## Surface Reconstruction by Voronoi Filtering (1998)

Venue: | Discrete and Computational Geometry |

Citations: | 342 - 11 self |

### BibTeX

@ARTICLE{Amenta98surfacereconstruction,

author = {Nina Amenta and Marshall Bern},

title = {Surface Reconstruction by Voronoi Filtering},

journal = {Discrete and Computational Geometry},

year = {1998},

volume = {22},

pages = {481--504}

}

### Years of Citing Articles

### OpenURL

### Abstract

We give a simple combinatorial algorithm that computes a piecewise-linear 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 two-dimensional manifold F embedded in three dimensions, and the desired output is a triangular mesh with vertex set equal to S that faithfully represen...

### Citations

796 | A volumetric method for building complex models from range images
- Curless, Levoy
- 1996
(Show Context)
Citation Context ...oint as a signed distance function. The zero set of this function is then contoured by a continuous piecewise-linear surface using the marching cubes algorithm. A later algorithm by Curless and Levoy =-=[12]-=- is designed for data samples collected by a laser range scanner. This algorithm sums anisotropically weighted contributions from the samples to compute a signed distance function, which is then discr... |

651 | Surface reconstruction from unorganized points
- HOPPE, DEROSE, et al.
- 1992
(Show Context)
Citation Context ...rithms with guarantees applied only to reconstructing curves in two dimensions. The first and most widely known reconstruction algorithm in the computer graphics community is the work of Hoppe et al. =-=[19, 20, 21]-=-. This algorithm estimates a tangent plane at each sample using the k nearest neighbors, and uses the distance to the plane of the closest sample point as a signed distance function. The zero set of t... |

435 | Three-dimensional alpha shapes
- Edelsbrunner, Mücke
- 1994
(Show Context)
Citation Context ...e algorithm of Hoppe et al. would fail. The most famous computational geometry construction for associating a polyhedral shape with an unorganized set of points is the ff-shape of Edelsbrunner et al. =-=[14, 15]-=-. Like our reconstructed surface, the ff-shape is a subcomplex of the Delaunay triangulation. A Delaunay simplex (edge, face, etc.) belongs to the ff-shape of S if its circumsphere has radius at most ... |

356 | A new Voronoi-based surface reconstruction algorithm
- Amenta, Bern, et al.
- 1998
(Show Context)
Citation Context ...ngulation, and Geomview [23] to visualize and print the results. We used vertices from pre-existing polyhedral models as inputs, in order to compare our results with "ground truth". A compan=-=ion paper [2]-=- reports on our experimental findings. The only tricky part of the implementation was the handling of degeneracies and near degeneracies. Our test examples, many of which started from approximately gr... |

270 | Piecewise smooth surface reconstruction, Proc
- Hoppe, DeRose, et al.
- 1994
(Show Context)
Citation Context ...rithms with guarantees applied only to reconstructing curves in two dimensions. The first and most widely known reconstruction algorithm in the computer graphics community is the work of Hoppe et al. =-=[19, 20, 21]-=-. This algorithm estimates a tangent plane at each sample using the k nearest neighbors, and uses the distance to the plane of the closest sample point as a signed distance function. The zero set of t... |

171 |
On the shape of a set of points in the plane
- Edelsbrunner, Kirkpatrick, et al.
- 1983
(Show Context)
Citation Context ...e algorithm of Hoppe et al. would fail. The most famous computational geometry construction for associating a polyhedral shape with an unorganized set of points is the ff-shape of Edelsbrunner et al. =-=[14, 15]-=-. Like our reconstructed surface, the ff-shape is a subcomplex of the Delaunay triangulation. A Delaunay simplex (edge, face, etc.) belongs to the ff-shape of S if its circumsphere has radius at most ... |

166 | Geometric structures for threedimensional shape representation
- Boissonnat
- 1984
(Show Context)
Citation Context ... resolve sharp dihedral angles. Edelsbrunner and Raindrop Geomagic [13] are continuing to develop ff-shape-based reconstruction along with proprietary extensions. An early algorithm due to Boissonnat =-=[9] is related to -=-ours. He proposed a "sculpting" heuristic for selecting a subset of Delaunay tetrahedra to represent the interior of an object. The heuristic is motivated by the observation that "typic... |

150 | Automatic reconstruction of surfaces and scalar fields from 3D scans - Bajaj, Bernardini, et al. - 1995 |

97 |
Triangulating topological spaces
- Edelsbrunner, Shah
- 1997
(Show Context)
Citation Context ...oint of the medial axis. We now give the proof of Theorem 2: the good triangles form a polyhedron homeomorphic to F . The proof relies on the lemmas above along with a result of Edelsbrunner and Shah =-=[16]-=-. Proof of Theorem 2: The theorem of Edelsbrunner and Shah tells us that it suffices to show that S has the following closed-ball property : the closure of each k-dimensional face, 1sks3, of the Voron... |

85 |
A framework for computational morphology
- Kirkpatrick, Radke
- 1985
(Show Context)
Citation Context ...he Delaunay triangulation in which each edge is included or excluded according to the angle between the circumcircles on either side. Our previous paper showed that both the crust and the fi-skeleton =-=[22]-=- (another emptyregion planar graph) correctly reconstruct curves even with nonuniform sampling. Our two-dimensional results [1] are in this way strictly stronger than those of the other authors. 3 Des... |

