Computational Geometry Algorithms Library Research Articles

3D Geometry-Based Indoor Network Extraction for Navigation Applications Using SFCGAL

This study is focused on indoor navigation network extraction for navigation applications based on available 3D building data and using SFCGAL library, e.g. simple features computational geometry algorithms library. In this study, special attention is given to 3D cadastre and BIM (building information modelling) datasets, which have been used as data sources for 3D geometric indoor modelling. SFCGAL 3D functions are used for the extraction of an indoor network, which has been modelled in the form of indoor connectivity graphs based on 3D geometries of indoor features. The extraction is performed by the integration of extract transform load (ETL) software and the spatial database to support multiple data sources and provide access to SFCGAL functions. With this integrated approach, the current lack of straightforward software support for complex 3D spatial analyses is addressed. Based on the developed methodology, we perform and discuss the extraction of an indoor navigation network from 3D cadastral and BIM data. The efficiency and performance of the network analyses were evaluated using the processing and query execution times. The results show that the proposed methodology for geometry-based navigation network extraction of buildings is efficient and can be used with various types of 3D geometric indoor data.

Method for generating high-quality tetrahedral meshes of geological models by utilizing CGAL

High-quality computational meshes are crucial in the analysis of displacements and stabilities of rock and soil masses. In this paper, we present a method for generating high-quality tetrahedral meshes of geological models to be used in stability analyses of rock and soil masses. The method is implemented by utilizing the Computational Geometry Algorithms Library (CGAL). The input is a geological model consisting of triangulated surfaces, and the output is a high-quality tetrahedral mesh of the geological model. To demonstrate the effectiveness of the presented method, we apply it to generate a series of computational meshes of geological model, and we then analyse the stabilities of the rock and soil slopes on the basis of the generated tetrahedral mesh models. The applications demonstrate the effectiveness and practicability of the present method.•A method for generating high-quality tetrahedral meshes of geological models is presented.•We evaluate the quality of the tetrahedral mesh of geological model using four metrics.•Three applications demonstrate the effectiveness and practicability of the presented method.

A polyhedra-based model for moving regions in databases

ABSTRACT Moving objects databases store and process objects with a focus on their spatiotemporal behaviour. To achieve this, the model of the data must be suitable to efficiently store and process moving objects. Currently, a unit-based model is widely used, where each moving object is divided into one or more time intervals, during which the object behaves uniformly. This model is also used for a data type called moving regions, which resembles moving and shape changing regions as, for example, forest fires or cloud fields. However, this model struggles to support operations like union, difference or intersection of two moving regions; the resulting objects are unnecessarily bloated and uncomfortable to handle because the resulting number of units is generally very high. In this paper, an alternative model for moving regions is proposed, which is based on polyhedra. Furthermore, this work develops an isomorphism between moving regions and polyhedra including all relevant operations, which has the additional advantage that several implementations for those are already readily available; this is demonstrated by a reference implementation using the existing and well-tested Computational Geometry Algorithms Library (CGAL).

Exact Minkowski sums of polygons with holes

We present an efficient algorithm that computes the Minkowski sum of two polygons, which may have holes. The new algorithm is based on the convolution approach. Its efficiency stems in part from a property for Minkowski sums of polygons with holes: Given two polygons with holes, for each input polygon we can fill up the holes that are relatively small compared to the other polygon. Specifically, we can always fill up all the holes of at least one polygon, transforming it into a polygon without holes, and still obtain exactly the same Minkowski sum. Obliterating holes in the input summands speeds up the computation of Minkowski sums.We introduce a robust implementation of the new algorithm, which follows the Exact Geometric Computation paradigm and thus guarantees exact results. We also present an empirical comparison of the performance of Minkowski sum construction of various input examples, where we show that the implementation of the new algorithm exhibits better performance than several other implementations in many cases.The software is available as part of the 2D Minkowski Sums package of Cgal (Computational Geometry Algorithms Library), starting from Release 4.7. Additional information and supplementary material is available at our project page http://acg.cs.tau.ac.il/projects/rc.

