Showing papers on "Point (geometry) published in 1992"

A method for registration of 3-D shapes

Abstract: The authors describe a general-purpose, representation-independent method for the accurate and computationally efficient registration of 3-D shapes including free-form curves and surfaces. The method handles the full six degrees of freedom and is based on the iterative closest point (ICP) algorithm, which requires only a procedure to find the closest point on a geometric entity to a given point. The ICP algorithm always converges monotonically to the nearest local minimum of a mean-square distance metric, and the rate of convergence is rapid during the first few iterations. Therefore, given an adequate set of initial rotations and translations for a particular class of objects with a certain level of 'shape complexity', one can globally minimize the mean-square distance metric over all six degrees of freedom by testing each initial registration. One important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model, prior to shape inspection. Experimental results show the capabilities of the registration algorithm on point sets, curves, and surfaces. >

Optical position determination

Abstract: This invention relates to a coordinate sensor for detecting the position of a moveable detector relative to a data space coded with coordinate information by detecting and processing the coordinate information. It may be used for obtaining both two and three dimensional position related information, such as might be used for determining the position of a pen/pencil on paper for handwriting data input. It also may be used for detecting angular, rotational, and linear motion. The coordinate sensor provides an apparatus for precisely locating the position of a movable element within a space. More particularly, it provides an input/output apparatus for use with a computer that includes a movable element, whose exact position within a space can be determined without any physical connection between the movable element and the space. It comprises a data space formatted with a code for designating the coordinates of a point in the data space, a detector for detecting the code, a processor for processing data from the detector to determine the position of the point in the data space, and a data output method for human interface to the coordinate sensor.

Measuring shapes by size functions

Abstract: We define the concept of size functions. They are functions from the real plane to the natural numbers which describe the `shape of the objects' (seen as submanifolds of a Euclidean space). We give two different techniques of computation of size functions and some actual examples of computation. Moreover, we present the concept of deformation distance between manifolds (i.e., curves, surfaces, etc.). It is a distance which measures the `difference in shape' of two manifolds. Finally we point out the link between deformation distances and size functions.

On dynamic Voronoi diagrams and the minimum Hausdorff distance for point sets under Euclidean motion in the plane

Abstract: We show that the dynamic Voronoi diagram of k sets of points in the plane, where each set consists of m points moving rigidly, has complexity O(n2k2ls(k)) for some fixed s, where ls(n) is the maximum length of a (n, s) Davenport-Schinzel sequence. This improves the result of Aonuma et al., who show an upper bound of O(n3k4 log* k) for the complexity of such Voronoi diagrams. We then apply this result to the problem of finding the minimum Hausdorff distance between two point sets in the plane under Euclidean motion. We show that this distance can be computed in time O((m + n)6 log (mn)), where the two sets contain m and n points respectively.

Parallel Computational Geometry

Abstract: Models of parallel computation convex hull intersection problems geometric searching visibility and separability nearest neighbours Vonoroi diagrams geometric optimization triangulation of polygons and point sets current trends future directions.

Apparatus for the registration of three-dimensional shapes

Abstract: An apparatus for identifying surfaces, comprising a point digitizing apparatus for obtaining data points of a surface to be located and an apparatus, such as a memory device, for storing a mathematical model of a model surface. A microprocessor control unit registers the mathematical model and the obtained data points, wherein the microprocessor control unit accomplishes the registration by (i) determining a data point set comprising various points from the obtained data points, (ii) finding a set of closest points comprising, for each point of the data point set, a closest point on the first mathematical model, (iii) registering the data point set to the set of closest points to obtain a registered point set, (iv) computing a mean square distance of points of the registered point set to the closest point set, (v) comparing the mean square distance to a predetermined threshold and (vi) if the mean square distance is above a threshold, (vi.i) redefining the data point set as the registered point set and (vi.ii) repeating parts (ii), (iii), (iv) and (v). Resultant from the registration, the control unit provides an output signal that can be used to indicate the surface position, quality and/or identification.

