Techniques in Computational Geometry

Topological sweep of the complete graph

Well known estimation techniques in computational geometry usually deal only with single geometric entities as unknown parameters and do not account for constrained observations within the estimation.The estimation model proposed in this paper is much more general, as it can handle multiple homogeneous vectors as well as multiple constraints. Furthermore, it allows the consistent handling of arbitrary covariance matrices for the observed and the estimated entities. The major novelty is the proper handling of singular observation covariance matrices made possible by additional constraints within the estimation. These properties are of special interest for instance in the calculus of algebraic projective geometry, where singular covariance matrices arise naturally from the non-minimal parameterizations of the entities.The validity of the proposed adjustment model will be demonstrated by the estimation of a fundamental matrix from synthetic data and compared to heteroscedastic regression [1], which is considered as state-of-the-art estimator for this task. As the latter is unable to simultaneously estimate multiple entities, we will also demonstrate the usefulness and the feasibility of our approach by the constrained estimation of three vanishing points from observed uncertain image line segments.KeywordsMahalanobis DistanceFundamental MatrixStraight Line SegmentHomogeneous EntityRandom Sample ConsensusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper describes applications of the optimized pattern discovery framework to text and Web mining. In particular, we introduce a class of simple combinatorial patterns over phrases, called proximity phrase association patterns, and consider the problem of finding the patterns that optimize a given statistical measure within the whole class of patterns in a large collection of unstructured texts. For this class of patterns, we develop fast and robust text mining algorithms based on techniques in computational geometry and string matching. Finally, we successfully apply the developed text mining algorithms to the experiments on interactive document browsing in a large text database and keyword discovery from Web bases.

Computational algebraic geometry has experienced significant advancements in recent years, driven by both theoretical breakthroughs and practical applications comprehensive review of the latest techniques in computational algebraic geometry, highlighting their development and impact on various domains. We begin by discussing foundational methods, including Gröbner bases, resultants, and elimination theory, which have been pivotal in solving polynomial systems and understanding algebraic varieties emerging algorithms that leverage improvements in computational efficiency, such as homotropy continuation methods and numerical algebraic geometry techniques. the applications of these advancements across different fields. In robotics and computer vision, for example, algebraic geometry methods are applied to solve problems related to motion planning and object recognition. In cryptography, recent developments in computational algebraic geometry offer new approaches to designing secure cryptographic systems resistant to quantum attacks. We also discuss the integration of algebraic geometry with machine learning and data science, emphasizing its role in improving predictive modeling and pattern recognition.

Maze solving has direct applications like Robot path finding, emergency evacuation, and drug discovery. Traditional methods of maze solving treat the problem of maze as a directional graph by connecting the key nodes. Because of the directional graph treatment, which is applied on the simplified form of the original maze, the maze solving algorithms at present cannot be directly applied in the critical domains of emergency evacuation etc. The proposed method uses the original complex maze without simplifying and hence can be directly applied in any critical application related to path finding. Proposed approach solves the mazes in a fully automated mode by combining the techniques of image processing, computational geometry and computational fluid dynamics. Moreover the high amount of human intervention needed for generating graphs from mazes in the traditional algorithms is eliminated altogether.

Tangible cultural heritage has provided visually appealing material for those researching in computer graphics, computational geometry, and interactive techniques. However, experiments were initially of more benefit to computer scientists as tests for their novel techniques than of any practical benefit to cultural-heritage professionals. Recently, however, tools have matured considerably, and technology is now enabling novel applications for documenting and preserving tangible cultural heritage and is continuing to inspire computer graphics researchers. Nevertheless, there's still a long way to go.

We explain the origin of the Veronese surface in the vacuum moduli space geometry of the MSSM electroweak sector. While this result appeared many years ago using techniques of computational algebraic geometry, it has never been demonstrated analytically. Here, we present an analytical derivation of the vacuum geometry of the electroweak theory by understanding how the F- and D-term relations lead to the Veronese surface. We moreover give a detailed description of this geometry, realising an extra branch as a zero-dimensional point when quadratic Higgs lifting deformations are incorporated into the superpotential.

We initiate the study of how to recommend deployment strategies to the task designers in crowdsourcing platforms. The work proposes the first ever optimization based formalism of the task deployment strategies based on different parameters that the task designers have in mind during deployment and present principled algorithms that are designed using computational geometry techniques.

