Showing papers on "Computational 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. >

Relative neighborhood graphs and their relatives

Abstract: Results of neighborhood graphs are surveyed. Properties, bounds on the size, algorithms, and variants of the neighborhood graphs are discussed. Numerous applications including computational morphology, spatial analysis, pattern classification, and databases for computer vision are described. >

Automatic Mesh Generation: Application to Finite Element Methods

Abstract: This textbook concentrates on mesh generation, an essential prerequisite for the numerical analysis of engineering problems, through the use of computational geometry. The author provides a detailed survey of existing methods and highlights the increasing need for automatic techniques. Traditional structured grids are considered alongside the unstructured meshes of triangles and tetrahedra. The text contains information on the algorithms for the creation of two-and three-dimensional cases, and details tools for the modification and exploitation of meshes.

Fast capacitance extraction of general three-dimensional structures

Abstract: K. Nabors and J. White (1991) presented a boundary-element-based algorithm for computing the capacitance of three-dimensional m-conductor structures whose computational complexity grows nearly as mn, where n is the number of elements used to discretize the conductor surfaces. In that algorithm, a generalized conjugate residual iterative technique is used to solve the n*n linear system arising from the discretization, and a multipole algorithm is used to compute the iterates. Several improvements to that algorithm are described which make the approach applicable and computationally efficient for almost any geometry of conductors in a homogeneous dielectric. Results using these techniques in a program which computes the capacitance of general 3D structures are presented to demonstrate that the new algorithm is nearly as accurate as the more standard direct factorization approach, and is more than two orders of magnitude faster for large examples. >

Dynamic algorithms in computational geometry

Abstract: Dynamic algorithms and data structures in the area of computational geometry are surveyed. The work has a twofold purpose: it introduces the area to the nonspecialist and reviews the state of the art for the specialist. Fundamental data structures, such as balanced search trees and general techniques for dynamization, are reviewed. Range searching, intersections, point location, convex hull, and proximity are discussed. Problems that do not fall into these categories are also discussed. Open problems are given. >

Interval arithmetic recursive subdivision for implicit functions and constructive solid geometry

Abstract: Recursive subdivision using interval arithmetic allows us to render CSG combinations of implicit function surfaces with or without anti -aliasing, Related algorithms will solve the collision detection problem for dynamic simulation, and allow us to compute mass. center of gravity, angular moments and other integral properties required for Newtonian dynamics. Our hidden surface algorithms run in ‘constant time.’ Their running times are nearly independent of the number of primitives in a scene, for scenes in which the visible details are not much smaller than the pixels. The collision detection and integration algorithms are utterly robust — collisions are never missed due 10 numerical error and we can provide guaranteed bounds on the values of integrals. CR Categories and Subject Descriptors: G. 1.0 [Numerical Analysis ] Numerical Algorithms 1.3.3 [Picture and Image Generation ] Display algorithms, Viewing algorithms, 1.3.5 [Computational Geometry and Object Modeling] Curve, surface, solid and object representations, 1.3,5 [Computational Geometry and Object Modeling] Hierarchy and geometric transformations. 1.3.7 [Three-Dimensional Graphics and Realism] Visible line/surface algorithms, Animation

Surface parametrization and curvature measurement of arbitrary 3-D objects: five practical methods

Abstract: Curvature sampling of arbitrary, fully described 3-D objects (e.g. tomographic medical images) is difficult because of surface patch parameterization problems. Five practical solutions are presented and characterized-the Sander-Zucker approach, two novel methods based on direct surface mapping, a piecewise linear manifold technique, and a turtle geometry method. One of the new methods, called the cross patch (CP) method, is shown to be very fast, robust in the presence of noise, and is always based on a proper surface parameterization, provided the perturbations of the surface over the patch neighborhood are isotropically distributed. >

Approximation by interval Bezier curves

Abstract: The interval Bezier curve, which, unlike other curve and surface approximation schemes, can transfer a complete description of approximation errors between diverse CAD/CAM systems that impose fundamentally incompatible constraints on their canonical representation schemes, is described. Interval arithmetic, which offers an essentially infallible way to monitor error propagation in numerical algorithms that use floating-point arithmetic is reviewed. Affine maps, the computations of which are key operations in the de Casteljau subdivision and degree-elevation algorithms for Bezier curves, the floating-point error propagation in such computations, approximation by interval polynomials, and approximation by interval Bezier curves are discussed. >