POINT CLOUD MANAGEMENT THROUGH THE REALIZATION OF THE INTELLIGENT CLOUD VIEWER SOFTWARE

Abstract. The paper presents a software dedicated to the elaboration of point clouds, called Intelligent Cloud Viewer (ICV), made in-house by AESEI software (Spin-Off of Politecnico di Bari), allowing to view point cloud of several tens of millions of points, also on of “no” very high performance systems. The elaborations are carried out on the whole point cloud and managed by means of the display only part of it in order to speed up rendering. It is designed for 64-bit Windows and is fully written in C ++ and integrates different specialized modules for computer graphics (Open Inventor by SGI, Silicon Graphics Inc), maths (BLAS, EIGEN), computational geometry (CGAL, Computational Geometry Algorithms Library), registration and advanced algorithms for point clouds (PCL, Point Cloud Library), advanced data structures (BOOST, Basic Object Oriented Supporting Tools), etc. ICV incorporates a number of features such as, for example, cropping, transformation and georeferencing, matching, registration, decimation, sections, distances calculation between clouds, etc. It has been tested on photographic and TLS (Terrestrial Laser Scanner) data, obtaining satisfactory results. The potentialities of the software have been tested by carrying out the photogrammetric survey of the Castel del Monte which was already available in previous laser scanner survey made from the ground by the same authors. For the aerophotogrammetric survey has been adopted a flight height of approximately 1000ft AGL (Above Ground Level) and, overall, have been acquired over 800 photos in just over 15 minutes, with a covering not less than 80%, the planned speed of about 90 knots.

Identifying EGFR mutation-induced drug resistance based on alpha shape model analysis of the dynamics.

BackgroundEpidermal growth factor receptor (EGFR) mutation-induced drug resistance is a difficult problem in lung cancer treatment. Studying the molecular mechanisms of drug resistance can help to develop corresponding treatment strategies and benefit new drug design.MethodsIn this study, Rosetta was employed to model the EGFR mutant structures. Then Amber was carried out to conduct molecular dynamics (MD) simulation. Afterwards, we used Computational Geometry Algorithms Library (CGAL) to compute the alpha shape model of the mutants.ResultsWe analyzed the EGFR mutation-induced drug resistance based on the motion trajectories obtained from MD simulation. We computed alpha shape model of all the trajectory frames for each mutation type. Solid angle was used to characterize the curvature of the atoms at the drug binding site. We measured the knob level of the drug binding pocket of each mutant from two ways and analyzed its relationship with the drug response level. Results show that 90 % of the mutants can be grouped correctly by setting a certain knob level threshold.ConclusionsThere is a strong correlation between the geometric properties of the drug binding pocket of the EGFR mutants and the corresponding drug responses, which can be used to predict the response of a new EGFR mutant to a drug molecule.

CGALmesh

CGALmesh is the mesh generation software package of the Computational Geometry Algorithm Library (CGAL). It generates isotropic simplicial meshes—surface triangular meshes or volume tetrahedral meshes—from input surfaces, 3D domains, and 3D multidomains, with or without sharp features. The underlying meshing algorithm relies on restricted Delaunay triangulations to approximate domains and surfaces and on Delaunay refinement to ensure both approximation accuracy and mesh quality. CGALmesh provides guarantees on approximation quality and on the size and shape of the mesh elements. It provides four optional mesh optimization algorithms to further improve the mesh quality. A distinctive property of CGALmesh is its high flexibility with respect to the input domain representation. Such a flexibility is achieved through a careful software design, gathering into a single abstract concept, denoted by the oracle, all required interface features between the meshing engine and the input domain. We already provide oracles for domains defined by polyhedral and implicit surfaces.

The computational geometry algorithms library CGAL

The Computational Geometry Algorithms Library (CGAL) is an open source software library that provides industrial and academic users with easy access to reliable implementations of efficient geometric algorithms.

The computational geometry algorithms library CGAL

The Computational Geometry Algorithms Library (CGAL) is an open source software library that provides industrial and academic users with easy access to reliable implementations of efficient geometric algorithms.

A Generic Implementation of dD Combinatorial Maps in CGAL

We present a generic implementation of dD combinatorial maps and linear cell complexes in CGAL, the Computational Geometry Algorithms Library. A combinatorial map describes an object subdivided into cells; a linear cell complex describes the linear geometry embedding of such a subdivision. In this paper, we show how generic programming and new techniques recently introduced in the C++11 standard allow a fully generic and customizable implementation of these two data structures, while maintaining optimal memory footprint and direct access to all information. We compare our implementation with existing 2D and 3D libraries implementing cellular structures, and illustrate its usage by two applications. To the best of our knowledge, the CGAL software package presented here offers the only available generic implementation of combinatorial maps in any dimension.

