Showing papers on "Constrained Delaunay triangulation published in 2009"

Robust and Efficient Surface Reconstruction From Range Data

Abstract: We describe a robust but simple algorithm to reconstruct a surface from a set of merged range scans. Our key contribution is the formulation of the surface reconstruction problem as an energy minimisation problem that explicitly models the scanning process. The adaptivity of the Delaunay triangulation is exploited by restricting the energy to inside/outside labelings of Delaunay tetrahedra. Our energy measures both the output surface quality and how well the surface agrees with soft visibility constraints. Such energy is shown to perfectly fit into the minimum s − t cuts optimisation framework, allowing fast computation of a globally optimal tetrahedra labeling, while avoiding the “shrinking bias” that usually plagues graph cuts methods. The behaviour of our method confronted to noise, undersampling and outliers is evaluated on several data sets and compared with other methods through different experiments: its strong robustness would make our method practical not only for reconstruction from range data but also from typically more difficult dense point clouds, resulting for instance from stereo image matching. Our effective modeling of the surface acquisition inverse problem, along with the unique combination of Delaunay triangulation and minimum s − t cuts, makes the computational requirements of the algorithm scale well with respect to the size of the input point cloud.

Challenges and Technologies in Reservoir Modeling

Abstract: Reservoir modeling is playing an increasingly important role in developing and producing hydrocarbon reserves. In this paper, we provide a brief overview of some main challenges in reservoir modeling, i.e., accurate and efficient modeling of complex reservoir geometry and heterogeneous reservoir properties. We then present modeling techniques we recently developed in addressing these challenges, including a method for generating constrained Voronoi grids and a generic global scale-up method. We focus on the Voronoi gridding method, which is based on a new constrained Delaunay triangulation algorithm and a rigorous method of adapting Voronoi grids to piecewise linear constraints. The global scale-up method based on generic flows is briefly described. Numerical examples are provided to demonstrate the techniques and the advantage of combining them in constructing accurate and efficient reservoir models. AMS subject classifications: 37M05, 76S05, 86A60, 65M50, 65N50

Incremental construction of the delaunay triangulation and the delaunay graph in medium dimension

Abstract: We describe a new implementation of the well-known incremental algorithm for constructing Delaunay triangulations in any dimension. Our implementation follows the exact computing paradigm and is fully robust. Extensive comparisons show that our implementation outperforms the best currently available codes for exact convex hulls and Delaunay triangulations, compares very well to the fast non-exact QHull implementation and can be used for quite big input sets in spaces of dimensions up to 6. To circumvent prohibitive memory usage, we also propose a modification of the algorithm that uses and stores only the Delaunay graph (the edges of the full triangulation). We show that a careful implementation of the modified algorithm performs only 6 to 8 times slower than the original algorithm while drastically reducing memory usage in dimension 4 or above.

Quality mesh generation for molecular skin surfaces using restricted union of balls

Abstract: Quality surface meshes for molecular models are desirable in the studies of protein shapes and functionalities. However, there is still no robust software that is capable to generate such meshes with good quality. In this paper, we present a Delaunay-based surface triangulation algorithm generating quality surface meshes for the molecular skin model. We expand the restricted union of balls along the surface and generate an @e-sampling of the skin surface incrementally. At the same time, a quality surface mesh is extracted from the Delaunay triangulation of the sample points. The algorithm supports robust and efficient implementation and guarantees the mesh quality and topology as well. Our results facilitate molecular visualization and have made a contribution towards generating quality volumetric tetrahedral meshes for the macromolecules.

Connectivity-Based Sensor Network Localization with Incremental Delaunay Refinement Method

Abstract: We study the anchor-free localization problem for a large-scale sensor network with a complex shape, knowing network connectivity information only. The main idea follows from our previous work in which a subset of the nodes are selected as landmarks and the sensor field is partitioned into Voronoi cells with all the nodes closest to the same landmark grouped into the same cell. We extract the combinatorial Delaunay complex as the dual complex of the landmark Voronoi diagram and embed the combinatorial Delaunay complex as a structural skeleton. In this paper we develop a new landmark selection algorithm with incremental Delaunay refinement method. This algorithm does not assume any knowledge of the network boundary and runs in a distributed manner to select landmarks incrementally until both the global rigidity property (the Delaunay complex is globally rigid and thus can be embedded uniquely) and the coverage property (every node is not far from the embedded Delaunay complex) are met. The new algorithm substantially improves the robustness and applicability of the original localization algorithm, especially in networks with very low average degree (even non- rigid networks) and complex shapes.

