Complexity of the Delaunay triangulation of points on polyhedral surfaces

It is well known that the complexity of the Delaunay triangulation of $n$ points in $R ^d$, i.e. the number of its simplices, can be $\Omega (n^\lceil \frac{d{2}\rceil })$. In particular, in $R ^3$, the number of tetrahedra can be quadratic. Differently, if the points are uniformly distributed in a cube or a ball, the expected complexity of the Delaunay triangulation is only linear. The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface. In this paper, we bound the complexity of the Delaunay triangulation of points distributed on the boundary of a given polyhedron. Under a mild uniform sampling condition, we provide deterministic asymptotic bounds on the complexity of the 3D Delaunay triangula- tion of the points when the sampling density increases. More precisely, we show that the complexity is $O(n^1.8)$ for general polyhedral surfaces and O(n\sqrtn) for convex polyhedral surfaces. Our proof uses a geometric result of independent interest that states that the medial axis of a surface is well approximated by a subset of the Voronoi vertices of the sample points. The proof extends easily to higher dimensions, leading to the first non trivial bounds for the problem when $d>3$

A Linear Bound on the Complexity of the Delaunay triangulation of points on polyhedral surfaces

Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time therefore depends on the complexity of the Delaunay triangulation of such point sets. Although the Delaunay triangulation of points in ^3 can be quadratic in the worst-case, we show that, under some mild sampling condition, the complexity of the 3D Delaunay triangulation of points distributed on a fixed number of facets of ^3 (e.g. the facets of a polyhedron) is linear. Our bound is deterministic and the constants are explicitly given

Spherical harmonics descriptor for 2D-image retrieval

In this paper, spherical harmonics are proposed as shape descriptors for 2D images. We introduce the concept of connectivity; 2D images are decomposed using connectivity, which is followed by 3D model construction. Spherical harmonics are obtained for 3D models and used as descriptors for the underlying 2D shapes. Difference between two images is computed as the Euclidean distance between their spherical harmonics descriptors. Experiments are performed to test the effectiveness of spherical harmonics for retrieval of 2D images. Item S8 within the MPEG-7 still images content set is used for performing experiments; this dataset consists of 3621 still images. Experimental results show that the proposed descriptors for 2D images are effective<br /

A Poisson sample of a smooth surface is a good sample

International audienceThe complexity of the 3D-Delaunay triangulation (tetrahedralization) of n points distributed on a surface ranges from linear to quadratic. When the points are a deterministic good sample of a smooth compact generic surface, the size of the Delaunay triangulation is O(n log n). Using this result, we prove that when points are Poisson distributed on a surface under the same hypothesis, whose expected number of vertices is λ, the expected size is O(λ log^2 λ)

Une approche hybride pour calculer l'enveloppe visuelle d'objets complexes

National audienceNous traitons de la reconstruction de l'enveloppe visuelle à partir des silhouettes de multiples images. Nous proposons une approche hybride qui tire avantage des approches surfaciques et fournit à complexité équivalente une précision beaucoup plus importante que les approches volumétriques. Celles-ci se basent en effet sur une discrétisation de l'espace sous la forme d'une grille régulière, dont beaucoup de cellules n'apportent que peu d'information. C'est pourquoi nous introduisons une discrétisation irrégulière, basée sur une grille constituée de points échantillonnés sur la surface de l'enveloppe visuelle, et de cellules tétraèdriques obtenues par triangulation de Delaunay sur les points échantillons. Ces cellules sont alors sculptées par cohérence avec les silhouettes. L'approche proposée conserve la robustesse d'une approche volumétrique mais s'avère nettement plus avantageuse en précision et en complexité. Elle rend donc possible la modélisation d'objets de géométrie complexe en temps-réel. Les applications à la modélisation d'objets en temps-réel sont discutées et des résultats préliminaires sur des données réelles et de synthèse sont présentés

Reconstruction locale et visualisation de nuages de points par surfaces de subdivision

National audienceLes surfaces de points, qu'elles soient directement acquises par scanner ou issues de la conversion d'autres modèles, permettent de stocker et de transmettre des objets complexes de manière économique, mais sont mal adaptées aux architectures matérielles existantes qui s'appuient sur une description géométrique à base de surfaces polygonales. Cet article propose une technique permettant d'obtenir une visualisation efficace des surfaces de points, entièrement à l'aide du pipeline de rendu matériel. L'idée centrale présentée ici est d'effectuer une reconstruction surfacique locale par triangulation d'un nuage de points, en générant un agrégat de 2-variétés se recouvrant, et en procurant une continuité visuelle dans les zones de recouvrements à l'aide des surfaces de subdivision

