Showing papers on "Computational geometry published in 1999"

Level set methods and fast marching methods : evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science

Abstract: Introduction 1 Formulations of interface propagation Part I Theory and Algorithms: 2 Theory of curve and surface evolution 3 Hamilton-Jacobi equations and associated theory 4 Numerical approximations: first attempt 5 Numerical schemes for hyperbolic conservation laws 6 Algorithms for the initial and boundary value formulations 7 Efficient schemes: adaptivity 8 Triangulated versions of level set and fast marching method: extensions and variations 9 Tests of basic methods Part II Applications: 10 Geometry 11 Grid generation 12 Image denoising 13 Computer vision: shape detection and recognition 14 Fluid mechanics and materials sciences: adding physics 15 Computational geometry and computer-aided-design 16 First arrivals, optimizations, and control 17 Applications to semi-conductor manufacturing 18 Comments, conclusions, future directions References Index

Fast computation of generalized Voronoi diagrams using graphics hardware

Abstract: We present a new approach for computing generalized 2D and 3D Voronoi diagrams using interpolation-based polygon rasterization hardware. We compute a discrete Voronoi diagram by rendering a three dimensional distance mesh for each Voronoi site. The polygonal mesh is a bounded-error approximation of a (possibly) non-linear function of the distance between a site and a 2D planar grid of sample points. For each sample point, we compute the closest site and the distance to that site using polygon scan-conversion and the Z-buffer depth comparison. We construct distance meshes for points, line segments, polygons, polyhedra, curves, and curved surfaces in 2D and 3D. We generalize to weighted and farthest-site Voronoi diagrams, and present efficient techniques for computing the Voronoi boundaries, Voronoi neighbors, and the Delaunay triangulation of points. We also show how to adaptively refine the solution through a simple windowing operation. The algorithm has been implemented on SGI workstations and PCs using OpenGL, and applied to complex datasets. We demonstrate the application of our algorithm to fast motion planning in static and dynamic environments, selection in complex user-interfaces, and creation of dynamic mosaic effects. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.3 [Computer Graphics]: Picture/Image Generation. Additional

Robust mesh watermarking

Abstract: We describe a robust method for watermarking triangle meshes. Watermarking provides a mechanism for copyright protection of digital media by embedding information identifying the owner in the data. The bulk of the research on digital watermarks has focused on media such as images, video, audio, and text. Robust watermarks must be able to survive a variety of “attacks”, including resizing, cropping, and filtering. For resilience to such attacks, recent watermarking schemes employ a “spread-spectrum” approach – they transform the document to the frequency domain and perturb the coefficients of the perceptually most significant basis functions. We extend this spread-spectrum approach to work for the robust watermarking of arbitrary triangle meshes. Generalizing spread spectrum techniques to surfaces presents two major challenges. First, arbitrary surfaces lack a natural parametrization for frequency-based decomposition. Our solution is to construct a set of scalar basis function over the mesh vertices using multiresolution analysis. The watermark perturbs vertices along the direction of the surface normal, weighted by the basis functions. The second challenge is that simplification and other attacks may modify the connectivity of the mesh. We use an optimization technique to resample an attacked mesh using the original mesh connectivity. Results show that our watermarks are resistant to common mesh operations such as translation, rotation, scaling, cropping, smoothing, simplification, and resampling, as well as malicious attacks such as the insertion of noise, modification of low-order bits, or even insertion of other watermarks. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Surface Representations.

Geometry-based watermarking of 3D models

Abstract: This article addresses the fundamentals of geometry-based watermarking. It presents a watermarking algorithm that modifies normal distribution to invisibly store information in the model's geometry.

Sliver exudation

Abstract: A sliver is a tetrahedron whose four vertices lie close to a plane and whose projection to that plane is a convex quadrilateral with no short edge. Slivers are notoriously common in 3-dimensional Delaunay triangulations even for well-spaced point sets. We show that if the Delaunay triangulation has the ratio property introduced in [15] then there is an assignment of weights so the weighted Delaunay triangulation contains no slivers. We also give an algorithm to compute such a weight assignment.

