Showing papers on "Computational geometry published in 2002"

Coverage control for mobile sensing networks

Abstract: This paper describes decentralized control laws for the coordination of multiple vehicles performing spatially distributed tasks. The control laws are based on a gradient descent scheme applied to a class of decentralized utility functions that encode optimal coverage and sensing policies. These utility functions are studied in geographical optimization problems and they arise naturally in vector quantization and in sensor allocation tasks. The approach exploits the computational geometry of spatial structures such as Voronoi diagrams.

Lectures on Discrete Geometry

Abstract: From the Publisher: Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization. This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces. Jiri Matousek is Professor of Computer Science at Charles University in Prague. His research has contributed to several of the considered areas and to their algorithmic applications. This is his third book.

Metamorphosis of Polyhedral Surfaces using Decomposition

Abstract: This paper describes an algorithm for morphing polyhedral surfaces based on their decompositions into patches. The given surfaces need neither be genus-zero nor two-manifolds. We present a new algorithm for decomposing surfaces into patches. We also present a new projection scheme that handles topologically cylinder-like polyhedral surfaces. We show how these two new techniques can be used within a general framework and result with morph sequences that maintain the distinctive features of the input models. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computational Geometry and Object Modeling]: Boundary representations I.3.7 [Three-Dimensional Graphics and Realism]: Animation

Bounded-distortion piecewise mesh parameterization

Abstract: Many computer graphics operations, such as texture mapping, 3D painting, remeshing, mesh compression, and digital geometry processing, require finding a low-distortion parameterization for irregular connectivity triangulations of arbitrary genus 2-manifolds. This paper presents a simple and fast method for computing parameterizations with strictly bounded distortion. The new method operates by flattening the mesh onto a region of the 2D plane. To comply with the distortion bound, the mesh is automatically cut and partitioned on-the-fly. The method guarantees avoiding global and local self-intersections, while attempting to minimize the total length of the introduced seams. To our knowledge, this is the first method to compute the mesh partitioning and the parameterization simultaneously and entirely automatically, while providing guaranteed distortion bounds. Our results on a variety of objects demonstrate that the method is fast enough to work with large complex irregular meshes in interactive applications.

Triangulations in CGAL

Abstract: This paper presents the main algorithmic and design choices that have been made to implement triangulations in the computational geometry algorithms library Cgal.

The many facets of linear programming

Abstract: We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interior-point, and other methods.

A Reflective Symmetry Descriptor

Abstract: Computing reflective symmetries of 2D and 3D shapes is a classical problem in computer vision and computational geometry. Most prior work has focused on finding the main axes of symmetry, or determining that none exists. In this paper, we introduce a new reflective symmetry descriptor that represents a measure of reflective symmetry for an arbitrary 3D voxel model for all planes through the model's center of mass (even if they are not planes of symmetry). The main benefits of this new shape descriptor are that it is defined over a canonical parameterization (the sphere) and describes global properties of a 3D shape. Using Fourier methods, our algorithm computes the symmetry descriptor in O(N4 logN) time for an N × N × N voxel grid, and computes a multiresolution approximation in O(N3 logN) time. In our initial experiments, we have found the symmetry descriptor to be useful for registration, matching, and classification of shapes.

Realistic input models for geometric algorithms

Abstract: The traditional worst-case analysis often fails to predict the actual behavior of the running time of geometric algorithms in practical situations. One reason is that worst-case scenarios are often very contrived and do not occur in practice. To avoid this, models are needed that describe the properties that realistic inputs have, so that the analysis can take these properties into account. We try to bring some structure to this emerging research direction. In particular, we present the following results: • We show the relations between various models that have been proposed in the literature. • For several of these models, we give algorithms to compute the model parameter(s) for a given (planar) scene; these algorithms can be used to verify whether a model is appropriate for typical scenes in some application area. • As a case study, we give some experimental results on the appropriateness of some of the models for one particular type of scene often encountered in geographic information systems, namely certain triangulated irregular networks.

Reconstruction of three-dimensional objects through matching of their parts

Abstract: The problem of re-assembling an object from its parts or fragments has never been addressed with a unified computational approach, which depends on the pure geometric form of the parts and not on application-specific features. We propose a method for the automatic reconstruction of a model based on the geometry of its parts, which may be computer-generated models or range-scanned models. The matching process can benefit from any other external constraint imposed by the specific application.

