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156
Point Set Surfaces
, 2001
"... We advocate the use of point sets to represent shapes. We provide a definition of a smooth manifold surface from a set of points close to the original surface. The definition is based on local maps from differential geometry, which are approximated by the method of moving least squares (MLS). We pre ..."
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Cited by 241 (34 self)
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We advocate the use of point sets to represent shapes. We provide a definition of a smooth manifold surface from a set of points close to the original surface. The definition is based on local maps from differential geometry, which are approximated by the method of moving least squares (MLS). We present tools to increase or decrease the density of the points, thus, allowing an adjustment of the spacing among the points to control the fidelity of the representation. To display the point set surface, we introduce a novel point rendering technique. The idea is to evaluate the local maps according to the image resolution. This results in high quality shading effects and smooth silhouettes at interactive frame rates.
The Power Crust
, 2001
"... The power crust is a construction which takes a sample of points from the surface of a threedimensional object and produces a surface mesh and an approximate medial axis. The approach is to first approximate the medial axis transform (MAT) of the object. We then use an inverse transform to produce ..."
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Cited by 201 (6 self)
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The power crust is a construction which takes a sample of points from the surface of a threedimensional object and produces a surface mesh and an approximate medial axis. The approach is to first approximate the medial axis transform (MAT) of the object. We then use an inverse transform to produce the surface representation from the MAT.
The Power Crust, Unions of Balls, and the Medial Axis Transform
 Computational Geometry: Theory and Applications
, 2000
"... The medial axis transform (or MAT) is a representation of an object as an infinite union of balls. We consider approximating the MAT of a threedimensional object, and its complement, with a finite union of balls. Using this approximate MAT we define a new piecewiselinear approximation to the objec ..."
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Cited by 172 (5 self)
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The medial axis transform (or MAT) is a representation of an object as an infinite union of balls. We consider approximating the MAT of a threedimensional object, and its complement, with a finite union of balls. Using this approximate MAT we define a new piecewiselinear approximation to the object surface, which we call the power crust. We assume that we are given as input a suficiently dense sample of points from the object surface. We select a subset of the Voronoi balls of the sample, the polar balls, as the union of balls representation. We bound the geometric error of the union, and of the corresponding power crust, and show that both representations are topologically correct as well. Thus, our results provide a new algorithm for surface reconstruction from sample points. By construction, the power crust is always the boundary of a solid, so we avoid the holefilling or manifold extraction steps used in previous algorithms. The union of balls representation and the power crust have corresponding piecewiselinear dual representations, which in some sense approximate the medial axis. We show a geometric relationship between these duals and the medial axis by proving that, as the sampling density goes to infinity, the set of poles, the centers of the polar balls, converge to the medial axis.
Computing and Rendering Point Set Surfaces
, 2002
"... We advocate the use of point sets to represent shapes. We provide a definition of a smooth manifold surface from a set of points close to the original surface. The definition is based on local maps from differential geometry, which are approximated by the method of moving least squares (MLS). The co ..."
Abstract

Cited by 167 (20 self)
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We advocate the use of point sets to represent shapes. We provide a definition of a smooth manifold surface from a set of points close to the original surface. The definition is based on local maps from differential geometry, which are approximated by the method of moving least squares (MLS). The computation of points on the surface is local, which results in an outofcore technique that can handle any point set.
Filling Holes In Complex Surfaces Using Volumetric Diffusion
, 2001
"... We address the problem of building watertight 3D models from surfaces that contain holesfor 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 al ..."
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Cited by 137 (1 self)
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We address the problem of building watertight 3D models from surfaces that contain holesfor 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 knownempty 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 noninterpenetrating 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 holefree surfaces that are plausible, visually acceptable, and usually close to the intended geometry.
OutofCore Simplification of Large Polygonal Models
, 2000
"... We present an algorithm for outofcore simplification of large polygonal datasets that are too complex to fit in main memory. The algorithm extends the vertex clustering scheme of Rossignac and Borrel [13] by using error quadric information for the placement of each cluster's representative vertex, ..."
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Cited by 133 (9 self)
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We present an algorithm for outofcore simplification of large polygonal datasets that are too complex to fit in main memory. The algorithm extends the vertex clustering scheme of Rossignac and Borrel [13] by using error quadric information for the placement of each cluster's representative vertex, which better preserves fine details and results in a low mean geometric error. The use of quadrics instead of the vertex grading approach in [13] has the additional benefits of requiring less disk space and only a single pass over the model rather than two. The resulting linear time algorithm allows simplification of datasets of arbitrary complexity. In order
Tight cocone: A watertight surface reconstructor
 JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING
, 2003
"... Surface reconstruction from unorganized sample points is an important problem in computer graphics, computer aided design, medical imaging and solid modeling. Recently a few algorithms have been developed that have theoretical guarantee of computing a topologically correct and geometrically close su ..."
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Cited by 106 (5 self)
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Surface reconstruction from unorganized sample points is an important problem in computer graphics, computer aided design, medical imaging and solid modeling. Recently a few algorithms have been developed that have theoretical guarantee of computing a topologically correct and geometrically close surface under certain condition on sampling density. Unfortunately, this sampling condition is not always met in practice due to noise, nonsmoothness or simply due to inadequate sampling. This leads to undesired holes and other artifacts in the output surface. Certain CAD applications such as creating a prototype from a model boundary require a watertight surface, i.e., no hole should be allowed in the surface. In this paper we describe a simple algorithm called Tight Cocone that works on an initial mesh generated by a popular surface reconstruction algorithm and fills up all holes to output a watertight surface. In doing so, it does not introduce any extra points and produces a triangulated surface interpolating the input sample points. In support of our method we present experimental results with a number of difficult data sets.
Building a Digital Model of Michelangelo's Florentine PietÃ
 IEEE Computer Graphics and Applications
, 2002
"... We describe a project to create a threedimensional digital model of Michelangelo's Florentine Piet a. The model is being used in a comprehensive arthistorical study of this sculpture that includes a consideration of historical records and artistic significance as well as scientific data. ..."
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Cited by 80 (4 self)
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We describe a project to create a threedimensional digital model of Michelangelo's Florentine Piet a. The model is being used in a comprehensive arthistorical study of this sculpture that includes a consideration of historical records and artistic significance as well as scientific data.
Surface Reconstruction based on Lower Dimensional Localized Delaunay Triangulation
, 2000
"... We present a fast, memory efficient algorithm that generates a manifold triangular mesh S passing through a set of unorganized points P #R 3 . Nothing is assumed about the geometry, topology or presence of boundaries in the data set except that P is sampled from a real manifold surface. The spe ..."
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Cited by 73 (5 self)
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We present a fast, memory efficient algorithm that generates a manifold triangular mesh S passing through a set of unorganized points P #R 3 . Nothing is assumed about the geometry, topology or presence of boundaries in the data set except that P is sampled from a real manifold surface. The speed of our algorithm is derived from a projectionbased approach we use to determine the incident faces on a point. We define our sampling criteria to sample the surface and guarantee a topologically correct mesh after surface reconstruction for such a sampled surface. We also present a new algorithm to find the normal at a vertex, when the surface is sampled according our given criteria. We also present results of our surface reconstruction using our algorithm on unorganized point clouds of various models. 1. Introduction The problem of surface reconstruction from unorganized point clouds has been, and continues to be, an important topic of research. The problem can be loosely stated ...