Showing papers on "Computational geometry published in 1995"

Almost optimal set covers in finite VC-dimension

Abstract: We give a deterministic polynomial-time method for finding a set cover in a set system (X, ?) of dual VC-dimensiond such that the size of our cover is at most a factor ofO(d log(dc)) from the optimal size,c. For constant VC-dimensional set systems, which are common in computational geometry, our method gives anO(logc) approximation factor. This improves the previous ?(log?X?) bound of the greedy method and challenges recent complexity-theoretic lower bounds for set covers (which do not make any assumptions about the VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those arising in three-dimensional polytope approximation and two-dimensional disk covering, we can quickly findO(c)-sized covers.

Curve and surface smoothing without shrinkage

Abstract: For a number of computational purposes, including visualization of scientific data and registration of multimodal medical data, smooth curves must be approximated by polygonal curves, and surfaces by polyhedral surfaces. An inherent problem of these approximation algorithms is that the resulting curves and surfaces appear faceted. Boundary-following and iso-surface construction algorithms are typical examples. To reduce the apparent faceting, smoothing methods are used. In this paper, we introduce a new method for smoothing piecewise linear shapes of arbitrary dimension and topology. This new method is in fact a linear low-pass filter that removes high-curvature variations, and does not produce shrinkage. Its computational complexity is linear in the number of edges or faces of the shape, and the required storage is linear in the number of vertices. >

Collision detection for interactive graphics applications

Abstract: Collision detection and response are important for interactive graphics applications such as vehicle simulators and virtual reality. Unfortunately, previous collision detection algorithms are too slow for interactive use. The paper presents a new algorithm for rigid or articulated objects that meets performance goals through a form of time critical computing. The algorithm supports progressive refinement, detecting collisions between successively tighter approximations to object surfaces as the application allows it more processing time. The algorithm uses simple four dimensional geometry to approximate motion, and hierarchies of spheres to approximate three dimensional surfaces at multiple resolutions. In a sample application, the algorithm allows interactive performance that is not possible with a good previous algorithm. In particular, the new algorithm provides acceptable accuracy while maintaining a steady and high frame rate, which in some cases improves on the previous algorithm's rate by more than two orders of magnitude. >

Geometric constraint solver

Abstract: The paper reports on the development of a 2D geometric constraint solver. The solver is a major component of a new generation of cad systems based on a high-level geometry representation. The solver uses a graph-reduction directed algebraic approach, and achieves interactive speed. The paper describes the architecture of the solver and its basic capabilities. Then, it discusses in detail how to extend the scope of the solver, with particular emphasis on the theoretical and human factors involved in finding a solution, in an exponentially large search space, so that the solution is appropriate to the application, and so that the way of finding it is intuitive for an untrained user.

A general surface approach to the integration of a set of range views

Abstract: This paper presents a new and general solution to the problem of range view integration. The integration problem consists in computing a connected surface model from a set of registered range images acquired from different viewpoints. The proposed method does not impose constraints on the topology of the observed surfaces, the position of the viewpoints, or the number of views that can be merged. The integrated surface model is piecewise estimated by a set of triangulations modeling each canonical subset of the Venn diagram of the set of range views. The connection of these local models by constrained Delaunay triangulations yields g non-redundant surface triangulation describing all surface elements sampled by the set of range views. Experimental results show that the integration technique can be used to build connected surface models of free-form objects. No integrated models built from objects of such complexity have yet been reported in the literature, It is assumed that accurate range views are available and that frame transformations between all pairs of views can be reliably computed. >

Computational Geometry in C.

Abstract: From the Publisher: This is the newly revised and expanded edition of a popular introduction to the design and implementation of geometry algorithms arising in areas such as computer graphics, robotics, and engineering design. The basic techniques used in computational geometry are all covered: polygon triangualtions, convex hulls, Voronoi diagrams, arrangements, geometric searching, and motion planning. The self-contained treatment presumes only an elementary knowledge of mathematics, but it reaches topics on the frontier of current research. Thus professional programmers will find it a useful tutorial.

Computing in Euclidean Geometry

Abstract: Mesh generation and optimal triangulation, M. Bern and D. Eppstein machine proofs of geometry theorems, S.C. Chou and M. Rethi randomized geometric algorithms, K. Clarkson Voronoi diagrams and Delanney triangulations, S. Fortune the state of art on Steiner ratio problems, D-Z. Du and F. Hwang on the development of quantitative geometry from Pythagoras to Grassmann, W-Y. Hsiang computational geometry and topological network designs, J. Smith and P. Winter polar forms and triangular B-spline surfaces, H-P. Seidel algebraic foundations of computational geometry, Chee Yap.

