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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 203 (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.
Curveskeleton properties, applications, and algorithms
 IEEE Transactions on Visualization and Computer Graphics
, 2007
"... Curveskeletons are thinned 1D representations of 3D objects useful for many visualization tasks including virtual navigation, reducedmodel formulation, visualization improvement, animation, etc. There are many algorithms in the literature describing extraction methodologies for different applicati ..."
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Cited by 69 (3 self)
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Curveskeletons are thinned 1D representations of 3D objects useful for many visualization tasks including virtual navigation, reducedmodel formulation, visualization improvement, animation, etc. There are many algorithms in the literature describing extraction methodologies for different applications; however, it is unclear how general and robust they are. In this paper, we provide an overview of many curveskeleton applications and compile a set of desired properties of such representations. We also give a taxonomy of methods and analyze the advantages and drawbacks of each class of algorithms.
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 ..."
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Cited by 68 (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.
Ray Tracing Point Set Surfaces
, 2003
"... Point set surfaces are a smooth manifold surface approximation from a set of sample points. The surface definition is based on a projection operation that constructs local polynomial approximations and respects a minimum feature size. We present techniques for ray tracing point set surfaces. For the ..."
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Cited by 54 (2 self)
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Point set surfaces are a smooth manifold surface approximation from a set of sample points. The surface definition is based on a projection operation that constructs local polynomial approximations and respects a minimum feature size. We present techniques for ray tracing point set surfaces. For the computation of raysurface intersection the properties of the projection operation are exploited: The surface is enclosed by a union of minimum feature size spheres. A ray is intersected with the spheres first and inside the spheres with local polynomial approximations. Our results show that 23 projections are sufficient to accurately intersect a ray with the surface.
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 52 (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.
Delaunay triangulation based surface reconstruction: Ideas and algorithms
 EFFECTIVE COMPUTATIONAL GEOMETRY FOR CURVES AND SURFACES
, 2006
"... ..."
On normals and projection operators for surfaces defined by point sets
 IN EUROGRAPHICS SYMP. ON POINTBASED GRAPHICS
, 2004
"... Levin’s MLS projection operator allows defining a surface from a set of points and represents a versatile procedure to generate points on this surface. Practical problems of MLS surfaces are a complicated nonlinear optimization to compute a tangent frame and the (commonly overlooked) fact that the ..."
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Cited by 48 (3 self)
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Levin’s MLS projection operator allows defining a surface from a set of points and represents a versatile procedure to generate points on this surface. Practical problems of MLS surfaces are a complicated nonlinear optimization to compute a tangent frame and the (commonly overlooked) fact that the normal to this tangent frame is not the surface normal. An alternative definition of Point Set Surfaces, inspired by the MLS projection, is the implicit surface version of Adamson & Alexa. We use this surface definition to show how to compute exact surface normals and present simple, efficient projection operators. The exact normal computation also allows computing orthogonal projections.
Nice Point Sets Can Have Nasty Delaunay Triangulations
 In Proc. 17th Annu. ACM Sympos. Comput. Geom
, 2001
"... We consider the complexity of Delaunay triangulations of sets of points in IR 3 under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of u points in ..."
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Cited by 47 (5 self)
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We consider the complexity of Delaunay triangulations of sets of points in IR 3 under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of u points in IR 3 with spread A has complexity il(min{A 3 , uA, u2}) and O (min{A 4, u2}). For the case A = D(v/), our lower bound construction consists of a gridlike sample of a right circular cylinder with constant height and radius. We also construct a family of smooth connected surfaces such that the Delaunay triangulation of any good point sample has nearquadratic complexity.
Retrieving articulated 3D models using medial surfaces
, 2008
"... We consider the use of medial surfaces to represent symmetries of 3D objects. This allows for a qualitative abstraction based on a directed acyclic graph of components and also a degree of invariance to a variety of transformations including the articulation of parts. We demonstrate the use of this ..."
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Cited by 39 (3 self)
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We consider the use of medial surfaces to represent symmetries of 3D objects. This allows for a qualitative abstraction based on a directed acyclic graph of components and also a degree of invariance to a variety of transformations including the articulation of parts. We demonstrate the use of this representation for 3D object model retrieval. Our formulation uses the geometric information