GENERATING 3D VOLUMETRIC MESHES OF INTERNAL AND EXTERNAL FRUIT STRUCTURE

Two essential functions in determining fruit quality are the transport and accumulation of water and dry matter to various fruit tissues. Since water and carbon are delivered to fruit tissues through a complex vasculature system, the internal fruit structure and pattern of vasculature may have a significant impact on their distribution within the fruit. The aim of this work is to provide methods for generating fruit structure that can be integrated with models of fruit function and used to investigate such effects. To this end, we have developed a modelling pipeline in the OpenAlea platform that involves two steps: (1) generating a 3D volumetric mesh representation of the entire fruit, and (2) generating a complex network of vasculature that is embedded within this mesh. To create the 3D volumetric mesh, we use reconstruction algorithms from the 3D mesh generation package of the Computational Geometry Algorithms Library. To generate the pattern of vasculature within this volumetric mesh, we use an algorithmic approach from PlantScan3D software that was designed to reconstruct tree architecture from laser scanner data. We have applied our modelling pipeline to generate the internal and external geometry of a cherry tomato fruit using Magnetic Resonance Imaging data as input. These kinds of applications of our pipeline demonstrate its ability to create species-specific models of fruit structure with relatively low effort. In another work, the volumetric meshes will be combined with models of function to form integrative computational fruit models, which will help to investigate the effects of fruit structure on quality.

INTEGRATION OF HETEROGENOUS DIGITAL SURFACE MODELS

Abstract. The application of extended digital surface models often reveals, that despite an acceptable global accuracy for a given dataset, the local accuracy of the model can vary in a wide range. For high resolution applications which cover the spatial extent of a whole country, this can be a major drawback. Within the Swiss National Forest Inventory (NFI), two digital surface models are available, one derived from LiDAR point data and the other from aerial images. Automatic photogrammetric image matching with ADS80 aerial infrared images with 25cm and 50cm resolution is used to generate a surface model (ADS-DSM) with 1m resolution covering whole switzerland (approx. 41000 km2). The spatially corresponding LiDAR dataset has a global point density of 0.5 points per m2 and is mainly used in applications as interpolated grid with 2m resolution (LiDAR-DSM). Although both surface models seem to offer a comparable accuracy from a global view, local analysis shows significant differences. Both datasets have been acquired over several years. Concerning LiDAR-DSM, different flight patterns and inconsistent quality control result in a significantly varying point density. The image acquisition of the ADS-DSM is also stretched over several years and the model generation is hampered by clouds, varying illumination and shadow effects. Nevertheless many classification and feature extraction applications requiring high resolution data depend on the local accuracy of the used surface model, therefore precise knowledge of the local data quality is essential. The commercial photogrammetric software NGATE (part of SOCET SET) generates the image based surface model (ADS-DSM) and delivers also a map with figures of merit (FOM) of the matching process for each calculated height pixel. The FOM-map contains matching codes like high slope, excessive shift or low correlation. For the generation of the LiDAR-DSM only first- and last-pulse data was available. Therefore only the point distribution can be used to derive a local accuracy measure. For the calculation of a robust point distribution measure, a constrained triangulation of local points (within an area of 100m2) has been implemented using the Open Source project CGAL. The area of each triangle is a measure for the spatial distribution of raw points in this local area. Combining the FOM-map with the local evaluation of LiDAR points allows an appropriate local accuracy evaluation of both surface models. The currently implemented strategy ("partial replacement") uses the hypothesis, that the ADS-DSM is superior due to its better global accuracy of 1m. If the local analysis of the FOM-map within the 100m2 area shows significant matching errors, the corresponding area of the triangulated LiDAR points is analyzed. If the point density and distribution is sufficient, the LiDAR-DSM will be used in favor of the ADS-DSM at this location. If the local triangulation reflects low point density or the variance of triangle areas exceeds a threshold, the investigated location will be marked as NODATA area. In a future implementation ("anisotropic fusion") an anisotropic inverse distance weighting (IDW) will be used, which merges both surface models in the point data space by using FOM-map and local triangulation to derive a quality weight for each of the interpolation points. The "partial replacement" implementation and the "fusion" prototype for the anisotropic IDW make use of the Open Source projects CGAL (Computational Geometry Algorithms Library), GDAL (Geospatial Data Abstraction Library) and OpenCV (Open Source Computer Vision).