Exploiting triangulated surface extraction using tetrahedral decomposition

Abstract: Beginning with digitized volumetric data, we wish to rapidly and efficiently extract and represent surfaces defined as isosurfaces in the interpolated data. The Marching Cubes algorithm is a standard approach to this problem. We instead perform a decomposition of each 8-cell associated with a voxel into five tetrahedra. We guarantee the resulting surface representation to be closed and oriented, defined by a valid triangulation of the surface of the body, which in turn is presented as a collection of tetrahedra. The entire surface is "wrapped" by a collection of triangles, which form a graph structure, and where each triangle is contained within a single tetrahedron. The representation is similar to the homology theory that uses simplices embedded in a manifold to define a closed curve within each tetrahedron. We introduce data structures based upon a new encoding of the tetrahedra that are at least four times more compact than the standard data structures using vertices and triangles. For parallel computing and improved cache performance, the vertex information is stored local to the tetrahedra. We can distribute the vertices in such a way that no tetrahedron ever contains more than one vertex, We give methods to evaluate surface curvatures and principal directions at each vertex, whenever these quantities are defined. Finally, we outline a method for simplifying the surface, that is reducing the vertex count while preserving the geometry. We compare the characteristics of our methods with an 8-cell based method, and show results of surface extractions from CT-scans and MR-scans at full resolution.

A theory of specular surface geometry

Abstract: A theoretical framework is introduced for the perception of specular surface geometry. When an observer moves in three-dimensional space, real scene features, such as surface markings, remain stationary with respect to the surfaces they belong to. In contrast, a virtual feature, which is the specular reflection of a real feature, travels on the surface. Based on the notion of caustics, a novel feature classification algorithm is developed that distinguishes real and virtual features from their image trajectories that result from observer motion. Next, using support functions of curves, a closed-form relation is derived between the image trajectory of a virtual feature and the geometry of the specular surface it travels on. It is shown that in the 2D case where camera motion and the surface profile are coplanar, the profile is uniquely recovered by tracking just two unknown virtual features. Finally, these results are generalized to the case of arbitrary 3D surface profiles that are travelled by virtual features when camera motion is not confined to a plane. An algorithm is developed that uniquely recovers 3D surface profiles using a single virtual feature tracked from the occluding boundary of the object. All theoretical derivations and proposed algorithms are substantiated by experiments. >

Closest-Point Problems in Computational Geometry

Abstract: A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in ℝ D . In particular, the closest pair problem, the exact and approximate post-office problem, and the problem of constructing spanners are discussed in detail.

The buffer tree: A new technique for optimal I/O-algorithms

Abstract: In this paper we develop a technique for transforming an internal memory tree data structure into an external storage structure We show how the technique can be used to develop a search-tree-like structure, a priority-queue, a (one-dimensional) range-tree and a segment-tree, and give examples of how these structures can be used to develop efficient I/O-algorithms All our algorithms are either extremely simple or straightforward generalizations of known internal memory algorithms — given the developed external data structures

A spherical representation for recognition of free-form surfaces

Abstract: Introduces a new surface representation for recognizing curved objects. The authors approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object. The mesh is computed from the data by deforming a standard shaped mesh, for example, an ellipsoid, until it fits the surface of the object. The authors define local regularity constraints that the mesh must satisfy. The authors then define a canonical mapping between the mesh describing the object and a standard spherical mesh. A surface curvature index that is pose-invariant is stored at every node of the mesh. The authors use this object representation for recognition by comparing the spherical model of a reference object with the model extracted from a new observed scene. The authors show how the similarity between reference model and observed data can be evaluated and they show how the pose of the reference object in the observed scene can be easily computed using this representation. The authors present results on real range images which show that this approach to modelling and recognizing 3D objects has three main advantages: (1) it is applicable to complex curved surfaces that cannot be handled by conventional techniques; (2) it reduces the recognition problem to the computation of similarity between spherical distributions; in particular, the recognition algorithm does not require any combinatorial search; and (3) even though it is based on a spherical mapping, the approach can handle occlusions and partial views. >

Automatic isosurface propagation using an extrema graph and sorted boundary cell lists

Abstract: A high-performance algorithm for generating isosurfaces is presented. In our method, guides to searching for cells intersected by an isosurface are generated as a pre-process. These guides are two kinds of cell lists: an extrema graph, and sorted lists of boundary cells. In an extrema graph, extremum points are connected by arcs, and each arc has a list of cells through which it passes. At the same time, all boundary cells are sorted according to their minimum and maximum values, and two sorted lists are then generated. Isosurfaces are generated by visiting adjacent intersected cells in order. Here, the starting cells for this process are found by searching in an extrema graph and in sorted boundary cell lists. In this process, isosurfaces appear to propagate themselves. Our algorithm is efficient, since it visits only cells that are intersected by an isosurface and cells whose IDs are included in the guides. It is especially efficient when many isosurfaces are interactively generated in a huge volume. Some benchmark tests described in this paper show the efficiency of the algorithm.

