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Reconstruction and Representation of 3D Objects with Radial Basis Functions
 Computer Graphics (SIGGRAPH ’01 Conf. Proc.), pages 67–76. ACM SIGGRAPH
, 2001
"... We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs al ..."
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Cited by 505 (1 self)
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We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBFpreviously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energyminimisation characterisation of polyharmonic splines result in a "smoothest" interpolant. This scaleindependent characterisation is wellsuited to reconstructing surfaces from nonuniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a noninterpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for realworld rangefinder data.
Surface Reconstruction by Voronoi Filtering
 Discrete and Computational Geometry
, 1998
"... We give a simple combinatorial algorithm that computes a piecewiselinear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled ..."
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Cited by 405 (11 self)
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We give a simple combinatorial algorithm that computes a piecewiselinear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on "local feature size", the output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We describe an implementation of the algorithm and show example outputs. 1 Introduction The problem of reconstructing a surface from scattered sample points arises in many applications such as computer graphics, medical imaging, and cartography. In this paper we consider the specific reconstruction problem in which the input is a set of sample points S drawn from a smooth twodimensional manifold F embedded in three dimensions, and the desired output is a triangular mesh with vertex set equal to S that faithfully represen...
The ballpivoting algorithm for surface reconstruction.
 IEEE TRansactions on Visualization and Computer Graphics,
, 1999
"... ..."
Multilevel Partition of Unity Implicits
 ACM TRANSACTIONS ON GRAPHICS
, 2003
"... We present a shape representation, the multilevel partition of unity implicit surface, that allows us to construct surface models from very large sets of points. There are three key ingredients to our approach: 1) piecewise quadratic functions that capture the local shape of the surface, 2) weighti ..."
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Cited by 218 (7 self)
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We present a shape representation, the multilevel partition of unity implicit surface, that allows us to construct surface models from very large sets of points. There are three key ingredients to our approach: 1) piecewise quadratic functions that capture the local shape of the surface, 2) weighting functions (the partitions of unity) that blend together these local shape functions, and 3) an octree subdivision method that adapts to variations in the complexity of the local shape. Our approach
Delaunay Based Shape Reconstruction from Large Data
, 2001
"... Surface reconstruction provides a powerful paradigm for modeling shapes from samples. For point cloud data with only geometric coordinates as input, Delaunay based surface reconstruction algorithms have been shown to be quite effective both in theory and practice. However, a major complaint against ..."
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Cited by 64 (5 self)
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Surface reconstruction provides a powerful paradigm for modeling shapes from samples. For point cloud data with only geometric coordinates as input, Delaunay based surface reconstruction algorithms have been shown to be quite effective both in theory and practice. However, a major complaint against Delaunay based methods is that they are slow and cannot handle large data. We extend the COCONE algorithm to handle supersize data. This is the first reported Delaunay based surface reconstruction algorithm that can handle data containing more than a million sample points on a modest machine.
Sampling and Reconstructing Manifolds Using AlphaShapes
 In Proc. 9th Canad. Conf. Comput. Geom
, 1997
"... There is a growing interest for the problem of reconstructing the shape of an object from multiple range images. Several methods, based on heuristics, have been described in the literature. We propose the use of alphashapes, which allow us to give a formal characterization of the reconstruction pro ..."
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Cited by 59 (7 self)
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There is a growing interest for the problem of reconstructing the shape of an object from multiple range images. Several methods, based on heuristics, have been described in the literature. We propose the use of alphashapes, which allow us to give a formal characterization of the reconstruction problem and to prove that, when certain sampling requirements are satisfied, the reconstructed alphashape is homeomorphic to the original object and approximate it within a fixed error bound. In a companion paper, we describe practical methods to automatically select an optimal alpha value, to deal with lessthanideal scans, and to fit smooth piecewise algebraic surface to the data points. 1 Introduction Cheaper, easiertouse 3D digitizers are fostering a growing interest for the problem of shapereconstruction. Automatic methods for reconstructing an accurate geometric model of an object from a set of digital scans have applications in reverse engineering, shape analysis, virtual worlds aut...