Mesh Generation from 3D Multi-material Images

Abstract: The problem of generating realistic computer models of objects represented by 3D segmented images is important in many biomedical applications. Labelled 3D images impose particular challenges for meshing algorithms because multi-material junctions form features such as surface pacthes, edges and corners which need to be preserved into the output mesh. In this paper, we propose a feature preserving Delaunay refinement algorithm which can be used to generate high-quality tetrahedral meshes from segmented images. The idea is to explicitly sample corners and edges from the input image and to constrain the Delaunay refinement algorithm to preserve these features in addition to the surface patches. Our experimental results on segmented medical images have shown that, within a few seconds, the algorithm outputs a tetrahedral mesh in which each material is represented as a consistent submesh without gaps and overlaps. The optimization property of the Delaunay triangulation makes these meshes suitable for the purpose of realistic visualization or finite element simulations.

Quality Triangulations with Locally Optimal Steiner Points

Abstract: We propose two novel ideas to improve the performance of Delaunay refinement algorithms which are used for computing quality triangulations. The first idea is an effective use of the Voronoi diagram and unifies previously suggested Steiner point insertion schemes (circumcenter, sink, off-center) together with a new strategy. The second idea is the integration of a new local smoothing strategy into the refinement process. These lead to two new versions of Delaunay refinement, where the second is simply an extension of the first. For a given input domain and a constraint angle $\alpha$, Delaunay refinement algorithms aim to compute triangulations that have all angles at least $\alpha$. The original Delaunay refinement algorithm of Ruppert is proven to terminate with size-optimal quality triangulations for $\alpha\le20.7^\circ$. In practice, the original and the consequent Delaunay refinement algorithms generally work for $\alpha\le34^\circ$ and fail to terminate for larger constraint angles. Our algorithms provide the same theoretical guarantees as the previous Delaunay refinement algorithms. The second of the proposed algorithms generally terminates for constraint angles up to $42^\circ$. Experiments also indicate that our algorithm computes significantly (about by a factor of two) smaller triangulations than the output of the previous Delaunay refinement algorithms. Moreover, the new algorithms are experimentally shown to outperform the previous algorithms even in the existence of additional constraints, such as the maximum area triangle constraint which is commonly used for computing uniform triangulations.

Off-centers: A new type of Steiner points for computing size-optimal quality-guaranteed Delaunay triangulations

Abstract: We introduce a new type of Steiner points, called off-centers, as an alternative to circumcenters, to improve the quality of Delaunay triangulations in two dimensions. We propose a new Delaunay refinement algorithm based on iterative insertion of off-centers. We show that this new algorithm has the same quality and size optimality guarantees of the best known refinement algorithms. In practice, however, the new algorithm inserts fewer Steiner points, runs faster, and generates smaller triangulations than the best previous algorithms. Performance improvements are significant especially when user-specified minimum angle is large, e.g., when the smallest angle in the output triangulation is 30^o, the number of Steiner points is reduced by about 40%, while the mesh size is down by about 30%. As a result of its shown benefits, the algorithm described here has already replaced the well-known circumcenter insertion algorithm of Ruppert and has been the default quality triangulation method in the popular meshing software Triangle.

Perturbing Slivers in 3D Delaunay Meshes

Abstract: Isotropic tetrahedron meshes generated by Delaunay refinement algorithms are known to contain a majority of well-shaped tetrahedra, as well as spurious sliver tetrahedra. As the slivers hamper stability of numerical simulations we aim at removing them while keeping the triangulation Delaunay for simplicity. The solution which explicitly perturbs the slivers through random vertex relocation and Delaunay connectivity update is very effective but slow. In this paper we present a perturbation algorithm which favors deterministic over random perturbation. The added value is an improved efficiency and effectiveness. Our experimental study applies the proposed algorithm to meshes obtained by Delaunay refinement as well as to carefully optimized meshes.