Iterative broadening

Abstract: Conventional blind search techniques generally assume that the goal nodes for a given problem are distributed randomly along the fringe of the search tree. We argue that this is often invalid in practice, suggest that a more reasonable assumption is that decisions made at each point in the search carry equal weight, and show that a new search technique that we call iterative broadening leads to orders-of-magnitude savings in the time needed to search a space satisfying this assumption. Both theoretical and experimental results are presented.

Space Efficient 3D Model Indexing

Abstract: : We show that the set of 2D images produced by the point features of a rigid 3D model can be represented with two lines in two high-dimensional spaces. These lines are the lowest-dimensional representation possible. We use this result to build a system for representing in a hash table at compile time, all the images that groups of model features can produce. Then at run time a group of image features can access the table and find all model groups that could match it. This table is efficient in terms of space, and is built and accessed through analytic methods that account for the effect of sensing error. In real images, it reduces the set of potential matches by a factor of several thousand. We also use this representation of a model's images to analyze two other approaches to recognition: invariants, and non-accidental properties. These are properties of images that some models always produce, and all other models either never produce (invariants) or almost never produce (non-accidental properties). In several domains we determine when invariants exist. In general we show that there is an infinite set of non-accidental properties that are qualitatively similar.... Object recognition, Non-accidental properties, Indexing, Hashing, Invariants, Space efficiency.

Faster line segment intersection

Abstract: Publisher Summary This chapter presents an algorithm to determine whether two line segments in 2-D space intersect or not and to determine the point of intersection. Rapid calculation of line segment intersections is important because this function is often called a primitive many number of times in the inner loops of other algorithms. To develop the algorithm, it is convenient to use vector representation. The chapter describes C implementation as one that involves calculation of the intersection point coordinates where the segments are found to intersect. The C implementation uses integer arithmetic. Calculation of the intersection point involves operations that are cubic in the input coordinates so that the limitation to input coordinates in the range [0, 1023] or other similar-sized range avoids overflow on 32-bit computers. When line segments do not intersect, input coordinates in the range [0, 16383] can be handled.

Lazy code motion

Abstract: We present a bit-vector algorithm for the optimal and economical placement of computations within flow graphs, which is as efficient as standard uni-directional analyses. The point of our algorithm...

Particle flux shadowing for three-dimensional topography simulation

Abstract: In a three-dimensional (3-D) topography simulator, a method for removing sources of particle flux because of neighboring topology, for a point on a workpiece undergoing a deposition or etch process step. The method is practiced in a Generalized Solids Modeling system that utilizes a boundary representation model for representing a workpiece as one or more material object solids. For any given point on the 3-D structure, the neighboring topography forms a complex shadowing mask with respect to sources of particle flux, thus making analytical determination of visible sources of incoming particle flux difficult. The method is comprised generally of the steps of: defining a numerical mesh in a space over a surface of the workpiece; specifying an intensity of incident flux for each mesh point, identifying a set of mesh points defining a visible range of mesh points with respect to a particular target point and identifying mesh points in said set of mesh points that are obscured by neighboring topology.

System and method of generating variable width lines within a graphics system

Abstract: A video graphics system and method for creating variable width lines such that the lines may be easily edited. As a line image is created, a center point of that image along its length is temporarily established having width information at various points. This width information is used to create an outline of image of the desired shape, the outline having point data spaced at intervals. This point data can then be edited by a user to create different shapes. In situations where a pressure sensitive stylus is used to create the variable length line, a look-back technique is employed to insure that the line remains wide at the end point even though the user is reducing pressure as the system is removed from the pad.

Triple density molecular quantum similarity measures: A general connection between theoretical calculations and experimental results

Abstract: A newquantum similarity measure is defined. It is proposed as a way to project density functions from infinite dimensional spaces, where density functions belong, to finite dimensional ones. The procedure allows the representation of a given molecule initially described in terms ofnth order density functions as a point in a finite dimensional Euclidean space: apoint-molecule. Further manipulation of the obtained information permits the representation of a molecular set as a cloud of points: amolecular point cloud, in a variety of graphical manners. A previous experience in the field is compared with this new approach.