Interpolation and approximation of surfaces from three–dimensional scattered data points
, 1997
"... There is a wide range of applications for which surface interpolation or approximation from scattered data points in space is important. Dependent on the field of application and the related properties of the data, many algorithms were developed in the past. This contribution gives a survey of exist ..."
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Cited by 46 (1 self)
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There is a wide range of applications for which surface interpolation or approximation from scattered data points in space is important. Dependent on the field of application and the related properties of the data, many algorithms were developed in the past. This contribution gives a survey of existing algorithms, and identifies basic methods common to independently developed solutions. We distinguish surface construction based on spatial subdivision, distance functions, warping, and incremental surface growing. The systematic analysis of existing approaches leads to several interesting open questions for further research. nothing is known about the surface from which the data originate. The task is to find the most reasonable solutions among usually several or even many possibilities. Surface reconstruction means that the surface from which the data are sampled is known, say in form of a real model, and the goal is to get a computerbased description of exactly this surface, cf. figure 1. This knowledge may be used in the selection of a favourable algorithm. A proper reconstruc1
Acquiring Input for Rendering at Appropriate Levels of Detail: Digitizing a Pietà
, 1998
"... We describe the design of a system to augment a light striping camera for three dimensional scanning with a photometric system to capture bump maps and approximate reflectances. In contrast with scanning an object with very high spatial resolution, this allows the relatively efficient and inexpe ..."
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Cited by 40 (6 self)
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We describe the design of a system to augment a light striping camera for three dimensional scanning with a photometric system to capture bump maps and approximate reflectances. In contrast with scanning an object with very high spatial resolution, this allows the relatively efficient and inexpensive acquistion of input for high quality rendering.
A Robust Procedure to Eliminate Degenerate Faces from Triangle Meshes
 VISION, MODELING AND VISUALIZATION (VMV01
, 2001
"... When using triangle meshes in numerical simulations or other sophisticated downstream applications, we have to guarantee that no degenerate faces are present since they have, e.g., no well defined normal vectors. In this paper we present a simple but effective algorithm to remove such artifacts from ..."
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Cited by 22 (1 self)
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When using triangle meshes in numerical simulations or other sophisticated downstream applications, we have to guarantee that no degenerate faces are present since they have, e.g., no well defined normal vectors. In this paper we present a simple but effective algorithm to remove such artifacts from a given triangle mesh. The central problem is to make this algorithm numerically robust because degenerate triangles are usually the source for all kinds of numerical instabilities. Our algorithm is based on a slicing technique that cuts a set of planes through the given polygonal model. The mesh slicing operator only uses numerically stable predicates and therefore is able to split faces in a controlled manner. In combination with a custom tailored mesh decimation scheme we are able to remove the degenerate faces from meshes like those typically generated by tesselation units in CAD systems.
Reconstructing surfaces and functions on surfaces from unorganized 3d data
 Algorithmica
, 1997
"... Creating a computer model from an existing part is a common problem in Reverse Engineering. The part might be scanned with a device like the laser range scanner, or points might be measured on its surface with a mechanical probe. Sometimes, not only the spatial location of points, but also some asso ..."
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Cited by 19 (0 self)
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Creating a computer model from an existing part is a common problem in Reverse Engineering. The part might be scanned with a device like the laser range scanner, or points might be measured on its surface with a mechanical probe. Sometimes, not only the spatial location of points, but also some associated physical property can be measured. The problem of automatically reconstructing from this data a topologically consistent and geometrically accurate model of the object and of the sampled scalar field is the subject of this paper. The algorithm proposed in this paper can deal with connected,orientable manifolds of unrestricted topological type, given a sufficiently dense and uniform sampling of the object’s surface. It is capable of automatically reconstructing both the model and a scalar field over its surface. It uses Delaunay triangulations, Voronoi diagrams and alphashapes for efficiency of computation and theoretical soundness. It generates a representation of the surface and the field based on BernsteinBézier polynomial implicit patches (Apatches), that are guaranteed to be smooth and singlesheeted.