contours

Abstract: Two methods are proposed for coding the discrete contour of binary images. A set of key pixels (guiding pixels) on the contour is defined for this purpose. The decoding schemes approximate the contour through the key pixels using the quadratic Bezier approximation technique. The amount of deviation of the decoded image from the original image is studied using the objective measures of percentage error and shape compactness. The decoded images are found to be faithful reproductions of the original image. A set of cleaning operations is also introduced as an intermediate step before final reproduction. It was found that the bit requirements and the compression ratios are also improved significantly as compared to the contour run length coding and discrete line segment coding techniques. >

Method and apparatus for decomposing a quadrilateral figure for display and manipulation by a computer system

Abstract: A circuit for decomposing a graphic figure to be displayed by a computer output display into pairs of line segments which line segments outline the figure comprising apparatus for providing signals indicative of the vertices of a figure to be displayed, apparatus for providing comparisons of the relative positions of such vertices as they are to be displayed, apparatus for selecting a starting vertex for determining the line segments which include common scan lines, apparatus for selecting a first line segment connected to the starting vertex, apparatus for selecting a second line segment connecting to the starting vertex; and apparatus for providing output signals indicative of line segments selected.

Method of drawing in graphics rendering system

Abstract: Method and means for drawing a convex geometric figure to framebuffer storage sequentially addressable as a plurality of update arrays of determined origins which tile the framebuffer. An array comprises pixel storage sites, each specifiable by an offset from array origin. The sites are concurrently updatable. A figure is specified as a set of directed lines; the line segments between mutual intersections comprise the figure boundary; the segments perambulate the boundary in a single sense. For an update array, concurrently for each pixel site, and concurrently for each directed line, the sidedness of the pixel is evaluated with respect to the line to derive a one-bit line discriminant signal. For each site, the line discriminant signals are ANDed to derive a result discriminant signal; a first value specifying outsidedness of the pixel with respect to the figure prevents writing to the site. The process is repeated for further arrays to tile the figure to be drawn.

An Analysis of Lowe's Model-based Vision System.

Abstract: Like several other model-based vision systems, the initial phase of Lowe's method involves applying an edge detector to the input image and extracting line segments from the result This gives a set of k observed segments These segments have to be matched to the n segments in the model This matching process can be viewed as a search through an "interpretation tree" containing (n + l) nodes For any reasonable values of k and n, this tree contains a very large number of nodes A sophisticated search strategy is needed to complete this search in a reasonable time The search strategy developed by Lowe starts by grouping the observed line segments into significant multisegment structures called perceptual groups These "observed" groups are then matched to groups known to exist in the model By initially matching groups instead of individual segments, the interpretation tree is pruned to a huge extent Furthermore, since perceptual groups containing three or more segments are used to initialize the search, at each stage of the search it is possible to solve for the object's pose parameters (ie location and orientation) Using this estimate of pose, the position of each model segment's image can be predicted Subbranches of the interpretation tree are only considered if the observed segments are in close proximity to the predicted image of their assigned model segment These two factors lead to a very efficient search strategy

Chords through a convex body generated from within an embedded body

Abstract: A body E is completely embedded within a convex body G. A line segment is generated by a measure depending only on E or on both E and G. This line segment is then projected to the surface of G in one or both directions. The distributions of selected line segments of this type are derived.

Method and apparatus for clipping figures against a clipping frame using line segment cross point and area detection

Abstract: In order to clip a figure using a predetermined frame, coordinate values of cross points defined by a line segment representing a figure to be clipped and each side of the clipping frame or each extension line thereof are detected. A positional relationship between the line segment and the clipping frame, and the presence/absence of the cross points on the clipping frame are obtained, respectively. By utilizing the above information, a new boundary line which does not exist in a figure before clipping can be extracted. Since the detection of cross point coordinates and the extraction of boundaries are independently processed, high-speed processing can be realized.

Interpreting engineering drawings of polyhedrons

Abstract: A system is discussed that restores solid models, or a set of polyhedra consistent with a given set of three orthographic views which are input by using a facsimile device. Emphasis is on the description of two novel techniques: the line-segment extraction method and the polyhedron restoration method. The line-segment extraction is performed by using a probe, a small fragment of line, which moves along the central axis of each line segment represented by a binary image. Therefore, the method does not require any preprocessing such as noise elimination, or thinning of the given binary image. The polyhedron restoration method uses a face-oriented approach, rather than the conventional wire-frame-oriented approach. The method is basically a combinatorial search procedure that finds all of the legal combinations of possible faces. >