Approximate range searching

Abstract: The range searching problem is a fundamental problem in computational geometry, with numerous important applications. Most research has focused on solving this problem exactly, but lower bounds show that if linear space is assumed, the problem cannot be solved in polylogarithmic time, except for the case of orthogonal ranges. In this paper we show that if one is willing to allow approximate ranges, then it is possible to do much better. In particular, given a bounded range Q of diameter w and "> 0, an approximate range query treats the range as a fuzzy object, meaning that points lying within distance "w of the boundary of Q either may or may not be counted. We show that in any fixed dimension d ,as et ofn points in R d can be preprocessed in O.nC logn/ time and O.n/ space, such that approximate queries can be answered in O.logn.1="/ d / time. The only assumption we make about ranges is that the intersection of a range and a d-dimensional cube can be answered in constant time (depending on dimension). For convex ranges, we tighten this to O.lognC.1="/ d 1 / time. We also present a lower bound for approximate range searching based on partition trees of .lognC.1="/ d 1 /, which implies optimality for convex ranges (assuming fixed dimensions). Finally, we give empirical evidence showing that allowing small relative errors can significantly improve query execution times. © 2000 Elsevier Science B.V. All rights reserved.

Geometric mesh partitioning: implementation and experiments

Abstract: We investigate a method of dividing an irregular mesh into equal-sized pieces with few interconnecting edges. The method's novel feature is that it exploits the geometric coordinates of the mesh vertices. It is based on theoretical work of Miller, Teng, Thurston, and Vavasis, who showed that certain classes of "well-shaped" finite element meshes have good separators. The geometric method is quite simple to implement: we describe a Matlab code for it in some detail. The method is also quite efficient and effective: we compare it with some other methods, including spectral bisection. >

Creating solid models from single 2D sketches

Abstract: We describe a method of constructing a B-rep solid model from a single hidden-line removed sketch view of a 3D object. The main steps of our approach are as follows. The sketch is first tidied in 2D (to remove digitisation errors). Line labelling is used to deduce the initial topology of the object and to locate hidden faces. Constraints are then produced from the line labelling and features in the drawing (such as probable symmetry) involving the unknown face coefficients and point depths. A least squares solution is found to the linear system and any grossly incompatible equations are rejected. Vertices are recalculated as the intersections of the faces to ensure we have a reconstructible solid. Any incomplete faces are then completed as far as possible from neighbouring faces, producing a solid model from the initial sketch, if successful. The current software works for polyhedral objects with trihedral vertices. CR Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modelling Geometric algorithms, languages and systems; I.2.10 [Artificial Intelligence]: Vision and Scene Understanding Perceptual reasoning; I.3.6 [Computer Graphics]: Methodology and Techniques Interaction techniques; J.6 [Computer Applications]: CAE CAD.

Terrain objects, their dynamics and their monitoring by the integration of GIS and remote sensing

Abstract: Geometrical and thematic data about terrain objects stored in a geographical information system (GIS) can be kept up-to-date by using remote sensing (RS) data. Geometrical and thematic data can be extracted from the RS data by segmentation and classification techniques respectively. The possibilities and reliability of the information extraction from RS data can be improved by the use of ancillary data and knowledge about the terrain objects. Object classification and aggregation hierarchies can be used to describe relationships between terrain objects; the categorization of the different types of terrain object dynamics that is presented will be based partly on these hierarchical relationships. Object models will be applied in a case study in which both the field geometry (field boundaries) and the crop type of agricultural fields are updated from a Landsat TM image. For that purpose, a three-stage strategy has been developed. In the first stage, the results of an edge detection procedure are integrated with fixed geometrical data already contained in the GIS by using knowledge about the aggregation structure and shape of the fields. In the next stage, the crop type of the fields is determined by means of object-based classification. Finally, conditional merging is performed to solve the problem of oversegmentation. The resulting field geometry was found to agree for 87% with the field geometry as determined by a photo-interpreter. >

A simple and efficient method for accurate collision detection among deformable polyhedral objects in arbitrary motion