Constructive neural network training algorithms for pattern classification using computational geometry techniques

This paper describes a new automated system for analyzing the stress intensity factors of three-dimensional cracks. Mesh refinement is well controlled by the fuzzy knowledge processing technique, while nodes and tetrahedral elements are generated quite rapidly by the computational geometry techniques. Material properties and boundary conditions are easily attached onto a generated mesh by clicking parts of geometrical model, such as loops or edges, using a mouse, and by inputting values. For accurate analyses of the stress intensity factors, the singular elements in which mid-point nodes near the crack front are shifted to quarter-points are automatically placed along the three-dimensional crack front. The complete finite element model generated is given to one of the commercial finite-element codes, MARC, and a stress anaysis is performed. Stress intensity factors are also automatically calculated, using the displacement extrapolation method. To demonstrate the practical performance of the present system, it is applied to the analysis of surface cracks in a plate subjected to uniform tension.

Three-dimensional particle shape characterizations from half particle geometries

Osteoporosis or 'porous bones' is the major cause of bone fracture in elderly post-menopausal women.As such, predicting fracture occurrence can significantly reduce the associated morbidity and reduce health-care expenditures.Examining local micro-structure of trabecular bone is becoming increasingly important for investigating bone fracture behavior.Conventional voxel based measures of bone architecture cannot provide morphology information regarding the fractures occurring at individual trabeculae.In this paper, the Delaunay triangulation, a computational geometry technique is used for assessing the effect of trabeculae structure on local micro-fracture, from sequential 2D images of the cross section of a hamster femur surface undergone a multistage loading until fracture occurrence.The most significant structural parameters to characterizethe strength of cancellous bone and the change in morphology of individual trabeculae during the fracture process are investigated.The results from different regions were compared to describe the possible architectural factors affecting the local micro-damages.

The problem of computing an optimal beam among weighted regions (called the optimal beam problem) arises in several applied areas such as radiation therapy, stereotactic brain surgery, medical surgery, geological exploration, manufacturing, and environmental engineering. In this paper, we present computational geometry techniques that enable us to develop efficient algorithms for solving various optimal beam problems among weighted regions in two and three dimensional spaces. In particular, we consider two types of problems: the covering problems (seeking an optimal beam to contain a specified target region), and the piercing problems (seeking an optimal beam of a fixed shape to pierce the target region). We investigate several versions of these problems, with a variety of beam shapes and target region shapes in 2-D and 3-D. Our algorithms are based on interesting combinations of computational geometry techniques and optimization methods, and transform the optimal beam problems to solving a collection of instances of certain special non-linear optimization problems. Our approach makes use of interesting geometric observations, such as utilizing some new features of Minkowski sums.

The aim of this research is to achieve spatial consistency of the UV map. We present an approach to produce a fully spatially consistent UV mapping based on the planar parameterisation of the mesh. We apply our method on a 3D digital replica of an ancient Greek Lekythos vessel. We parameterise the mesh of a 3D model onto a unit square 2D plane using computational conformal geometry techniques. The proposed method is genus independent, due to an iterative 3D mesh cutting procedure. Having now the texture of a 3D model depicted on a spatially continuous two dimensional structure enables us to efficiently apply a vast range of image processing based techniques and algorithms.

Investigation of three-dimensional (3-D) geometry and fluid-dynamics in human arteries is an important issue in vascular disease characterization and assessment. Thanks to recent advances in magnetic resonance (MR) and computed tomography (CT), it is now possible to address the problem of patient-specific modeling of blood vessels, in order to take into account interindividual anatomic variability of vasculature. Generation of models suitable for computational fluid dynamics is still commonly performed by semiautomatic procedures, in general based on operator-dependent tasks, which cannot be easily extended to a significant number of clinical cases. In this paper, we overcome these limitations making use of computational geometry techniques. In particular, 3-D modeling was carried out by means of 3-D level sets approach. Model editing was also implemented ensuring harmonic mean curvature vectors distribution on the surface, and model geometric analysis was performed with a novel approach, based on solving Eikonal equation on Voronoi diagram. This approach provides calculation of central paths, maximum inscribed sphere estimation and geometric characterization of the surface. Generation of adaptive-thickness boundary layer finite elements is finally presented. The use of the techniques presented here makes it possible to introduce patient-specific modeling of blood vessels at clinical level.