Self-Adapting Point Location

Point location in spatial subdivision is one of the most studied problems in computational geometry. In the case of triangulations of Rd, we revisit the problem to exploit a possible coherence between the query-points. For a single query, walking in the triangulation is a classical strategy with good practical behavior and expected complexity O(n^(1/d)) if the points are evenly distributed. For a batch of query-points, the main idea is to use previous queries to improve the current one; we compare various strategies that have an influence on the constant hidden in the big-O notation. Still regarding the complexity of a query, we show how the Delaunay hierarchy can be used to answer, under some hypotheses, a query q with a O(log #(pq) ) randomized expected complexity, where #(.) indicates the number of simplices crossed by the line pq, and p is a previously located query. The data structure has O(n log n) construction complexity and O(n) memory complexity

Medial Axis Approximation and Regularization

Medial axis is a classical shape descriptor. Among many good properties, medial axis is thin, centered in the shape, and topology preserving. Therefore, it is constantly sought after by researchers and practitioners in their respective domains. However, two barriers remain that hinder wide adoption of medial axis. First, exact computation of medial axis is very difficult. Hence, in practice medial axis is approximated discretely. Though abundant approximation methods exist, they are either limited in scalability, insufficient in theoretical soundness, or susceptible to numerical issues. Second, medial axis is easily disturbed by small noises on its defining shape. A majority of current works define a significance measure to prune noises on medial axis. Among them, local measures are widely available due to their efficiency, but can be either too aggressive or conservative. While global measures outperform local ones in differentiating noises from features, they are rarely well-defined or efficient to compute. In this dissertation, we attempt to address these issues with sound, robust and efficient solutions. In Chapter 2, we propose a novel medial axis approximation called voxel core. We show voxel core is topologically and geometrically convergent to the true medial axis. We then describe a straightforward implementation as a result of our simple definition. In a variety of experiments, our method is shown to be efficient and robust in delivering topological promises on a wide range of shapes. In Chapter 3, we present Erosion Thickness (ET) to regularize instability. ET is the first global measure in 3D that is well-defined and efficient to compute. To demonstrate its usefulness, we utilize ET to generate a family of shape revealing and topology preserving skeletons. Finally, we point out future directions, and potential applications of our works in real world problems

Efficient computation of discrete Voronoi diagram and homotopy-preserving simplified medial axis of a 3d polyhedron

The Voronoi diagram is a fundamental geometric data structure and has been well studied in computational geometry and related areas. A Voronoi diagram defined using the Euclidean distance metric is also closely related to the Blum medial axis, a well known skeletal representation. Voronoi diagrams and medial axes have been shown useful for many 3D computations and operations, including proximity queries, motion planning, mesh generation, finite element analysis, and shape analysis. However, their application to complex 3D polyhedral and deformable models has been limited. This is due to the difficulty of computing exact Voronoi diagrams in an efficient and reliable manner. In this dissertation, we bridge this gap by presenting efficient algorithms to compute discrete Voronoi diagrams and simplified medial axes of 3D polyhedral models with geometric and topological guarantees. We apply these algorithms to complex 3D models and use them to perform interactive proximity queries, motion planning and skeletal computations. We present three new results. First, we describe an algorithm to compute 3D distance fields of geometric models by using a linear factorization of Euclidean distance vectors. This formulation maps directly to the linearly interpolating graphics rasterization hardware and enables us to compute distance fields of complex 3D models at interactive rates. We also use clamping and culling algorithms based on properties of Voronoi diagrams to accelerate this computation. We introduce surface distance maps, which are a compact distance vector field representation based on a mesh parameterization of triangulated two-manifolds, and use them to perform proximity computations. Our second main result is an adaptive sampling algorithm to compute an approximate Voronoi diagram that is homotopy equivalent to the exact Voronoi diagram and preserves topological features. We use this algorithm to compute a homotopy-preserving simplified medial axis of complex 3D models. Our third result is a unified approach to perform different proximity queries among multiple deformable models using second order discrete Voronoi diagrams. We introduce a new query called N-body distance query and show that different proximity queries, including collision detection, separation distance and penetration depth can be performed based on Nbody distance query. We compute the second order discrete Voronoi diagram using graphics hardware and use distance bounds to overcome the sampling errors and perform conservative computations. We have applied these queries to various deformable simulations and observed up to an order of magnitude improvement over prior algorithms