A fast and efficient projection-based approach for surface reconstruction

Abstract: We present a fast and memory efficient algorithm that generates a manifold triangular mesh S with or without boundary passing through a set of unorganized points P/spl sub//spl Rscr//sup 3/ with no other additional information. Nothing is assumed about the geometry or topology of the sampled manifold model, except for its reasonable smoothness. The speed of our algorithm is derived from a projection-based approach we use to determine the incident faces on a point. Our algorithm has successfully reconstructed the surfaces of unorganized point clouds of sizes varying from 10,000 to 100,000 in about 3-30 seconds on a 250 MHz, R10000 SGI Onyx2. Our technique can be specialized for different kinds of input and applications. For example, our algorithm can be specialized to handle data from height fields like terrain and range scan, even in the presence of noise. We have successfully generated meshes for range scan data of size 900,000 points in less than 40 seconds.

Coverage control for mobile sensing networks: variations on a theme

Abstract: This paper presents control and coordination algorithms for networks of autonomous vehicles. We focus on groups of vehicles performing distributed sensing tasks where each vehicle plays the role of a mobile tunable sensor. We design distributed gradient descent algorithms for a class of utility functions which encodes optimal coverage and sensing policies. These utility functions are studied in geographical optimization, vector quantization, and sensor allocation contexts. The algorithms exploit the computational geometry of spatial structures such as Voronoi diagrams.

Geometric Algebra: A computational framework for geometrical applications Part 1

Abstract: Every vector space with an inner product has a geometric algebra, whether or not you choose to use it. This article shows how to call on this structure to define common geometrical constructs, ensuring a consistent computational framework. The goal is to show you that this can be done and that it is compact, directly computational, and transcends the dimensionality of subspaces. We do not use geometric algebra to develop new algorithms for graphics, but hope to demonstrate that one can automatically take care of some of the lower level algorithmic aspects, without tricks, exceptions, or hidden degenerate cases by using geometric algebra as a language.

Efficient computation of the topology of level sets

Abstract: This paper introduces two efficient algorithms that compute the Contour Tree of a 3D scalar field F and its augmented version with the Betti numbers of each isosurface. The Contour Tree is a fundamental data structure in scientific visualization that is used to preprocess the domain mesh to allow optimal computation of isosurfaces with minimal overhead storage. The Contour Tree can also be used to build user interfaces reporting the complete topological characterization of a scalar field, as shown in Figure 1.The first part of the paper presents a new scheme that augments the Contour Tree with the Betti numbers of each isocontour in linear time. We show how to extend the scheme introduced in [3] with the Betti number computation without increasing its complexity. Thus, we improve on the time complexity from our previous approach [10] from O(m log m) to O(n log n + m), where m is the number of tetrahedra and n is the number of vertices in the domain of F.The second part of the paper introduces a new divide-and-conquer algorithm that computes the Augmented Contour Tree with improved efficiency. The central part of the scheme computes the output Contour Tree by merging two intermediate Contour Trees and is independent of the interpolant. In this way we confine any knowledge regarding a specific interpolant to an oracle that computes the tree for a single cell. We have implemented this oracle for the trilinear interpolant and plan to replace it with higher order interpolants when needed. The complexity of the scheme is O(n + t log n), where t is the number of critical points of F. For the first time we can compute the Contour Tree in linear time in many practical cases when t = O(n1 - e).Lastly, we report the running times for a parallel implementation of our algorithm, showing good scalability with the number of processors.

A multi-resolution scheme ICP algorithm for fast shape registration

Abstract: The iterative closest point (ICP) algorithm is widely used for the registration of geometric data. One of its main drawbacks is its quadratic time complexity O(N/sup 2/) with the shape number of points N, which implies long processing time, especially when using high resolution data. This paper proposes to accelerate the process by a coarse to fine multiresolution approach in which a solution at a coarse level is successively improved at a finer level of representation. Specifically, it investigates this multiresolution ICP approach when coupled with the tree search or the neighbor search closest point algorithms. A theoretical and practical analysis and a comparison of the considered algorithms are presented. Confirming the success of the multiresolution scheme, the results also show that this combination permits us to create a very fast ICP algorithm, gaining speed up to a factor of 27 over a standard fast ICP algorithm.