Using generic programming for designing a data structure for polyhedral surfaces

Abstract: Software design solutions are presented for combinatorial data structures, such as polyhedral surfaces and planar maps, tailored for program libraries in computational geometry. Design issues considered are flexibility, time and space efficiency, and ease-of-use. We focus on topological aspects of polyhedral surfaces and evaluate edge-based representations with respect to our design goals. A design for polyhedral surfaces in a halfedge data structure is developed following the generic programming paradigm known from the Standard Template Library STL for C++. Connections are shown to planar maps and face-based structures.

Reachability Analysis Using Polygonal Projections

Abstract: This paper presents Coho, a reachability analysis tool for systems modeled by non-linear, ordinary differential equations. Coho represents high-dimensional objects using projections onto planes corresponding to pairs of variables. This representation is compact and allows efficient algorithms from computational geometry to be exploited while also capturing dependencies in the behaviour of related variables. Reachability is performed by integration where methods from linear programming and linear systems theory are used to bound trajectories emanating from each face of the object. This paper has two contributions: first, we describe the implementation of Coho and, second, we present analysis results obtained by using Coho on several simple models.

Feature extraction of separation and attachment lines

Abstract: Separation and attachment lines are topologically significant curves that exist on 2D surfaces in 3D vector fields. Two algorithms are presented, one point-based and one element-based, that extract separation and attachment lines using eigenvalue analysis of a locally linear function. Unlike prior techniques based on piecewise numerical integration, these algorithms use robust analytical tests that can be applied independently to any point in a vector field. The feature extraction is fully automatic and suited to the analysis of large-scale numerical simulations. The strengths and weaknesses of the two algorithms are evaluated using analytic vector fields and also results from computational fluid dynamics (CFD) simulations. We show that both algorithms detect open separation lines-a type of separation that is not captured by conventional vector field topology algorithms.

Orthogonal Polyhedra: Representation and Computation

Abstract: In this paper we investigate orthogonal polyhedra, i.e. polyhedra which are finite unions of full-dimensional hyper-rectangles. We define representation schemes for these polyhedra based on their vertices, and show that these compact representation schemes are canonical for all (convex and non-convex) polyhedra in any dimension. We then develop efficient algorithms for membership, face-detection and Boolean operations for these representations.

Parametric person identification from the EEG using computational geometry

Abstract: Person identification based on features extracted parametrically from the EEG spectrum is investigated in this work. The method proposed utilizes computational geometry algorithms (convex polygon intersections), appropriately modified, in order to classify unknown EEGs. The signal processing step includes EEG spectral analysis for feature extraction, by fitting a linear model of the AR type on the alpha rhythm EEG signal. The correct classification scores obtained on real EEG data experiments (91% in the worst case) are promising in that they corroborate existing evidence that EEG carries genetically specific information and is therefore appropriate as a basis for person identification methods.

Geometric Applications of a Randomized Optimization Technique

Abstract: We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching) Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal k -point subsets, matching point sets under translation, computing rectilinear p -centers and discrete 1-centers, and solving linear programs with k violations

A shape-preserving data embedding algorithm for NURBS curves and surfaces

Abstract: Existing data embedding algorithms for polygonal meshes and their attributes cannot be applied to the majority of (geometric) computer aided design (CAD) applications for two major reasons. First, these CAD systems employ parametric curves and surfaces, not polygonal meshes, as their main shape-defining primitives. Second, most CAD applications do not tolerate modifications of model topology and/or geometry that are introduced by existing data embedding algorithms. We propose a new data embedding algorithm for non-uniform rational B-spline (NURBS) curves and surfaces, which employs rational linear reparameterization for embedding messages. The algorithm exactly preserves the shape, that is, the geometry and topology of its embedding targets. Furthermore, it preserves the data size of the model. We consider how these two properties, preservation of shape and preservation of data size, can be significant with regard to the use of data embedding in CAD applications. In addition to the shape- and data size-preserving data embedding algorithm for NURBS curves and surfaces, we outline additional methods for embedding data in various types of parametric curves and surfaces.

