Showing papers on "Point (geometry) published in 1994"

Self-calibration from multiple views with a rotating camera

Abstract: A new practical method is given for the self-calibration of a camera. In this method, at least three images are taken from the same point in space with different orientations of the camera and calibration is computed from an analysis of point matches between the images. The method requires no knowledge of the orientations of the camera. Calibration is based on the image correspondences only. This method differs fundamentally from previous results by Maybank and Faugeras on selfcalibration using the epipolar structure of image pairs. In the method of this paper, there is no epipolar structure since all images are taken from the same point in space. Since the images are all taken from the same point in space, determination of point matches is considerably easier than for images taken with a moving camera, since problems of occlusion or change of aspect or illumination do not occur. The calibration method is evaluated on several sets of synthetic and real image data.

Simulated annealing for constrained global optimization

Abstract: Hide-and-Seek is a powerful yet simple and easily implemented continuous simulated annealing algorithm for finding the maximum of a continuous function over an arbitrary closed, bounded and full-dimensional body. The function may be nondifferentiable and the feasible region may be nonconvex or even disconnected. The algorithm begins with any feasible interior point. In each iteration it generates a candidate successor point by generating a uniformly distributed point along a direction chosen at random from the current iteration point. In contrast to the discrete case, a single step of this algorithm may generateany point in the feasible region as a candidate point. The candidate point is then accepted as the next iteration point according to the Metropolis criterion parametrized by anadaptive cooling schedule. Again in contrast to discrete simulated annealing, the sequence of iteration points converges in probability to a global optimum regardless of how rapidly the temperatures converge to zero. Empirical comparisons with other algorithms suggest competitive performance by Hide-and-Seek.

Detection of 3-D simple points for topology preserving transformations with application to thinning

Abstract: The problems of 3-D digital topology preservation under binary transformations and 3-D object thinning are considered in this correspondence. First, the authors establish the conditions under which transformation of an object voxel to a non-object voxel, or its inverse does not affect the image topology. An efficient algorithm to detect a simple point has been proposed on the basis of those conditions. In this connection, some other interesting properties of 3-D digital geometry are also discussed. Using these properties and the simple point detection algorithm, the authors have proposed an algorithm to generate a surface-skeleton so that the topology of the original image is preserved, the shape of the image is maintained as much as possible, and the results are less affected by noise. >

Simple constrained deformations for geometric modeling and interactive design

Abstract: Deformations are a powerful tool for shape modeling and design. We present a new model for producing controlled spatial deformations, which we term Simple Constrained Deformations (Scodef). The user defines a set of constraint points, giving a desired displacement and radius of influence for each. Each constraint point determines a local B-spline basis function centered at the constraint point, falling to zero for points beyond the radius. The deformed image of any point in space is a blend of these basis functions, using a projection matrix computed to satisfy the constraints. The deformation operates on the whole space regardless of the representation of the objects embedded inside the space. The constraints directly influence the final shape of the deformed objects, and this shape can be fine-tuned by adjusting the radius of influence of each constraint point. The computations required by the technique can be done very efficiently, and real-time interactive deformation editing on current workstations is possible.

Shortest paths in the plane with polygonal obstacles

Abstract: We present a practical algorithm for finding minimum-length paths between points in the Euclidean plane with (not necessarily convex) polygonal obstacles. Prior to this work, the best known algorithm for finding the shortest path between two points in the plane required O(n2 log n) time and O(n2) space, where n denotes the number of obstacle edges. Assuming that a triangulation or a Voronoi diagram for the obstacle space is provided with the input (if is not, either one can be precomputed in O(n log n) time), we present an O(kn) time algorithm, where k denotes the number of “islands” (connected components) in the obstacle space. The algorithm uses only O(n) space and, given a source point s, produces an O(n) size data structure such that the distance between s and any other point x in the plane (x) is not necessarily an obstacle vertex or a point on an obstacle edge) can be computed in O(1) time. The algorithm can also be used to compute shortest paths for the movement of a disk (so that optimal movement for arbitrary objects can be computed to the accuracy of enclosing them with the smallest possible disk).