Parallel geometric algorithms for multi-core computers

Computers with multiple processor cores using shared memory are now ubiquitous. In this paper, we present several parallel geometric algorithms that specifically target this environment, with the goal of exploiting the additional computing power. The algorithms we describe are (a) 2-/3-dimensional spatial sorting of points, as is typically used for preprocessing before using incremental algorithms, (b) d-dimensional axis-aligned box intersection computation, and finally (c) 3D bulk insertion of points into Delaunay triangulations, which can be used for mesh generation algorithms, or simply for constructing 3D Delaunay triangulations. For the latter, we introduce as a foundational element the design of a container data structure that both provides concurrent addition and removal operations and is compact in memory. This makes it especially well-suited for storing large dynamic graphs such as Delaunay triangulations. We show experimental results for these algorithms, using our implementations based on the Computational Geometry Algorithms Library (CGAL). This work is a step towards what we hope will become a parallel mode for CGAL, where algorithms automatically use the available parallel resources without requiring significant user intervention.

Contributing vertices-based Minkowski sum computation of convex polyhedra

Minkowski sum is an important operation. It is used in many domains such as: computer-aided design, robotics, spatial planning, mathematical morphology, and image processing. We propose a novel algorithm, named the Contributing Vertices-based Minkowski Sum (CVMS) algorithm for the computation of the Minkowski sum of convex polyhedra. The CVMS algorithm allows to easily obtain all the facets of the Minkowski sum polyhedron only by examining the contributing vertices—a concept we introduce in this work, for each input facet. We exploit the concept of contributing vertices to propose the Enhanced and Simplified Slope Diagram-based Minkowski Sum (ESSDMS) algorithm, a slope diagram-based Minkowski sum algorithm sharing some common points with the approach proposed by Wu et al. [Wu Y, Shah J, Davidson J. Improvements to algorithms for computing the Minkowski sum of 3-polytopes. Comput Aided Des. 2003; 35(13): 1181–92]. The ESSDMS algorithm does not embed input polyhedra on the unit sphere and does not need to perform stereographic projections. Moreover, the use of contributing vertices brings up more simplifications and improves the overall performance. The implementations for the mentioned algorithms are straightforward, use exact number types, produce exact results, and are based on CGAL, the Computational Geometry Algorithms Library. More examples and results of the CVMS algorithm for several convex polyhedra can be found at http://liris.cnrs.fr/hichem.barki/mksum/CVMS-convex.

An experimental study of point location in planar arrangements in CGAL

We study the performance in practice of various point-location algorithms implemented in CGAL (the Computational Geometry Algorithms Library), including a newly devised landmarks algorithm. Among the other algorithms studied are: a naïve approach, a “walk along a line” strategy, and a trapezoidal decomposition-based search structure. The current implementation addresses general arrangements of planar curves, including arrangements of nonlinear segments (e.g., conic arcs) and allows for degenerate input (for example, more than two curves intersecting in a single point or overlapping curves). The algorithms use exact geometric computation and thus result in the correct point location. In our landmarks algorithm (a.k.a. jump & walk), special points, “landmarks,” are chosen in a preprocessing stage, their place in the arrangement is found, and they are inserted into a data structure that enables efficient nearest-neighbor search. Given a query point, the nearest landmark is located and a “walk” strategy is applied from the landmark to the query point. We report on various experiments with arrangements composed of line segments or conic arcs. The results indicate that compared to the other algorithms tested, the landmarks approach is the most efficient, when the overall (amortized) cost of a query is taken into account, combining both preprocessing and query time. The simplicity of the algorithm enables an almost straightforward implementation and rather easy maintenance. The generic programming implementation allows versatility both in the selected type of landmarks and in the choice of the nearest-neighbor search structure. The end result is an efficient point-location algorithm that bypasses the alternative CGAL implementations in most practical aspects.

Algorithm 889

Surfaces of R 3 are ubiquitous in science and engineering, and estimating the local differential properties of a surface discretized as a point cloud or a triangle mesh is a central building block in computer graphics, computer aided design, computational geometry, and computer vision. One strategy to perform such an estimation consists of resorting to polynomial fitting, either interpolation or approximation, but this route is difficult for several reasons: choice of the coordinate system, numerical handling of the fitting problem, and extraction of the differential properties. This article presents a generic C++ software package solving these problems. On the theoretical side and as established in a companion paper, the interpolation and approximation methods provided achieve the best asymptotic error bounds known to date. On the implementation side and following state-of-the-art coding rules in computational geometry, genericity of the package is achieved thanks to four template classes accounting for, (a) the type of the input points, (b) the internal geometric computations, (c) a conversion mechanism between these two geometries, and (d) the linear algebra operations. An instantiation within the Computational Geometry Algorithms Library (CGAL, version 3.3) and using LAPACK is also provided.

