We address the problem of building watertight 3D models from surfaces that contain holes---for example, sets of range scans that observe most but not all of a surface. We specifically address situations in which the holes are too geometrically and topologically complex to fill using triangulation algorithms. Our solution begins by constructing a signed distance function, the zero set of which defines the surface. Initially, this function is defined only in the vicinity of observed surfaces. We then apply a diffusion process to extend this function through the volume until its zero set bridges whatever holes may be present. If additional information is available, such as known-empty regions of space inferred from the lines of sight to a 3D scanner, it can be incorporated into the diffusion process. Our algorithm is simple to implement, is guaranteed to produce manifold non-interpenetrating surfaces, and is efficient to run on large datasets because computation is limited to areas near holes. By showing results for complex range scans, we demonstrate that our algorithm produces hole-free surfaces that are plausible, visually acceptable, and usually close to the intended geometry.

Point sets become an increasingly popular shape representation. Most shape processing and rendering tasks require the approximation of a continuous surface from the point data. We present a surface approximation that is motivated by an efficient iterative ray intersection computation. On each point on a ray, a local normal direction is estimated as the direction of smallest weighted co-variances of the points. The normal direction is used to build a local polynomial approximation to the surface, which is then intersected with the ray. The distance to the polynomials essentially defines a distance field, whose zero-set is computed by repeated ray intersection. Requiring the distance field to be smooth leads to an intuitive and natural sampling criterion, namely, that normals derived from the weighted co-variances are well defined in a tubular neighborhood of the surface. For certain, well-chosen weight functions we can show that well-sampled surfaces lead to smooth distance fields with non-zero gradients and, thus, the surface is a continuously differentiable manifold. We detail spatial data structures and efficient algorithms to compute ray-surface intersections for fast ray casting and ray tracing of the surface.

It is well known that the complexity of the Delaunay trian-gulation of N points in 3, i.e. the number of its faces, can be (N2). The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms rst construct the Delau-nay 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 generic smooth sur-faces of 3. Under a mild uniform sampling condition, we show that the complexity of the 3D Delaunay triangulation of the points is O(N log N). Categories and Subject Descriptors F.2.2 [Theory of Computation]: Analysis of Algorithms and Problem Complexity|Geometrical problems and com-

Abstract 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. Usage. CGAL is used in a diverse range of domains requiring geometric computation such as computer graphics, scientific visualization, computer aided design and modeling, geographic information systems, molecular biology, medical imaging, and many more. Since CGAL provides a wide range of components, we restrict ourselves to mentioning just a few here. As an example application of CGAL, a series of packages are provided which are useful in robotics and automation: Minkowski sums, offset polygons, Boolean operations on curved regions. The high precision of CGAL allows users to solve geometric problems involving motion in restricted environments, such as those arising in assembly planning. The robustness and efficiency of components such as the Delaunay triangulation and mesh construction and manipulation packages makes CGAL attractive for simulations, in particular those involving proteins, particle physics, fluid dynamics, medical modeling, biophysics, geophysics, and astronomy. Indeed, the aforementioned components are largely used in these areas. Some support for manipulations of polynomials and for solving univariate polynomial equations and bivariate polynomial systems is also provided, as well as handling for convex quadratic programs. History of the CGAL Open Source Project. Several European research groups started to develop their own small geometry libraries in the early 90's. In 1996, a consortium of eight sites was created to gather the work of these groups into a single software library, namely CGAL. Their main goal was to promote research in computational geometry and to translate the results into robust software suitable for industrial applications. Around this time the Computational Geometry Impact Task Force Report [C + 96, C + 99] made a series of recommendations. Amongst these recommendations, the production and distribution of usable (and useful) geometric software, and the need to establish a reward structure for software implementations in academia, were key. On November 2003, when version 3.0 was released, CGAL officially became an Open Source project, allowing new contributors to join the project. License. CGAL is distributed under the GPL license (apart from a few basic parts, which are distributed under the LGPL license). In particular, it is publicly and freely available for academic use. Commercial licenses are offered by Geometry Factory, a company founded in 2003 mainly for this purpose. *

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Within the field of computer graphics and visualization, it is often necessary to visualize polygonal models with large number of polygons. Display quality is mandatory, but it is also desirable to have the ability to rapidly update the display in order to facilitate interactive use. Point based rendering methods have been shown effective for this task. Building on this paradigm we introduce the PMR system which uses a hierarchy both in points and triangles for rendering. This hierarchy is fundamentally different from the ones used in existing methods. It is based on the feature geometry in the object space rather than its projection in the screen space. This provides certain advantages over the existing methods.

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In this paper we present techniques for the efficient generation of a level-of-detail (LOD) data structure for large scale point-based surface representation and rendering. Our approach generates a spatial partitioning hierarchy of irregular point samples in 3D space, and we provide an efficient point-octree LOD generation algorithm. Using the concept of transformation-invariant homogeneous covariance matrices we show how bounding ellipsoid information can efficiently be computed for all LODs. Furthermore, we present an efficient data structure for the representation of the LOD hierarchy.

Modern scanning devices allow to obtain a dense sample of discrete points from the surface of a physical object. A piecewise linear surface interpolating these points is computed to reconstruct the sampled surface. Typically such surfaces have a large combinatorial description since the input is usually too dense. In this paper we present an algorithm to decimate the samples to eliminate oversampling. The algorithm decimates the sample with the guarantee that the remaining points are sufficient to reconstruct the surface and has a density controlled by an user input.

3D scanners developed over the past several decades have fa-cilitated the reconstruction of complicated engineering parts. Typically the boundary representation of a part is recon-structed from its scanned cloud of points. This approach, however, is still limited and cannot be applied to a family of objects such as thin parts. Recently, new 3D scanning de-vices have been developed. These devices capture additional information, such as normals and texture, as well as conven-tional information, including clouds of sampled points. This paper describes a new and fast reverse engineer-ing method for creating a 3D computerized model from data captured by contemporary 3D scanning devices. The proposed method aggregates large-scale 3D scanned data into an extended Hierarchical Space Decomposition Model