Introduction to Braided Geometry and $q$-Minkowski Space

Abstract: We present a systematic introduction to the geometry of linear braided spaces. These are versions of $\R^n$ in which the coordinates $x_i$ have braid-statistics described by an R-matrix. From this starting point we survey the author's braided-approach to $q$-deformation: braided differentiation, exponentials, Gaussians, integration and forms, i.e. the basic ingredients for $q$-deformed physics are covered. The braided approach includes natural $q$-Euclidean and $q$-Minkowski spaces in R-matrix form.

Undetermined sets of point-open games

Abstract: We show that a set of reals is undetermined in Galvin’s point-open game iff it is uncountable and has property C′′, which answers a question of Gruenhage. Let X be a topological space. The point-open game G(X) of Galvin [G] is played as follows. Black chooses a point x0 ∈ X, then White chooses an open set U0 3 x0, then B chooses a point x1 ∈ X, then W chooses an open set U1 3 x1, etc. B wins the play (x0, U0, x1, U1, . . .) iff X = ⋃ n Un. Galvin [G] showed that the Continuum Hypothesis yields a Lusin set X which is undetermined (i.e. for which the game G(X) is undetermined). (A Lusin set is an uncountable set of reals which has countable intersection with every meager set.) Recently Rec law [R] showed that every Lusin set is undetermined. Motivated by Rec law’s result we prove the following. Theorem. Let X be a topological space in which every point is Gδ. Then G(X) is undetermined iff X is uncountable and has property C ′′. Property C ′′ was introduced by Rothberger (see [M]). A topological space X has property C ′′ if for every sequence Un (n ∈ ω) of open covers of X there exist Un ∈ Un such that X = ⋃ n Un. It is known (see [M] or [FM]) that every Lusin set has property C ′′. Clearly, a space with property C ′′ must be Lindelöf. Martin’s Axiom implies that every Lindelöf space of size less than 2א0 has property C ′′ and that there are sets of reals of size 2א0 with property C ′′ (see [M]). Thus, Martin’s Axiom yields undetermined sets of reals of size 2א0 (Theorem 4 of [G]). On the other hand, in Laver’s [L] model for Borel’s conjecture all metric spaces with property C ′′ are countable (see Note 1). Thus, consistently, all metric spaces are determined. 1991 Mathematics Subject Classification: 03E15, 54G15. Supported by KBN grant PB 2 1017 91 01.

Euclidean structure from uncalibrated images

Abstract: A number of recent papers have demonstrated that camera “self-calibration” can be accomplished purely from image measurements, without requiring special calibration objects or known camera motion. We describe a method, based on self-calibration, for obtaining (scaled) Euclidean structure from multiple uncalibrated perspective images using only point matches between views. The method is in two stages. First, using an uncalibrated camera, structure is recovered up to an affine ambiguity from two views. Second, from one or more further views of this affine structure the camera intrinsic parameters are determined, and the structure ambiguity reduced to scaled Euclidean. The technique is independent of how the affine structure is obtained. We analyse its limitations and degeneracies. Results are given for images of real scenes. An application is described for active vision, where a Euclidean reconstruction is obtained during normal operation with an initially uncalibrated camera. Finally, it is demonstrated that Euclidean reconstruction can be obtained from a single perspective image of a repeated structure

Error propagation in machine vision

Abstract: Machine vision systems that perform inspection tasks must be capable of making measurements. A vision system measures an image to determine a measurement of the object being viewed. The image measurement depends on several factors, including sensing, image processing, and feature extraction. We consider the error that can occur in measuring the distance between two corner points of the 2D image. We analyze the propagation of the uncertainty in edge point position to the 2D measurements made by the vision system, from 2D curve extraction, through point determination, to measurement. We extend earlier work on the relationship between random perturbation of edge point position and variance of the least squares estimate of line parameters and analyze the relationship between the variance of 2D points.

Computational Analysis of Visual Motion

Abstract: Introduction: Image Motion in Visual Function. Geometry in Euclidean Space R3: Some Basic Notions. Rigid Body Kinematics: Some Basic Notions. Interpretation of Point Correspondences. Optical Flow Interpretation. Interpretation of Straight Line Correspondences. Interpretation of Image Motion of Straight Lines. Detection, Computation, and Segmentation of Visual Motion. Conclusion: Current Issues in Analysis of Visual Motion.