Iterated snap rounding with bounded drift

Snap Rounding and its variant, Iterated Snap Rounding, are methods for converting arbitrary-precision arrangements of segments into a fixed-precision representation (we call them SR and ISR for short). Both methods approximate each original segment by a polygonal chain, and both may lead, for certain inputs, to rounded arrangements with undesirable properties: in SR the distance between a vertex and a non-incident edge of the rounded arrangement can be extremely small, inducing potential degeneracies. In ISR, a vertex and a non-incident edge are well separated, but the approximating chain may drift far away from the original segment it approximates. We propose a new variant, Iterated Snap Rounding with Bounded Drift, which overcomes these two shortcomings of the earlier methods. The new solution augments ISR with simple and efficient procedures that guarantee the quality of the geometric approximation of the original segments, while still maintaining the property that a vertex and a non-incident edge in the rounded arrangement are well separated. We investigate the properties of the new method and compare it with the earlier variants. We have implemented the new scheme on top of CGAL, the Computational Geometry Algorithms Library, and report on experimental results.

Exact and efficient construction of Minkowski sums of convex polyhedra with applications

We present an exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in R 3 . Our implementation is complete in the sense that it does not assume general position. Namely, it can handle degenerate input, and it produces exact results. We also present applications of the Minkowski-sum computation to answer collision and proximity queries about the relative placement of two convex polyhedra in R 3 . The algorithms use a dual representation of convex polyhedra, and their implementation is mainly based on the Arrangement package of Cgal, the Computational Geometry Algorithm Library. We compare our Minkowski-sum construction with the only three other methods that produce exact results we are aware of. One is a simple approach that computes the convex hull of the pairwise sums of vertices of two convex polyhedra. The second is based on Nef polyhedra embedded on the sphere, and the third is an output-sensitive approach based on linear programming. Our method is significantly faster. The results of experimentation with a broad family of convex polyhedra are reported. The relevant programs, source code, data sets, and documentation are available at http://www.cs.tau.ac.il/~efif/CD and a short movie [Fogel E, Halperin D. Video: Exact Minkowski sums of convex polyhedra. In: Proceedings of 21st annual ACM symposium on computational geometry. 2005. p. 382–3] that describes some of the concepts portrayed in this paper can be downloaded from http://www.cs.tau.ac.il/~efif/CD/Mink3d.avi.

Boolean operations on 3D selective Nef complexes: Data structure, algorithms, optimized implementation and experiments

Nef polyhedra in d-dimensional space are the closure of half-spaces under boolean set operations. In consequence, they can represent non-manifold situations, open and closed sets, mixed-dimensional complexes, and they are closed under all boolean and topological operations, such as complement and boundary. They were introduced by W. Nef in his seminal 1978 book on polyhedra. The generality of Nef complexes is essential for some applications. In this paper, we present a new data structure for the boundary representation of three-dimensional Nef polyhedra and efficient algorithms for boolean operations. We use exact arithmetic to avoid well-known problems with floating-point arithmetic and handle all degeneracies. Furthermore, we present important optimizations for the algorithms, and evaluate this optimized implementation with extensive experiments. The experiments supplement the theoretical runtime analysis and illustrate the effectiveness of our optimizations. We compare our implementation with the Acis CAD kernel. Acis is mostly faster, by a factor up to six. There are examples on which Acis fails. The implementation was released as Open Source in the Computational Geometry Algorithm Library ( Cgal) release 3.1 in December 2004.

Advanced programming techniques applied to Cgal's arrangement package

Arrangements of planar curves are fundamental structures in computational geometry. Recently, the arrangement package of Cgal, the Computational Geometry Algorithms Library, has been redesigned and re-implemented exploiting several advanced programming techniques. The resulting software package, which constructs and maintains planar arrangements, is easier to use, to extend, and to adapt to a variety of applications. It is more efficient space- and time-wise, and more robust. The implementation is complete in the sense that it handles degenerate input, and it produces exact results. In this paper we describe how various programming techniques were used to accomplish specific tasks within the context of computational geometry in general and arrangements in particular. These tasks are exemplified by several applications, whose robust implementation is based on the arrangement package. Together with a set of benchmarks they assured the successful application of the various programming techniques.