An introduction to line geometry with applications

Abstract: The article presents a brief tutorial on classical line geometry and investigates new aspects of line geometry which arise in connection with a computational treatment. These mainly concern approximation and interpolation problems in the set of lines or line segments in Euclidean three-space. In particular, we study the approximation of data lines by, in a certain sense, ‘linear’ families of lines. These sets are, for instance linear complexes and linear congruences. An application is the reconstruction of helical surfaces or surfaces of revolution from scattered data points. This is based on the fact that the normals of these surfaces lie in linear complexes; in particular, normals of surfaces of revolution intersect the axis of revolution. Approximation with linear complexes or congruences is also useful in detecting singular positions of serial or parallel robots. These are positions where the robot should be a rigid system but possesses an undesirable and unexpected instantaneous self motion.

Accessibility analysis for planning of dimensional inspection with coordinate measuring machines

Abstract: Computer-controlled dimensional inspection is typically performed with coordinate measuring machines (CMMs), which are very precise Cartesian robots that use touch probes to measure the coordinates of points on a workpiece's surfaces. Automatic planning and programming of inspection tasks with a CMM involve spatial reasoning, to determine how to orient the part on the CMM, which probes to use, how to orient the probes, and so on. This paper introduces the notions of accessibility and approachability, which are important for inspection planning, and describes two sets of implemented algorithms for computing accessibility information. One of these sets of algorithms performs exact computations on polyhedral objects and is relatively slow, whereas the other uses discrete approximations and achieves high speed by exploiting standard computer graphics hardware. The discretized algorithm has been tested on real-world parts, and is sufficiently fast for industrial applications.

Mathematics for Computer Graphics Applications

Abstract: This completely revised Second Edition of ""Computer Graphics"" includes valuable information on major organizational changes within the last few years. This edition brings to the fore the basic mathematical tools of computer graphics, including vectors, matrices, and transformations. Additionally, it provides a strong, comprehensive base in exploring math, computer science, physics, engineering, and in special subjects such as algebraic and computational geometry, geometric modeling, and CAD/CAM. A highly diversified book that can be utilized as a primary textbook, supplemental teaching resource, individual tutorial, or key reference text. Includes new chapters on symmetry, limit and continuity, constructive solid geometry, and the Bezier curve. Provides many new figures and exercises. Contains an annotated suggested reading list with exercises and answers in each chapter. Appeals to both academics and professionals. Offers a new solutions manual for instructors.

On some geometric optimization problems in layered manufacturing

Abstract: Efficient geometric algorithms are given for optimization problems arising in layered manufacturing, where a 3D object is built by slicing its CAD model into layers and manufacturing the layers successively. The problems considered include minimizing the stair-step error on the surfaces of the manufactured object under various formulations, minimizing the volume of the so-called support structures used, and minimizing the contact area between the supports and the manufactured object—all of which are factors that affect the speed and accuracy of the process. The stair-step minimization algorithm is valid for any polyhedron, while the support minimization algorithms are applicable only to convex polyhedra. The techniques used to obtain these results include construction and searching of certain arrangements on the sphere, 3D convex hulls, halfplane range searching, and constrained optimization.

A framework for streaming geometry in VRML

Abstract: We introduce a framework for streaming geometry in VRML that eliminates the need to perform complete downloads of geometric models before starting to display them. This framework for the progressive transmission of geometry has three main parts, as follows: 1) a process to generate multiple levels-of-detail (LODs); 2) a transmission process (preferably in compressed form); and 3) a data structure for receiving and exploiting the LODs generated in the first part and transmitted in the second. The processes in parts 1 and 2 have already received considerable attention. We concentrate on a solution for part 3. Our basic contribution is a flexible LOD storage scheme, which we refer to as a progressive multilevel mesh. This scheme, primarily intended as a data structure in memory, has a low memory footprint and provides easy access to the various LODs (thus suitable for efficient rendering). This representation is not tied to a particular automated polygon reduction tool. In fact, we can use the output of any polygon reduction algorithm based on vertex clustering (including the edge collapse operations used in several algorithms). The progressive multilevel mesh complements compression techniques such as those developed by M. Deering (1995), H. Hoppe (1996) or G. Taubin et al. (1998). We discuss the integration of some of these compression techniques. However, for the sake of simplicity, we use a simple file format to describe the algorithm.