Range searching with efficient hierarchical cuttings

Abstract: We present an improved space/query time tradeoff for the general simplex range searching problem, matching known lower bounds up to small polylogarithmic factors. In particular, we construct a linear-space simplex range searching data structure with O(n1–1/d) query time, which is optimal for d=2 and probably also for d>2. Further we show that multilevel range searching data structures can be built with only a polylogarithmic overhead in space and query time per level (the previous solutions require at least a small fixed power of n). We show that Hopcroft's problem (detecting an incidence among n lines and n points) can be solved in time n4/32O(log n). In all these algorithms, we apply Chazelle's results on computing optimal cuttings.

Applications of random sampling to on-line algorithms in computational geometry

Abstract: This paper presents a general framework for the design and randomized analysis of geometric algorithms. These algorithms are on-line and the framework provides general bounds for their expected space and time complexities when averaging over all permutations of the input data. The method is general and can be applied to various geometric problems. The power of the technique is illustrated by new efficient on-line algorithms for constructing convex hulls and Voronoi diagrams in any dimension, Voronoi diagrams of line segments in the plane, arrangements of curves in the plane, and others.

New foundations for the geometry of interaction

Abstract: A new formal embodiment of J.-Y. Girard's (1989) geometry of interaction program is given. The geometry of interaction interpretation considered is defined, and the computational interpretation is sketched in terms of dataflow nets. Some examples that illustrate the key ideas underlying the interpretation are given. The results, which include the semantic analogue of cut-elimination, stated in terms of a finite convergence property, are outlined. >

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.

Systolic parallel processing

Abstract: The Systolic Mode of Parallel Processing. Introduction to the Underlying Concept. The Original Motivation: VSLI Implementation. The Present Trend: Efficient Algorithms for Massively Parallel Computers. A List of Known Applications. Defining and Expressing Systolic Arrays and Algorithms. Using Automata Notions. Defining Systolic Automata, Arrays, and Algorithms. Expressing Systolic Algorithms. Analysis and Comparison of Systolic Algorithms. Matrix-Vector and Matrix Multiplication. Introduction to Vectors and Matrices. Matrix-Vector Multiplication. Systolic Simulation of Feedforward Artificial Neural Networks. Matrix Multiplication. Solving Systems of Linear Algebraic Equations. Introduction to Linear Systems. Gaussian Elimination. Systolic Arrays for Triangularization and LU/QR Decomposition. Systolic Algorithms for Back Substitution. Systolic Implementation of Iterative Methods. Further Problems of Linear Algebra. Computing the Inverse of a Matrix. Generalized Elimination. Computing the Characteristic Polynomial. Matrix Transposition and Related Operations. Convolution and Linear Filters. Convolution, Correlation, FIR and IIR Filters. Semi-Systolic Realizations. Unidirectional Full-Systolic Arrays. Systolic Arrays with Bidirectional Data Flow. Bit-Level Systolic Convolver. Operations with Polynomials. Introduction. Multiplication of Polynomials and Integers. Division of Polynomials. Computing the Greatest Common Divisor. Polynomial Interpolation. Evaluation of Polynomials. Comparison Problems. Sorting. Selection and Running Order Statistics. Sorting and Order Statistics for Rank Filtering. A Data Structure: Priority Queue. Dynamic Programming and its Applications. Introduction. Implementing the Dynamic Programming Recurrence in a Two-Dimensional Systolic Array. Implementation in One-Dimensional Arrays. Further Dynamic Programming Recurrences. Computational Geometry. Convex Hull. Nearest-Neighbours Problems. Systematic Design of Systolic Algorithms. Dependence Graphs. Systolic Array Dependence Graphs. Extracting Systolic Algorithms from Dependence Graphs. Modifying the Properties of Systolic Algorithms. Partitioning of Systolic Algorithms. Partitioning, Algorithm Mapping, Design of Flexible Systolic Structures, Time Sharing. Application of c-Slow Automata to the Realization of Parallel Structures. Examples: Mapping Different Filter Banks onto the Same Fixed-Size Processor Array. A Summary of the Technique and Alternative Approaches. References and Additional Literature. Subject Index.

