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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 ..."
Abstract

Cited by 240 (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.
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.
Approximating and Intersecting Surfaces from Points
, 2003
"... 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 poin ..."
Abstract

Cited by 67 (3 self)
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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 covariances 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 zeroset 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 covariances are well defined in a tubular neighborhood of the surface. For certain, wellchosen weight functions we can show that wellsampled surfaces lead to smooth distance fields with nonzero gradients and, thus, the surface is a continuously differentiable manifold. We detail spatial data structures and efficient algorithms to compute raysurface intersections for fast ray casting and ray tracing of the surface.
Approximating Bounded, Nonorientable Surfaces from Points
 In Shape Modeling International
, 2004
"... We present an approach to surface approximation from points that allows reconstructing surfaces with boundaries, including globally nonorientable surfaces. The surface is defined implicitly using directions of weighted covariances and weighted averages of the points. Specifically, a point belongs ..."
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Cited by 23 (3 self)
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We present an approach to surface approximation from points that allows reconstructing surfaces with boundaries, including globally nonorientable surfaces. The surface is defined implicitly using directions of weighted covariances and weighted averages of the points. Specifically, a point belongs to the surface, if its direction to the weighted average has no component into the direction of smallest covariance. For bounded surfaces, we require in addition that any point on the surface is close to the weighted average of the input points. We compare this definition to alternatives and discuss the details and parameter choices. Points on the surface can be determined by intersection computations. We show that the computation is local and, therefore, no globally consistent orientation of normals is needed. Continuity of the surfaces is not affected by the particular choice of local orientation. We demonstrate our approach by rendering several bounded (and nonorientable) surfaces using ray casting.