Application Challenges to Computational Geometry: CG Impact Task Force Report

Abstract: With rapid advances in computer hardware and visualization systems, geometric computing is creeping into virtually every corner of science and engineering, from design and manufacturing to astrophysics to molecular biology to uid dynamics. This report assesses the opportunities and challenges this presents for the eld of computational geometry in the years ahead. Can CG meet the algorithmic needs of practitioners? Should it look to applied areas for new sources of problems? Can CG live up to its potential and become a key player in the vast and diverse world of geometric computing? These are some of the questions addressed in this document. It was prepared by a group of computer scientists, engineers, and mathematicians with extensive experience in geometric computing. This report is intended as a wake-up call rather than an agenda setter. It is hoped it will engage a community-wide discussion on the future of computational geometry. The fraction of computing falling under the loosely deened rubric of \geometric computation" has been on the rise and is likely to become dominant in the next decade. Computer graphics, manufacturing, scientiic visualization, computer vision, astrophysics, molecular biology, and uid mechanics are just a few in a crowd of avid users of geometric computing. Where does computational geometry t into all this? Twenty-odd years ago, the nascent eld of computational geometry set sail on a mission to build general tools | analytical and computational | to satisfy the algorithmic needs of The intention was to create a body of knowledge to which computer programmers could turn for help when wrestling with geometric

On the computational geometry of ruled surfaces

Abstract: This article presents a brief introduction to the classical geometry of ruled surfaces with emphasis on the Klein image and studies aspects which arise in connection with a computational treatment of these surfaces. As ruled surfaces are one parameter families of lines, one can apply curve theory and algorithms to the Klein image, when handling these surfaces. We study representations of rational ruled surfaces and get efficient algorithms for computation of planar intersections and contour outlines. Further, low degree boundary curves, useful for tensor product representations, are studied and illustrated at hand of several examples. Finally, we show how to compute efficiently low degree rational G 1 ruled surfaces.

Output sensitive extraction of silhouettes from polygonal geometry

Abstract: An algorithm to allow real time interactive extraction and orthographic display of silhouettes of complex two-manifold polygonal object(s) is presented. An off-line pre-processing of all the edges of all polygons enables the efficient extraction of the silhouette edges in real time, once a viewing direction is prescribed. During the interactive session, the time complexity of extracting the silhouette edges is linear in the number of edges in the silhouette, and is typically in the order of O(/spl radic/n), where n is the number of polygons in the scene. The time complexity of the pre-processing stage is linear in n.

Determining the lines through four lines

Abstract: This paper describes how to compute the line or lines which intersect four given lines in three dimensions. This intersection computation arises in computer graphics (for visibility computations), computational geometry (for line traversals), and computer vision (for object recognition). Given four distinct lines in three dimensions, there exist zero, one, two, or various infinities of lines intersecting the given lines. We use the Pliicker coordinatization of lines to cast this problem as a null-space computation in five dimensions, and show how the singular value decomposition (SVD) yields a simple, stable characterization of the incident lines, and an efficient algorithm to determine them. Finally, we enumerate the types of input degeneracies that may arise, show how to detect each type in practice, and describe for each case the solution set of lines that arises.

Tools for computing tangent curves for linearly varying vector fields over tetrahedral domains

Abstract: We present some very efficient and accurate methods for computing tangent curves for three-dimensional flows. Our methods work directly in physical coordinates, eliminating the usual need to switch back and forth with computational coordinates. Unlike conventional methods, such as Runge-Kutta, for computing tangent curves which give only approximations, our methods produce exact values based upon piecewise linear variation over a tetrahedrization of the domain of interest. We use balycentric coordinates in order to efficiently track cell-to-cell movement of the tangent curves.