Study of ambiguities of linear arrays

Abstract: The paper is concerned with the ambiguities inherent in the manifold of any uniform or nonuniform linear-array. Ambiguities arise when the manifold repeats itself or when a point on the array manifold can be written as a linear combination of other points. So far only "trivial" ambiguities associated with uniform linear arrays have been studied in the literature. These ambiguities are viewed as appearance of grating lobes when beamforming is used. In the paper the ambiguity problem is extended to any linear geometry and to "non-trivial" ambiguities. Then a new technique is proposed for identifying the ambiguous sets associated with a given geometry. >

Boundary Values of Berezin Symbols

Abstract: A functional Hilbert space is a collection H of complex-valued functions on some set S such that H is a Hilbert space with respect to the usual vector operations on functions and which has the property that point evaluations are continuous (i.e., for each z ∊ S, the map f → f (z) is a continuous linear functional on H).

Competitive searching in a generalized street

Abstract: We consider the problem of a robot which has to find a path in an unknown simple polygon from one point s to another point t, based only on what it has seen so far. A Street is a polygon for which the two boundary chains from s to t are mutually weakly visible, and the set of streets was the only class of polygons for which a competitive search algorithm was known.We define a new, strictly larger class of polygons, called 𝒢-streets which are characterized by the property that every point on the boundary of a 𝒢-street is visible from a point on a horizontal line segment connecting the two boundary chains from s to t. We present an on-line strategy for a robot placed at s to find t in an unknown rectilinear 𝒢-street; the length of the path created is at most 9 times the length of the shortest path in the L1 metric. This is optimal since we show that no strategy can achieve a smaller competitive factor for all rectilinear 𝒢-streets. Compared to the L2-shortest path, the strategy is 9.06-competitive which leaves only a very small gap to the lower bound of 9.

Halftoning method using space filling curves

Abstract: An image processing system for preparing a document image for printing, each discrete area or pixel in the image described by a signal having a number of possible states greater than can be rendered by a selected printer. In such a system a screening pattern is used which distributes screen values through the image in accordance with the pattern of a space filling curve. Each signal is thresholded against a threshold determined by such ordering and output a signal renderable by the selected printer. A straight forward function relates threshold value in a screen matrix of values and the length of a space filling curve at any point in an image. The nature of certain space filling curves, allows the generation of two reasonably sized lookup tables providing the length of the curve at any point on the page.

Matching shapes with a reference point

Abstract: For two given point sets, we present a very simple (almost trivial) algorithm to translate one set so that the Hausdorff distance between the two sets is not larger than a constant factor times the minimum Hausdorff distance which can be achieved in this way. The algorithm just matches the so-called Steiner points of the two sets.The focus of our paper is the general study of reference points (like the Steiner point) and their properties with respect to shape matching.For more general transformations than translations, our method eliminates several degrees of freedom from the problem and thus yields good matchings with improved time bounds.

Medial axis transform to boundary representation conversion

Abstract: The medial axis transform (MAT) has potential as a powerful representation for a conceptual design tool for objects with inherent symmetry or near-symmetry. The medial axis of two-dimensional objects or medial surface of three-dimensional objects provides a conceptual design base, with transition to a detailed design occurring when the radius function is added to the medial axis or surface, since this additional information completely specifies a particular object. To make such a design tool practicable, however, it is essential to be able to convert from an MAT format to a boundary representation of an object. In this thesis, we provide the details for the conversion of the MAT of a set of two- and three-dimensional objects to a boundary representation. We demonstrate certain smoothness properties of the MAT and show the relationship between the tangent to the MAT at a point and the boundary points related to that MAT point. We classify the MAT points based on the tangency conditions at the point, and for each type of point, we detail the method for obtaining the boundary points related to it. We discuss requirements for an MAT to be locally valid in the sense that the given curves could actually be the MAT of an allowable object. We also provide a theoretical error bound on the computation for the two-dimensional case. Finally, we discuss an implementation of our algorithm both for piecewise linear two-dimensional MATs and for piecewise planar and linear three-dimensional MATs, and demonstrate some results we have obtained.