Quasi-optimal upper bounds for simplex range searching and new zone theorems

Abstract: This paper presents quasi-optimal upper bounds for simplex range searching. The problem is to preprocess a setP ofn points in źd so that, given any query simplexq, the points inP źq can be counted or reported efficiently. Ifm units of storage are available (n 0. To fine-tune our results in the reporting case we also establish new zone theorems for arrangements and merged arrangements of planes in 3-space, which are of independent interest.

Diameter, width, closest line pair, and parametric searching

Abstract: We apply Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improve solutions for them. We obtain, for any fixed e > 0, an O(n1+e) algorithm for computing the diameter of a point set in 3-space, an O(n8/5+e) algorithm for computing the closest pair in a set of n lines in space. All these algorithms are deterministic. We also look at the problem of computing the k-th smallest slope formed by the lines joining n points in the plane. In 1989 Cole, Salowe, Steiger, and Szemere´di gave an optimal but very complicated O(n log n) solution based on Megiddo's technique. We follow a different route and give a very simple O(n log2n) solution which bypasses parametric searching altogether.

Delaunay's mesh of a convex polyhedron in dimension d. application to arbitrary polyhedra

Abstract: This paper presents a method for creating a Delaunay triangulation connected to a set of specified points. The theoretical aspect is recalled for an arbitrary dimension and the method is discussed in order to derive a practical approach, valid for dimensions 2 and 3, which is simple, robust and well adapted to computation. Convex polyhedral and arbitrary polyhedral situations are introduced.

A collision-based model of spiral phyllotaxis

Abstract: Plant organs are often arranged in spiral patterns. This effect is termed spiral phyllotaxis. Well known examples include the layout of seeds in a sunflower head and the arrangement of scales on a pineapple. This paper presents a method for modeling spiral phyllotaxis based on detecting and eliminating collisions between the organs while optimizing their packing. In contrast to geometric models previously used for computer graphics purposes, the new method arranges organs of varying sizes on arbitrary surfaces of revolution. Consequently, it can be applied to synthesize a wide range of natural plant structures. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling: Curve, surface, solid and object representation. I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism. J.3 [Life and Medical Sciences]: Biology.

A rational rotation method for robust geometric algorithms

Abstract: Algorithms in computational geometry often use the real-RAM model of computation. This model assumes that exact real numbers can be stored in memory and retreived in constant time, and that field operations (+, –, *, /) and certain other operations, square root, sine, and cosine for instance, are exact, and can be applied in constant time. These assumptions are often difficult to discharge at implementation time. Even well-understood algorithms, like line-sweep for polygon union [PS85], present much trouble. Why? Such algorithms obtain good combinatorial complexity bounds by exploiting geometric orders on the input. These relations are often implicit, so an algorithm can only probe them pointwise with a collection of predicate functions answering, e.g., “Is point p left of edge e?”. In implementations, the reply must depend on arithmetic with finitely represented numbers. The trouble originates here.

Approximate decision algorithms for point set congruence

Abstract: This paper considers the computer vision problem of testing whether two equal cardinality points sets A and B in the plane are e-congruent. We say that A and B are e-congruent if there exists an isometry I and bijection l:A → B such that dist(I(a), l(a)) ⩽ e, for all a ϵ A. Since known methods for this problem are expensive, we develop approximate decision algorithms that are considerably faster than the known decision algorithms, and have bounds on their imprecision. Our approach reduces the problem to that of computing maximum flows on a series of graphs with integral capacities.

An efficient approach to removing geometric degeneracies

Abstract: Our aim is to perturb the input so that an algorithm designed under the hypothesis of input non-degeneracy can execute on arbitrary instances. The deterministic scheme of [EmCa] was the first efficient method and was applied to two important predicates. Here it is extended in a consistent manner to another two common predicates, thus making it valid for most algorithms in computational geometry. It is shown that this scheme incurs no extra algebraic complexity over the original algorithm while it increases the bit complexity by a factor roughly proportional to the dimension of the geometric space. The second contribution of this paper is a variant scheme for a restricted class of algorithms that is asymptotically optimal with respect to the algebraic as well as the bit complexity. Both methods are simple to implement and require no symbolic computation. They also conform to certain criteria ensuring that the solution to the original input can be restored from the output on the perturbed input. This is immediate when the input to solution mapping obeys a continuity property and requires some case-specific work otherwise. Finally we discuss extensions and limitations to our approach.