Advances in Digital and Computational Geometry

Abstract: From the Publisher: Computational geometry deals with the construction of algorithms and their complexity related to problems in geometry arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographic information systems. Digital geometry deals with geometric properties of subsets of digital images or, equivalently, with geometric properties of finite sets of lattice points. Digital geometry can anticipate progress in imaging technology allowing higher and higher spatial resolution. It seems that the input data in both fields will "converge" to data embedded in digital arrays of very high spatial resolution. This book covers the important developments in digital and computational geometry including methods of approximating geometric objects.

Predicting geometrical properties of random packed beds from computer simulation

Abstract: Using advances in computational geometry and collision-detection algorithms, an algorithm was developed to analyze and predict the geometrical properties of a randomly packed structure using packing objects of arbitrary shape. A 3-D polygonal model was used to describe an arbitrarily complex packing object and a simple container object. The dynamics of the packing process is not simulated, but a search algorithm finds stable equilibrium positions for the packing objects of arbitrary shapes using a collision-detection algorithm in a 3-D space. A modified conjugate gradient optimization method determines the packing object's final packing location and orientation. Once the bed is packed, both macroscopic quantities like the overall porosity, the specific surface area, and the number of packing objects per unit volume and microscopic properties like the porosity variation in any direction could be determined. For accurate porosity calculation inside a given 3-D polygonal sample space, the Sutherland-Hodgman polygon clipper algorithm was used. Predicted results are validated against available experimental data for spheres, Raschig rings, Pall rings, and cascade minirings.

Self-calibration of a light striping system by matching multiple 3-D profile maps

Abstract: A novel method is proposed for refining the calibration of a light striping system including a projective transformation between the image plane of the camera and the plane of the laser sheet, and also the direction of the scanning with respect to the plane of the laser sheet. The refinement is obtained through weighted least squares matching of multiple profile maps acquired from different viewpoints and registered previously using an approximate calibration. Testing with synthetically generated profile maps shows that if the geometry of the object is appropriate and the registration parameters and the intrinsic parameters of the system are known exactly, then a calibration accuracy of 0.003...0.00003% relative to the scene dimensions can be achieved as the average noise level in the maps used for the calibration decreases from 0.3 down to zero pixels. It is also possible to adjust several calibrations at the same time. The registration and calibration parameters can be refined simultaneously, but a close initial estimate and rather complex object geometry are needed for an accuracy of 0.03% when the average noise level is 0.03 pixels. Determining the corresponding points by interpolation on the parametric domains of the maps yields higher accuracy than perpendicular projection to the tangent planes at the closest points in 3D in both registration and calibration tasks. The highest accuracy is achieved when the interpolation errors are as equal as possible within the overlapping areas.

Multiresolution rendering with displacement mapping

Abstract: In this paper, we present for the first time an approach for hardware accelerated displacement mapping The displaced surface is generated from a 2D displacement map by remeshing a coarse triangle mesh according to the screen projection of the surface The remeshing algorithm is implemented in hardware Filtered access to the displacement map makes our approach competitive with available view dependent multiresolution techniques The advantage of displacement mapping is the compact representation A displacement mapped surface consumes together with all filter levels only a fraction of the storage space needed for a hardware compatible representation of an equivalent triangle mesh A possible design of the displacement mapping rendering pipeline is proposed Previously described hardware components are used as often as possible Our approach can be smoothly integrated into all available graphics application programming interfaces Most existing graphics applications can be extended to the new feature with marginal effort CR Categories: I31 [Computer Graphics]: Hardware Architecture—Raster display devices I33 [Computer Graphics]: Picture/Image Generation—Bitmap and framebuffer operations, Display algorithms, Viewing algorithms I35 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations I37 [Computer Graphics]: Three-Dimensional Graphics and Realism—Color, shading, shadowing, and texture

Applied Geometry for Computer Graphics

Abstract: From the Publisher: Focussing on the manipulation and representation of geometrical objects, this book explores the application of geometry to computer graphics and computer-aided design (CAD).. "Over 260 exercises are included, many of which encourage the reader to implement the techniques and algorithms discussed through use of a computer package with graphing and computer algebra capabilities. Suitable for students of engineering and computer science as well as of mathematics, the book provides a foundation in the extensive applications of geometry in real world situations.