Apparatus and method for evaluating shape of three-dimensional object

Abstract: An apparatus for evaluating the shape of a three-dimensional object, comprises means for setting an ideal surface of the three-dimensional object within a given space based on reference data, means for setting a worked surface of the three-dimensional object corresponding to the ideal surface within said space as a set of space coordinates obtained from measurements at a plurality of measurement points, intersection point calculating means for determining space coordinates at a point at which a line extending from each of the measurement points perpendicularly intersects the ideal surface, and distance calculating means for determining the distance between the intersection point and the measurement point based on the space coordinates of intersection and measurement points, whereby the distance determined by said distance calculating means can be used to evaluate the precision of the worked surface. Measurement data of the work surface previously measured is compared with the ideal surface based on reference data such as CAD shape data or the like to determine a distance from each of the measurement points to the ideal surface in the perpendicular direction.

Method and system for comparing free-form geometries using high density point data models

Abstract: A method is provided for verifying the accuracy of a part geometry with respect to a master geometry. The method begins with the step of providing a computer workstation including a computer and an output device attached to the computer. The method also includes the steps of generating a master HDPDM describing the geometry of the master and generating a part HDPDM describing the geometry of the part. The master HDPDM includes a plurality of points. The method further includes the step of aligning the master HDPDM and the part HDPDM within a coordinate frame. The method continues with the step of computing the distance from each point of the master HDPDM to the surface described by the part HDPDM to obtain signed distance data. The method concludes with the step of displaying an image on the output device based on the signed distance data. The image represents the non-conformance between the master and the part and is used to verify the accuracy of the geometry of the part with respect to the geometry of the master.

The Grothendieck theory of dessins d'enfants: Dessins from a geometric point of view

Abstract: Abstract In this paper we study the topological aspects of dessins (via analytic description) with two distinct goals. Firstly we are interested in fields of definition and fields of moduli. We give a topological proof that there exist some dessins with no model defined over their field of moduli. This answers explicitly a question asked in [Har87]. Our second motivation is to collect practical and theoretical data for the explicit computation of covers given by some topological description, following ideas of Atkin [ASD71] Oesterle and ourselves. This leads to a method for the computation of the linear space associated to a divisor on a given dessin. Introduction This paper develops some practical applications of the archimedean analytic description of coverings through Puiseux series. In the second section, we recall a classical result due to Klein concerning the classification of genus zero Galois coverings, and related to the classification of regular polytopes. In the third section we give a review of many possible definitions of what a moduli field is. We do not claim to exhaust the list of various contradictory notions denoted by these words, but simply to avoid the frequent confusion about it. The fourth section is an illustration of what knowledge can be provided by local considerations at infinity. We show that such a study leads to interesting examples of coverings with strange rationality properties, which we can state by mere combinatorial considerations.

Observability/identifiability of rigid motion under perspective projection

Abstract: The "visual motion estimation" problem concerns the reconstruction of the motion of an object viewed under projection. This paper addresses the feasibility of such a problem when the object is represented as a "rigid" set of point-features in the Euclidean 3D space. We represent rigid motion as a point on the so-called "essential manifold" and show that it is globally observable from perspective projections under some general position conditions. Such conditions hold when the path of the viewer and the visible objects cannot be embedded in a quadric surface of R/sup 3/. >

Utilization of ground control points for image orientation without point identification in image space

Abstract: The classical photogrammetric point determination makes use of ground control points (GCP), which includes the point identification in the images and the interactive measurement of image coordinates. The new approach proposed in this paper utilizes 3D GCP, but without any point identification in image space. Besides the control point coordinates additional object point coordinates of conjugate points are required. The conjugate points are acquired in such a way, that at least three object points are arranged in the surroundings of each control point. The approach takes advantage of the knowledge that all these points lie on the terrain surface. Thus, a mathematical condition can be set up which describes that the control point is located in an inclined plane, defined by the three neighboring object points. THe paper describes the mathematical model in detail and demonstrates the efficiency of the new approach using simulated data.© (1994) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.