Image-based photo hulls

Abstract: We present an efficient image-based rendering algorithm that computes photo hulls of a scene photographed from multiple viewpoints. Our algorithm, called image-based photo hulls (IBPH), like the image-based visual hulls (IBVH) algorithm from Matusik et. al. (2000) on which it is based, takes advantage of epipolar geometry to efficiently reconstruct the geometry and visibility of a scene. Our IBPH algorithm differs from IBVH in that it utilizes the color information of the images to identify the scene geometry. These additional color constraints often result in a more accurately reconstructed geometry, which projects to better synthesized virtual views of the scene. We demonstrate our algorithm running in a real-time 3D telepresence application using video data acquired from four viewpoints.

Robust Geometric Computing in Motion

Abstract: Transforming a geometric algorithm into an effective computer program is a difficult task. This transformation is particularly made hard by the basic assumptions of most theoretical geometric algor...

Identifying faces in a 2D line drawing representing a manifold object

Abstract: A straightforward way to illustrate a 3D model is to use a line drawing. Faces in a 2D line drawing provide important information for reconstructing its 3D geometry. Manifold objects belong to a class of common solids and most solid systems are based on manifold geometry. In this paper, a new method is proposed for finding faces from single 2D line drawings representing manifolds. The face identification is formulated based on a property of manifolds: each edge of a manifold is shared exactly by two faces. The two main steps in our method are (1) searching for cycles from a line drawing and (2) searching for faces from the cycles. In order to speed up the face identification procedure, a number of properties, most of which relate to planar manifold geometry in line drawings, are presented to identify most of the cycles that are or are not real faces in a drawing, thus reducing the number of unknown cycles in the second searching. Schemes to deal with manifolds with curved faces and manifolds each represented by two or more disjoint graphs are also proposed. The experimental results show that our method can handle manifolds previous methods can handle, as well as those they cannot.

Geometric algorithms for the analysis of 2D-electrophoresis gels.

Abstract: In proteomics, two-dimensional gel electrophoresis (2-DE) is a separation technique for proteins. The resulting protein spots can be identified either by using picking robots and subsequent mass spectrometry or by visual cross inspection of a new gel image with an already analyzed master gel. Difficulties especially arise from inherent noise and irregular geometric distortions in 2-DE images. Aiming at the automated analysis of large series of 2-DE images, or at the even more difficult interlaboratory gel comparisons, the bottleneck is to solve the two most basic algorithmic problems with high quality: Identifying protein spots and computing a matching between two images. For the development of the analysis software CAROl at Freie Universitat Berlin, we have reconsidered these two problems and obtained new solutions which rely on methods from computational geometry. Their novelties are: 1. Spot detection is also possible for complex regions formed by several "merged" (usually saturated) spots; 2. User-defined landmarks are not necessary for the matching. Furthermore, images for comparison are allowed to represent different parts of the entire protein pattern, which only partially "overlap." The implementation is done in a client server architecture to allow queries via the internet. We also discuss and point at related theoretical questions in computational geometry.

Discrete differential error metric for surface simplification

Abstract: In this paper we propose a new discrete differential error metric for surface simplification. Many surface simplification algorithms have been developed in order to produce rapidly high quality approximations of polygonal models, and the quadric error metric based on the distance error is the most popular and successful error metric so far Even though such distance based error metrics give visually pleasing results with a reasonably fast speed, it is hard to measure an accurate geometric error on a highly curved and thin region since the error measured by the distance metric on such a region is usually small and causes a loss of visually important features. To overcome such a drawback, we define a new error metric based on the theory of local differential geometry in such a way that the first and the second order discrete differentials approximated locally on a discrete polygonal surface are integrated into the usual distance error metric. The benefits of our error metric are preservation of sharp feature regions after a drastic simplification, small geometric errors, and fast computation comparable to the existing methods.

Computational surface flattening: a voxel-based approach

Abstract: A voxel-based method for flattening a surface in 3D space into 2D while best preserving distances is presented. Triangulation or polyhedral approximation of the voxel data are not required. The problem is divided into two main parts: Voxel-based calculation of the minimal geodesic distances between points on the surface and finding a configuration of points in 2D that has Euclidean distances as close as possible to these distances. The method suggested combines an efficient voxel-based hybrid distance estimation method, that takes the continuity of the underlying surface into account, with classical multidimensional scaling (MDS) for finding the 2D point configuration. The proposed algorithm is efficient, simple, and can be applied to surfaces that are not functions. Experimental results are shown.