Constructing Delaunay Triangulations along Space-Filling Curves

Abstract: Incremental construction con BRIO using a space-filling curve order for insertion is a popular algorithm for constructing Delaunay triangulations So far, it has only been analyzed for the case that a worst-case optimal point location data structure is used which is often avoided in implementations In this paper, we analyze its running time for the more typical case that points are located by walking We show that in the worst-case the algorithm needs quadratic time, but that this can only happen in degenerate cases We show that the algorithm runs in O(n logn) time under realistic assumptions Furthermore, we show that it runs in expected linear time for many random point distributions

Filling holes in triangular meshes by curve unfolding

Abstract: We propose a novel approach to automatically fill holes in triangulated models. Each hole is filled using a minimum energy surface that is obtained in three steps. First, we unfold the hole boundary onto a plane using energy minimization. Second, we triangulate the unfolded hole using a constrained Delaunay triangulation. Third, we embed the triangular mesh as a minimum energy surface in ℝ3. The running time of the method depends primarily on the size of the hole boundary and not on the size of the model, thereby making the method applicable to large models. Our experiments demonstrate the applicability of the algorithm to the problem of filling holes bounded by highly curved boundaries in large models.

A Self-stabilizing and Local Delaunay Graph Construction

Abstract: This paper studies the construction of self-stabilizing topologies for distributed systems. While recent research has focused on chain topologies where nodes need to be linearized with respect to their identifiers, we go a step further and explore a natural 2-dimensional generalization. In particular, we present a local self-stabilizing algorithm that constructs a Delaunay graph from any initial connected topology and in a distributed manner. This algorithm terminates in time O(n 3) in the worst-case. We believe that such self-stabilizing Delaunay networks have interesting applications and give insights into the necessary geometric reasoning that is required for higher-dimensional linearization problems.

Constrained CVT meshes and a comparison of triangular mesh generators

Abstract: Mesh generation in regions in Euclidean space is a central task in computational science, and especially for commonly used numerical methods for the solution of partial differential equations, e.g., finite element and finite volume methods. We focus on the uniform Delaunay triangulation of planar regions and, in particular, on how one selects the positions of the vertices of the triangulation. We discuss a recently developed method, based on the centroidal Voronoi tessellation (CVT) concept, for effecting such triangulations and present two algorithms, including one new one, for CVT-based grid generation. We also compare several methods, including CVT-based methods, for triangulating planar domains. To this end, we define several quantitative measures of the quality of uniform grids. We then generate triangulations of several planar regions, including some having complexities that are representative of what one may encounter in practice. We subject the resulting grids to visual and quantitative comparisons and conclude that all the methods considered produce high-quality uniform grids and that the CVT-based grids are at least as good as any of the others.

Computing 3D Periodic Triangulations

Abstract: This work is motivated by the need for software computing 3D periodic triangulations in numerous domains including astronomy, material engineering, biomedical computing, fluid dynamics etc. We design an algorithmic test to check whether a partition of the 3D flat torus into tetrahedra forms a triangulation (which subsumes that it is a simplicial complex). We propose an incremental algorithm that computes the Delaunay triangulation of a set of points in the 3D flat torus without duplicating any point, whenever possible; our algorithmic test detects when such a duplication can be avoided, which is usually possible in practical situations. Even in cases where point duplication is necessary, our algorithm always computes a triangulation that is homeomorpic to the flat torus. To the best of our knowledge, this is the first algorithm of this kind whose output is provably correct. The implementation will be released in Cgal[7].