80 | Nonlinear manifold learning for visual speech recognition
- Bregler, S
- 1995
(Show Context)
Citation Context ...ensions. Manifold learning is the problem of reconstructing a smooth k-dimensional manifold embedded in IR d . This problem arises in modeling unknown dynamical systems from experimental observations =-=[10]-=-. Even if Voronoi filtering can be generalized to this problem, its running time for the important case in which ksd would not be competitive with algorithms that compute triangulations only in k-dime... |

77 | r-regular shape reconstruction from unorganized points
- Attali
- 1998
(Show Context)
Citation Context ...an minimum spanning tree can be used to reconstruct uniformly sampled curves in the plane. Bernardini and Bajaj [6] prove that ff-shapes also reconstruct uniformly sampled curves in the plane. Attali =-=[3]-=- gives yet another provably correct reconstruction algorithm for uniformly sampled curves in the plane, using a subgraph of the Delaunay triangulation in which each edge is included or excluded accord... |

65 | Piecewise-linear interpolation between polygonal slices - Barequet, Sharir - 1994 |

34 | Automatic reconstruction of 3d cad models from digital scans
- Bernardini, Bajaj, et al.
- 1997
(Show Context)
Citation Context ...t the optimal value of ff depends on the sampling density, which often varies over different 2 parts of the surface. For uniformly sampled surfaces, however, ff-shapes are workable. Bernardini et al. =-=[7]-=- follow ff-shape-based reconstruction with a clean-up phase to resolve sharp dihedral angles. Edelsbrunner and Raindrop Geomagic [13] are continuing to develop ff-shape-based reconstruction along with... |

26 | Closed Object Boundaries from Scattered Points - Veltkamp - 1992 |

19 |
Constructing discrete medial axis of 3-d objects
- Goldak, Yu, et al.
- 1991
(Show Context)
Citation Context ...an give Delaunay tetrahedra with circumspheres that are arbitrarily far from the medial axis; indeed it is this second observation which motivates our definition of poles. Goldak, Yu, Knight and Dong =-=[18]-=- made a similar oversight, asserting incorrectly that the Voronoi diagram vertices asymptotically approach the medial axis as the sampling density goes to infinity. Finally, for the two-dimensional pr... |

19 | A survey of parametric scattered data fitting using triangular interpolants. Curve and surface Design - SIAM - Mann, Lounsbery, et al. - 1992 |

16 |
Miranda Gomes, “Computational morphology of curves
- Figueiredo, de
- 1995
(Show Context)
Citation Context ...ly approach the medial axis as the sampling density goes to infinity. Finally, for the two-dimensional problem there are a few recent algorithms with provable guarantees. Figueiredo and Miranda Gomes =-=[17]-=- prove that the Euclidean minimum spanning tree can be used to reconstruct uniformly sampled curves in the plane. Bernardini and Bajaj [6] prove that ff-shapes also reconstruct uniformly sampled curve... |

15 | Surface reconstruction by wrapping finite sets in space
- Edelsbrunner
- 1996
(Show Context)
Citation Context ...ampled surfaces, however, ff-shapes are workable. Bernardini et al. [7] follow ff-shape-based reconstruction with a clean-up phase to resolve sharp dihedral angles. Edelsbrunner and Raindrop Geomagic =-=[13] are conti-=-nuing to develop ff-shape-based reconstruction along with proprietary extensions. An early algorithm due to Boissonnat [9] is related to ours. He proposed a "sculpting" heuristic for selecti... |

10 |
The crust and the fi-skeleton: Combinatorial curve reconst ruc tion
- Amenta, Bern, et al.
- 1998
(Show Context)
Citation Context ... error tolerance. The algorithm relies on the well-known constructions of the Delaunay triangulation and the Voronoi diagram. This paper is an extension of previous work by Amenta, Bern, and Eppstein =-=[1] on recons-=-tructing curves in two dimensions. Our previous work defined a planar graph on the sample points called the "crust". The crust is the set of edges in the Delaunay triangulation of the sample... |

10 | Triangulation-based object reconstruction methods, in - Bernardini, Bajaj, et al. - 1997 |

4 |
Sampling and reconstructing manifolds using ff-shapes
- Bernardini, Bajaj
- 1997
(Show Context)
Citation Context ...ithms with provable guarantees. Figueiredo and Miranda Gomes [17] prove that the Euclidean minimum spanning tree can be used to reconstruct uniformly sampled curves in the plane. Bernardini and Bajaj =-=[6]-=- prove that ff-shapes also reconstruct uniformly sampled curves in the plane. Attali [3] gives yet another provably correct reconstruction algorithm for uniformly sampled curves in the plane, using a ... |

4 |
A program for convex hulls. http://cm.bell-labs.com/netlib/voronoi /hull.html
- Clarkson
(Show Context)
Citation Context ...point on F . 6 Implementation and Examples Manolis Kamvysselis, an undergraduate from MIT, implemented steps 1--4 of the crust algorithm during a summer at Xerox PARC. We used Clarkson's Hull program =-=[11] for Delau-=-nay triangulation, and Geomview [23] to visualize and print the results. We used vertices from pre-existing polyhedral models as inputs, in order to compare our results with "ground truth". ... |

2 | Scattered Data Techniques for Surfaces. To appear in Geometry Detection, Estimation and Synthesis for Scientific Visualization - Lodha |