Abstract: We propose an accurate collision detection algorithm for use in virtual reality applications. The algorithm works for three-dimensional graphical environments where multiple objects, represented as polyhedra (boundary representation), are undergoing arbitrary motion (translation and rotation). The algorithm can be used directly for both convex and concave objects and objects can be deformed (non-rigid) during motion. The algorithm works efficiently by first reducing the number of face pairs that need to be checked accurately for interference by first localizing possible collision regions using bounding box and spatial subdivision techniques; face pairs that remain after this pruning stage are then accurately checked for interference. The algorithm is efficient, simple to implement, and does not require any memory intensive auxiliary data structures to be precomputed and updated. Since polyhedral shape representation is one of the most common shape representation schemes, this algorithm should be useful to a wide audience. Performance results are given to show the efficiency of the proposed method.

Voronoi diagrams for planar shapes

Abstract: Although many algorithms compute Voronoi diagrams for polygons, few do so for shapes bounded by arbitrary closed curves. The paper presents an algorithm which does this. It also traces the diagrams directly from their differential properties. >

Unstructured mesh generation and adaptivity

Abstract: An overview of current unstructured mesh generation and adaptivity techniques is given. Basic building blocks taken from the field of computational geometry are first described. Various practical mesh generation techniques based on these algorithms are then constructed and illustrated with examples. Issues of adaptive meshing and stretched mesh generation for anisotropic problems are treated in subsequent sections. The presentation is organized in an educational manner, for readers familiar with computational fluid dynamics, wishing to learn more about current unstructured mesh techniques.

Evolutions of planar polygons

Abstract: Evolutions of closed planar polygons are studied in this work. In the first part of the paper, the general theory of linear polygon evolutions is presented, and two specific problems are analyzed. The first one is a polygonal analog of a novel affine-invariant differential curve evolution, for which the convergence of planar curves to ellipses was proved. In the polygon case, convergence to polygonal approximation of ellipses, polygo nal ellipses, is proven. The second one is related to cyclic pursuit problems, and convergence, either to polygonal ellipses or to polygonal circles, is proven. In the second part, two possible polygonal analogues of the well-known Euclidean curve shortening flow are presented. The models follow from geometric considerations. Experimental results show that an arbitrary initial polygon converges to either regular or irregular polygonal approximations of circles when evolving according to the proposed Euclidean flows.

Calibration-free visual control using projective invariance

Abstract: Much of the previous work on hand-eye coordination has emphasized the reconstructive aspects of vision. Recently, techniques that avoid explicit reconstruction by placing visual feedback into a control loop have been developed. When properly defined, these methods lead to calibration insensitive hand-eye coordination. Recent work on projective geometry as applied to vision is used to extend this paradigm in two ways. First, it is shown how results from projective geometry can be used to perform online calibration. Second, results on projective invariance are used to define setpoints for visual control that are independent of viewing location. These ideas are illustrated through a number of examples and have been tested on an implemented system. >

A unified algorithm for finding maximum and minimum object enclosing rectangles and cuboids

Abstract: Given a set of n points in R 2 bounded within a rectangular floor F, and a rectangular plate P of specified size, we consider the following two problems: find an isothetic position of P such that it encloses (i) maximum and (ii) minimum number of points, keeping P totally contained within F. For both of these problems, a new algorithm based on interval tree data structure is presented, which runs in O(nlogn) time and consumes O(n) space. If polygonal objects of arbitrary size and shape are distributed in R 2, the proposed algorithm can be tailored for locating the position of the plate to enclose maximum or minimum number of objects with the same time and space complexity. Finally, the algorithm is extended for identifying a cuboid, i.e., a rectangular parallelepiped that encloses maximum number of polyhedral objects in R 3. Thus, the proposed technique serves as a unified paradigm for solving a general class of enclosure problems encountered in computational geometry and pattern recognition.

Visualization of geometric algorithms

Abstract: Investigates the visualization of geometric algorithms We discuss how limiting the domain makes it possible to create a system that enables others to use it easily Knowledge about the domain can be very helpful in building a system which automates large parts of the user's task A system can be designed to isolate the user from any concern about how graphics is done The application need only specify "what" happens and need not be concerned with "how" to make it happen on the screen We develop a conceptual model and a framework for experimenting with it We also present a system, GASP (Geometric Animation System, Princeton), which implements this model GASP allows quick generation of 3D geometric algorithm visualizations, even for highly complex algorithms It also provides a visual debugging facility for geometric computing We show the utility of GASP by presenting a variety of examples >