Theory of a Practical Delaunay Meshing Algorithm for a Large Class of Domains

Abstract: Recently a Delaunay refinement algorithm has been proposed that can mesh domains as general as piecewise smooth complexes. These domains include polyhedra, smooth and piecewise smooth surfaces, volumes enclosed by them, and above all non-manifold spaces. The algorithm is guaranteed to capture the input topology at the expense of four tests, some of which are computationally intensive and hard to implement. The goal of this paper is to present the theory that justifies a refinement algorithm with a single disk test in place of four tests of the previous algorithm. The algorithm is supplied with a resolution parameter that controls the level of refinement. We prove that, when the resolution is fine enough (this level is reached very fast in practice), the output mesh becomes homeomorphic to the input while preserving all input features. Moreover, regardless of the refinement level, each k-manifold element in the input complex is meshed with a triangulated k-manifold. Boundary incidences among elements maintain the input structure. Implementation results reported in a companion paper corroborate our claims. Research supported by NSF, USA (CCF-0430735 and CCF-0635008) and RGC, Hong Kong, China (HKUST 6181/04E). Department of Computer Science and Engineering, HKUST, Hong Kong. Email: scheng@cse.ust.hk Department of Computer Science and Engineering, Ohio State University, Ohio, USA. Email: tamaldey@cse.ohio-state.edu Department of Computer Science and Engineering, Ohio State University, Ohio, USA. Email: levinej@cse.ohio-state.edu

Feature preserving Delaunay mesh generation from 3D multi-material images

Abstract: Generating realistic geometric models from 3D segmented images is an important task in many biomedical applications. Segmented 3D images impose particular challenges for meshing algorithms because they contain multi-material junctions forming features such as surface patches, edges and corners. The resulting meshes should preserve these features to ensure the visual quality and the mechanical soundness of the models. We present a feature preserving Delaunay refinement algorithm which can be used to generate high-quality tetrahedral meshes from segmented images. The idea is to explicitly sample corners and edges from the input image and to constrain the Delaunay refinement algorithm to preserve these features in addition to the surface patches. Our experimental results on segmented medical images have shown that, within a few seconds, the algorithm outputs a tetrahedral mesh in which each material is represented as a consistent submesh without gaps and overlaps. The optimization property of the Delaunay triangulation makes these meshes suitable for the purpose of realistic visualization or finite element simulations.

Automatic rigging for animation characters with 3D silhouette

Abstract: Animating an articulated 3D character requires the specification of its interior skeleton structure which defines how the skin surface is deformed during animation. Currently this task is to a large extent accomplished manually, which consumes a large amount of animators' time. This paper presents an automatic rigging method making use of a new geometry entity called the 3D silhouette. The first step is to extract a coarse 3D curve skeleton and some skeletal joints of a character. This curve skeleton is then refined with a perpendicular silhouette. According to the connectivity of the skeletal joints, the hierarchical animation skeleton is finally constructed. By avoiding complicated computation such as voxelization and pruning, this method is simple and efficient, much faster than existing methods. It proves very useful for quick animation production, with applications including games design and prototype graphical systems. Copyright © 2009 John Wiley & Sons, Ltd. Generating animation skeleton of a hand model with our method: (a) Original mode, (b) Primary 3D silhouette, (c) 3D medial axis of hand through constrained Delaunay triangulation, (d) Decomposition result, (e) Curve skeleton and key skeleton points, (f) Animation skeleton.

The spanning ratio of the Delaunay triangulation is greater than pi/2.

Abstract: Consider the Delaunay triangulation T of a set P of points in the plane. The spanning ratio of T , i.e. the maximum ratio between the length of the shortest path between this pair on the graph of the triangulation and their Euclidean distance. It has long been conjectured that the spanning ratio of T can be at most π/2. We show in this note that there exist point sets in convex position with a spanning ratio > 1.5810 and in general position with a spanning ratio > 1.5846, both of which are strictly larger than π/2 ≈ 1.5708. Furthermore, we show that any set of points drawn independently from the same distribution will, with high probability, have a spanning ratio larger than π/2.

CAD data visualization on mobile devices using sequential constrained Delaunay triangulation

Abstract: 3D graphic rendering in mobile application programs is becoming increasingly popular with rapid advances in mobile device technology. Current 3D graphic rendering engines for mobile devices do not provide triangulation capabilities for surfaces; therefore, mobile 3D graphic applications have been dealing only with pre-tessellated geometric data. Since triangulation is comparatively expensive in terms of computation, real-time tessellation cannot be easily implemented on mobile devices with limited resources. No research has yet been reported on real-time triangulation on mobile devices. In this paper, we propose a real-time triangulation algorithm for visualization on mobile devices based on sequential constrained Delaunay triangulation. We apply a compact data structure and a sequential triangulation process for visualization of CAD data on mobile devices. In order to achieve a high performance and compact implementation of the triangulation, the nature of the CAD data is fully considered in the computational process. This paper also presents a prototype implementation for a mobile 3D CAD viewer running on a handheld Personal Digital Assistant (PDA).