Optimized compression of triangle mesh geometry using prediction trees

Abstract: Almost all triangle mesh compression algorithms to date are driven by the mesh connectivity code. The geometry code usually employs a straightforward prediction method applied to the vertex sequence as dictated by the connectivity code. This generates a suboptimal geometry code, which results in significant loss in code efficiency, since the geometry dominates the mesh information content. The paper proposes a manifold mesh code which optimizes the geometric component, at the slight expense of the connectivity code. This mesh geometry code is shown to be up to 50% more compact than the state-of-the-art geometry code of Touma and Gotsman (1998), especially for models with non-smooth geometry, such as CAD models.

Evaluation of sphericity error from form data using computational geometric techniques

Abstract: The measurement data for evaluation of sphericity error can be obtained from inspection devices such as form measuring instruments/set-ups Due to misalignment and size-suppression inherent in these measurements, sphericity data obtained will be distorted Hence, the sphericity error is evaluated with reference to an assessment feature, referred to as a limacoid Appropriate methods based on the computational geometry have been developed to establish Minimum Circumscribed, Maximum Inscribed and Minimum Zone Limacoids The present methods start with the construction of 3-D hulls A 3-D convex outer hull is established using computational geometric concepts presently available A heuristic method is followed in this paper to establish a 3-D inner hull Based on a new concept of 3-D equi-angular line, 3-D farthest or nearest equi-angular diagrams are constructed for establishing the assessment limacoids Algorithms proposed in the present work are implemented and validated with the simulated data and the data available in the literature

Range Searching in Categorical Data: Colored Range Searching on Grid

Abstract: Range searching, a fundamental problem in numerous applications areas, has been widely studied in computational geometry and spatial databases. Given a set of geometric objects, a typical range query asks for reporting all the objects that intersect a query object. However in many applications, including databases and network routing, input objects are partitioned into categories and a query asks for reporting the set of categories of objects that intersect a query object. Moreover in many such applications, objects lie on a grid. We abstract the category of an object by associating a color with each object. In this paper, we present efficient data structures for solving the colored range-searching and colored point-enclosure problem on U × U grid. Our data structures use near- linear space and answer a query in O(log log U + k) time, where k is the output size. As far as we know, this is the first result on colored range-searching for objects lying on a grid.

High-Level Filtering for Arrangements of Conic Arcs

Abstract: Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for implementing robust geometric algorithms is to use exact algebraic number types -- yet this may lead to a very slow, inefficient program. In this paper we suggest a simple technique for filtering the computations involved in the arrangement construction: when constructing an arrangement vertex, we keep track of the steps that lead to its construction and the equations we need to solve to obtain its coordinates. This construction history can be used for answering predicates very efficiently, compared to a naive implementation with an exact number type. Furthermore, using this representation most arrangement vertices may be computed approximately at first and can be refined later on in cases of ambiguity. Since such cases are relatively rare, the resulting implementation is both efficient and robust.

Interface Tracking for Axisymmetric Flows

Abstract: A front tracking method for inviscid gas dynamics is presented. The key constructions and algorithms used in the code are described and the interrelations between shock capturing, interface dynamics, computational geometry, grid construction, and parallelism are discussed for the code as a whole. Validation is carried out by comparing the single mode bubble velocity for Rayleigh--Taylor instability with theoretical models and experimental results. The calculations are validated by mesh refinement studies and by the comparison of the asymptotic limit of the minimum radius $r_{\rm min} \rightarrow \infty$ to a pure planar computation in two dimensions.

Ellipsoid decomposition of 3D-models

Abstract: In this paper we present a simple technique to approximate the volume enclosed by a given triangle mesh with a set of overlapping ellipsoids. This type of geometry representation allows us to approximately reconstruct 3D-shapes from a very small amount of information being transmitted. The two central questions that we address are: how can we compute optimal fitting ellipsoids that lie in the interior of a given triangle mesh and how do we select the most significant (least redundant) subset from a huge number of candidate ellipsoids. Our major motivation for computing ellipsoid decompositions is the robust transmission of geometric objects where the receiver can reconstruct the 3D-shape even if part of the data gets lost during transmission.