On good triangulations in three dimensions

Abstract: In this paper, we give an algorithm that triangulates the convex hull of a three dimensional point set with guaranteed quality tetrahedra. Good triangulations of convex polyhedra are a special case of this problem. We also give a bound on the number of additional points used to achieve these guarantees and report on the techniques we use to produce a robust implementation of this algorithm under finite precision arithmetic.

Optimal trajectory planning of manipulators with collision detection and avoidance

Abstract: This article presents an optimal trajectory-planning method for robot manipulators with collision detection and avoidance. The obstacles and robot segments are represented by a set of convex polyhedra. The collision detection is performed at each discretized robot configuration by an efficient procedure devel oped with the computational geometry method, which computes a distance function of the robot segments and the obstacles. By introducing this function for specifying the collision-free con straint, the path-planning problem is formulated as an optimal control problem using the augmented Lagrangian, which may be considered as a combination of the duality, penalty and con straint relaxation methods. The problem is solved by a robust UZAWA-like algorithm, where a subgradient method is applied for the primal optimization, as the distance function is not ev erywhere differentiable. An example is given for the trajectory planning of a robot arm with three revolute joints.

Computing curvilinear structure by token-based grouping

Abstract: A computational framework for computing curvilinear structure on the edge data of images is presented. The method is symbolic, operating on geometric entities/tokens. It is also constructive, hierarchical, parallel, and locally distributed. Computation proceeds independently at each token and at each stage interleaves the discovery of structure with its careful description. The process yields a hierarchy of descriptions at multiple scales. These multiscale descriptions provide efficient feature indexing both for the grouping process itself as well as for subsequent recognition processes. Experimental results are presented to demonstrate the effectiveness of the approach with respect to curvilinear structure, and its application to more general grouping problems is discussed. >

Motion constraints from contact geometry: representation and analysis

Abstract: A method to determine constraints on translational and rotational motion of planar and 3-D objects from their contact geometry is presented. Translations are represented by spatial vectors and rotations by axes in space. For each of these, a geometric realization (M/sub a/) of the space of motion parameters is created. Subspaces in M/sub a/ that represent the range of values of motion parameters that are disallowed due to the contact are identified. The geometric realization makes it easier to visualize results, provides a good measure of the extent of restraints between objects, reduces computations by eliminating redundant constraints, and simplifies computation of new constraints. The proposed representation can be used effectively to automate the evaluation of motion constraints. >

Recognition of contact state based on geometric model

Abstract: The estimation of contact states by using force information acquired in the mating process is discussed, and a method for generating the state classifiers based on geometric models of workpieces on a computer is developed. A symbolic representation of contact states is addressed. Static behavior of workpieces at each contact state is analyzed by applying the theory of polyhedral convex cones. State classifiers that discriminate contact states are formulated by using the polyhedral convex cones, which directly provide a set of discriminant functions. To reduce real-time computations, the classifiers are simplified to a minimum set by using reduction rules of polyhedral convex cones. The algorithm to generate the state classifiers was implemented on a computer. An experiment to identify the current contact state from the measured reaction force is described to demonstrate the usefulness of the approach. >

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.

Data reduction using cubic rational B-splines

Abstract: A geometric method for fitting rational cubic B-spline curves to data representing smooth curves, such as intersection curves or silhouette lines, is presented. The algorithm relies on the convex hull and on the variation diminishing properties of Bezier/B-spline curves. It is shown that the algorithm delivers fitting curves that approximate the data with high accuracy even in cases with large tolerances. The ways in which the algorithm computes the end tangent magnitudes and inner control points, fits cubic curves through intermediate points, checks the approximate error, obtains optimal segmentation using binary search, and obtains appropriate final curve form are discussed. >

Dynamic half-space reporting, geometric optimization, and minimum spanning trees

Abstract: The authors describe dynamic data structures for half-space range reporting and for maintaining the minima of a decomposable function. Using these data structures, they obtain efficient dynamic algorithms for a number of geometric problems, including closest/farthest neighbor searching, fixed dimension linear programming, bi-chromatic closest pair, diameter, and Euclidean minimum spanning tree. >