Repairing and meshing imperfect shapes with Delaunay refinement

Abstract: As a direct consequence of software quirks, designer errors, and representation flaws, often three-dimensional shapes are stored in formats that introduce inconsistencies such as small gaps and overlaps between surface patches. We present a new algorithm that simultaneously repairs imperfect geometry and topology while generating Delaunay meshes of these shapes. At the core of this approach is a meshing algorithm for input shapes that are piecewise smooth complexes (PSCs), a collection of smooth surface patches meeting at curves non-smoothly or in non-manifold configurations. Guided by a user tolerance parameter, we automatically merge nearby components while building a Delaunay mesh that has many of these errors fixed. Experimental evidence is provided to show the results of our algorithm on common computer-aided design (CAD) formats. Our algorithm may also be used to simplify shapes by removing small features which would require an excessive number of elements to preserve them in the output mesh.

Image compression with anisotropic triangulations

Abstract: We propose a new image compression method based on geodesic Delaunay triangulations. Triangulations are generated by a progressive geodesic meshing algorithm which exploits the anisotropy of images through a farthest point sampling strategy. This seeding is performed according to anisotropic geodesic distances which force the anisotropic Delaunay triangles to follow the geometry of the image. Geodesic computations are performed using a Riemannian Fast Marching, which recursively updates the geodesic distance to the seed points. A linear spline approximation on this triangulation allows to approximate faithfully sharp edges and directional features in images. The compression is achieved by coding both the coefficients of the spline approximation and the deviation of the geodesic triangulation from an Euclidean Delaunay triangulation. Numerical results show that taking into account the anisotropy improves the approximation by isotropic triangulations of complex images. The resulting encoder competes well with wavelet-based encoder such as JPEG-2000 on geometric images.

Filtering relocations on a Delaunay triangulation

Abstract: Updating a Delaunay triangulation when its vertices move is a bottleneck in several domains of application. Rebuilding the whole triangulation from scratch is surprisingly a very viable option compared to relocating the vertices. This can be explained by several recent advances in efficient construction of Delaunay triangulations. However, when all points move with a small magnitude, or when only a fraction of the vertices move, rebuilding is no longer the best option. This paper considers the problem of efficiently updating a Delaunay triangulation when its vertices are moving under small perturbations. The main contribution is a set of filters based upon the concept of vertex tolerances. Experiments show that filtering relocations is faster than rebuilding the whole triangulation from scratch under certain conditions.

Exact Delaunay graph of smooth convex pseudo-circles: general predicates, and implementation for ellipses

Abstract: We examine the problem of computing exactly the Delaunay graph (and the dual Voronoi diagram) of a set of, possibly intersecting, smooth convex pseudo-circles in the Euclidean plane, given in parametric form. Pseudo-circles are (convex) sites, every pair of which has at most two intersecting points. The Delaunay graph is constructed incrementally. Our first contribution is to propose robust end efficient algorithms for all required predicates, thus generalizing our earlier algorithms for ellipses, and we analyze their algebraic complexity, under the exact computation paradigm. Second, we focus on InCircle, which is the hardest predicate, and express it by a simple sparse 5 X 5 polynomial system, which allows for an efficient implementation by means of successive Sylvester resultants and a new factorization lemma. The third contribution is our cgal-based c++ software for the case of ellipses, which is the first exact implementation for the problem. Our code spends about 98 sec to construct the Delaunay graph of 128 non-intersecting ellipses, when few degeneracies occur. It is faster than the cgal segment Delaunay graph, when ellipses are approximated by k-gons for k > 15.