Numerical solution of phase change problems: an eulerian-lagrangian approach

Abstract: The problem of solid-liquid phase change is tackled from art Eulerian-Lagrangian kinematic point of view. The theory is first presented for a one-dimensional space and then generalized to a two-dimensional space. The irregular shape of the phase front is treated with generalized curvilinear coordinates. A feature of the resulting finite-difference scheme is that the current field values and front position are solved simultaneously. As a result, numerical solutions show excellent agreement with the analytical solution for the one-dimensional melting problem and with experimental data for a two-dimensional buoyancy-driven phase-change problem.

Method and apparatus for changing the perspective view of a three-dimensional object image displayed on a display screen

Abstract: An apparatus and method for rotating a perspective view of a three-dimensional object image displayed on a display screen from an original view point to a desired view point, and for moving the perspective view in a parallel manner from the original view point to the desired view point. When the perspective view is rotated, a circle is generated that has a first point corresponding to the desired view point. A line segment is generated from the first point to the center of the circle. A hemisphere is generated that has a top surface, a vertex, and the circle as a bottom surface. A second point on the top surface represents the first point projected to the top surface. The desired view point is a projection from the second point to the bottom surface of the hemisphere and the original view point is a projection from the vertex of the hemisphere. The perspective view displayed on the display screen is then changed to the desired view point. When the perspective view is moved in a parallel manner, a boundary is generated which represents a displayable range of the three-dimensional object image on the display screen. A line segment is generated from the current cursor position to the center of the boundary. The displayed perspective view is changed by moving the perspective view along the line segment by a distance equal to the length of the line segment.

Pareto optimality, anonymity, and strategy-proofness in location problems

Abstract: Generalized location problems withn agents are considered, who each report a point inm-dimensional Euclidean space. A solution assigns a compromise point to thesen points, and the individual utilities for this compromise point are equal to the negatives of the Euclidean distances to the individual positions. Form = 2 andn odd, it is shown that a solution is Pareto optimal, anonymous, and strategy-proof if, and only if, it is obtained by taking the coordinatewise median with respect to a pair of orthogonal axes. Further, for all other situations withm≥2, such a solution does not exist. A few results concerning other solution properties, as well as different utility functions, are discussed.

An analysis of vector space models based on computational geometry

Abstract: This paper analyzes the properties, structures and limitations of vector-based models for information retrieval from the computational geometry point of view. It is shown that both the pseudo-cosine and the standard vector space models can be viewed as special cases of a generalized linear model. More importantly, both the necessary and sufficient conditions have been identified, under which ranking functions such as the inner-product, cosine, pseudo-cosine, Dice, covariance and product-moment correlation measures can be used to rank the documents. The structure of the solution region for acceptable ranking is analyzed and an algorithm for finding all the solution vectors is suggested.

On Local Matching of Free-Form Curves

Abstract: Geometric matching in general is a difficult unsolved problem in computer vision. Fortunately, in many practical applications, some a priori knowledge exists which considerably simplifies the problem. In visual navigation, for example, the motion between successive positions is usually either small or approximately known, but a more precise registration is required for environment modeling. The algorithm described in this paper meets this need. Objects are represented by free-form curves, i.e., arbitrary space curves of the type found in practice. A curve is available in the form of a set of chained points. The proposed algorithm is based on iterativeiy matching points on one curve to the closest points on the other. A least-squares technique is used to estimate 3-D motion from the point correspondences, which reduces the average distance between curves in the two sets. Both synthetic and real data have been used to test the algorithm, and the results show that it is efficient and robust, and yields an accurate motion estimate.

The farthest point Delaunay triangulation minimizes angles

Abstract: The planar dual to the Euclidean farthest point Voronoi diagram for the set of vertices of a convex polygon has the lexicographic minimum possible sequence of triangle angles, sorted from sharpest to least sharp. As a consequence, the sharpest angle determined by three vertices of a convex polygon can be found in linear time.

Locations and strengths of point sources: stability estimates

Abstract: The author proves Lipschitz stability results in the problem of the determination of locations and strengths of point sources in three-dimensional Euclidean spaces from measurement of generated potentials.