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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 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.
Finding the homology of submanifolds with high confidence from random samples
, 2004
"... Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn ” the hom ..."
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Cited by 117 (7 self)
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Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn ” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learningtheoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to selfintersection of the submanifold. We are also able to treat the situation where the data is “noisy ” and lies near rather than on the submanifold in question.
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 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.
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 53 (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.
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 49 (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.
Delaunay triangulation based surface reconstruction: Ideas and algorithms
 EFFECTIVE COMPUTATIONAL GEOMETRY FOR CURVES AND SURFACES
, 2006
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SmoothSurface Reconstruction in Near Linear Time
, 2001
"... A surface reconstruction algorithm takes as input a set of sample points from an unknown closed and smooth surface in 3d space, and produces a piecewise linear approximation of the surface that contains the sample points. Variants of this problem have received considerable attention in computer ..."
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Cited by 42 (6 self)
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A surface reconstruction algorithm takes as input a set of sample points from an unknown closed and smooth surface in 3d space, and produces a piecewise linear approximation of the surface that contains the sample points. Variants of this problem have received considerable attention in computer vision and computer graphics and more recently in computational geometry. In the latter area, three different algorithms (Amenta and Bern `98, and refined in Amenta, Choi, Dey and Leekha `00; Amenta, Choi and Kolluri `00; Boissonnat and Cazals `00) have been proposed. These algorithms have a correctness guarantee: if the sample is sufficiently dense then the output is a good approximation to the original surface. They have unfortunately a worstcase running time that is quadratic in the size of the input. This is so because they are based on the construction of 3d Voronoi diagrams or Delaunay tetrahedrizations, which can have quadratic size. Even worse, according to recent work (Erickson `01), there are surfaces for which this is the case even when the sample set is "locally uniform" on the surface. In this paper, we describe a new algorithm that also has a correctness guarantee but whose worstcase running time is almost linear. In fact, O(n log n) where n is the input size. As in some of the previous algorithms, the piecewise linear approximation produced by the new algorithm is a subset of the 3d Delaunay tetrahedrization; however, this is obtained by computing only the relevant parts of the 3d Delaunay structure. The algorithm first estimates for each sample point the surface normal and a parameter that is then used to "decimate" the set of samples. The resulting subset of sample points is locally uniform and so a reconstruction based on it can be compu...
Isosurface stuffing: Fast tetrahedral meshes with good dihedral angles
 Special issue on Proceedings of SIGGRAPH 2007
, 2007
"... org/10.1145/1239451.1239508. Copyright Notice Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profi t or direct commercial advantage and that copies show this notice on the ..."
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Cited by 42 (3 self)
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org/10.1145/1239451.1239508. Copyright Notice Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profi t or direct commercial advantage and that copies show this notice on the fi rst page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specifi c permission and/or a fee. Permissions may be