Natural Neighbor Interpolation - Critical Assessment and New Contributions

Abstract: In engineering and science, a multitude of problems exhibit an inherently geometric nature. The computational assessment of such problems requires an adequate representation by means of data structures and processing algorithms. One of the most widely adopted and recognized spatial data structures is the Delaunay triangulation which has its canonical dual in the Voronoi diagram. While the Voronoi diagram provides a simple and elegant framework to model spatial proximity, the core of which is the concept of natural neighbors, the Delaunay triangulation provides robust and efficient access to it. This combination explains the immense popularity of Voronoi- and Delaunay-based methods in all areas of science and engineering. This thesis addresses aspects from a variety of applications that share their affinity to the Voronoi diagram and the natural neighbor concept. First, an idea for the generalization of B-spline surfaces to unstructured knot sets over Voronoi diagrams is investigated. Then, a previously proposed method for \(C^2\) smooth natural neighbor interpolation is backed with concrete guidelines for its implementation. Smooth natural neighbor interpolation is also one of many applications requiring derivatives of the input data. The generation of derivative information in scattered data with the help of natural neighbors is described in detail. In a different setting, the computation of a discrete harmonic function in a point cloud is considered, and an observation is presented that relates natural neighbor coordinates to a continuous dependency between discrete harmonic functions and the coordinates of the point cloud. Attention is then turned to integrating the flexibility and meritable properties of natural neighbor interpolation into a framework that allows the algorithmically transparent and smooth extrapolation of any known natural neighbor interpolant. Finally, essential properties are proved for a recently introduced novel finite element tessellation technique in which a Delaunay triangulation is transformed into a unique polygonal tessellation.

Generalized Two-Dimensional Delaunay Mesh Refinement

Abstract: Delaunay refinement is a popular mesh generation method which makes it possible to derive mathematical guarantees with respect to the quality of the elements. Traditional Delaunay refinement algorithms insert Steiner points in a small enumerable number (one or two) of specific positions inside circumscribed circles of poor quality triangles and on encroached segments. In this paper we prove that there exist entire two-dimensional and one-dimensional regions that can be used for the insertion of Steiner points (innumerable number of choices), while the guarantees on mesh quality can be preserved. This result opens up the possibility to use multiple point placement strategies, all covered by a single proof. In addition, the parallelization of this generalized algorithm immediately implies the parallelization for each individual point placement method.

Variational Characterisation of Gibbs Measures with Delaunay Triangle Interaction

Abstract: This paper deals with stationary Gibbsian point processes on the plane with an interaction that depends on the tiles of the Delaunay triangulation of points via a bounded triangle potential. It is shown that the class of these Gibbs processes includes all minimisers of the associated free energy density and is therefore nonempty. Conversely, each such Gibbs process minimises the free energy density, provided the potential satisfies a weak long-range assumption.

Delaunay refinement algorithms for numerical methods

Abstract: Ruppert's algorithm is an elegant solution to the mesh generation problem for non-acute domains in two dimensions. This thesis develops a three-dimensional Delaunay refinement algorithm which produces a conforming Delaunay tetrahedralization, ensures a bound on the radius-edge ratio of nearby all tetrahedra, generates tetrahedra of a size related to the local feature size and the size of nearby small input angles, and is simple enough to admit an implementation. To do this, Delaunay refinement algorithms for estimating local feature size are constructed. These estimates are then used to determine an appropriately sized protection region around acutely adjacent features of the input. Finally, a simple variant of Ruppert's algorithm can be applied to produce a quality mesh. Additionally, some finite element interpolation results pertaining to Delaunay refinement algorithms in two dimensions are considered.

Delaunay Triangulation of Imprecise Points Simplified and Extended

Abstract: Suppose we want to compute the Delaunay triangulation of a set P whose points are restricted to a collection ${\mathcal R}$ of input regions known in advance. Building on recent work by Loffler and Snoeyink[21], we show how to leverage our knowledge of ${\mathcal R}$ for faster Delaunay computation. Our approach needs no fancy machinery and optimally handles a wide variety of inputs, eg, overlapping disks of different sizes and fat regions.

CAD Tools for Creating Space-filing 3D Escher Tiles

Abstract: We discuss the design and implementation of CAD tools for creating decorative solids that tile 3-space in a regular, isohedral manner. Starting with the simplest case of extruded 2D tilings, we describe geometric algorithms used for maintaining boundary representations of 3D tiles, including a Java implementation of an interactive constrained Delaunay triangulation library and a mesh-cutting algorithm used in layering extruded tiles to create more intricate designs. Finally, we demonstrate a CAD tool for creating 3D tilings that are derived from cubic lattices. The design process for these 3D tiles is more constrained, and hence more difficult, than in the 2D case, and it raises additional user interface issues.