## Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee (2004)

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Venue: | Algorithmica |

Citations: | 34 - 7 self |

### BibTeX

@ARTICLE{Dey04approximatingthe,

author = {Tamal K. Dey and Wulue Zhao},

title = {Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee},

journal = {Algorithmica},

year = {2004},

volume = {38},

pages = {387--398}

}

### OpenURL

### Abstract

The medial axis of a surface in 3D is the closure of all points that have two or more closest points on the surface. It is an essential geometric structure in a number of applications involving 3D geometric shapes. Since exact computation of the medial axis is difficult in general, efforts continue to improve their approximations. Voronoi diagrams turn out to be useful for this approximation. Although it is known that Voronoi vertices for a sample of points from a curve in 2D approximate its medial axis, similar result does not hold in 3D. Recently, it has been discovered that only a subset of Voronoi vertices converge to the medial axis as sample density approaches infinity. However, most applications need a non-discrete approximation as opposed to a discrete one. To date no known algorithm can compute this approximation straight from the Voronoi diagram with a guarantee of convergence. We present such an algorithm and its convergence analysis in this paper. One salient feature of the algorithm is that it is scale and density independent. Experimental results corroborate our theoretical claims.

### Citations

434 | Three-dimensional alpha shapes
- Edelsbrunner, Mucke
- 1994
(Show Context)
Citation Context ...y a key role in capturing information about shapes. This observation has led to a number of algorithms for the related problem of surface reconstruction which exploit the structures of these diagrams =-=[1, 3, 7, 14, 15]. The ¢ Vo-=-ronoi diagram VP for a point £ 3 set P is a cell complex consisting of � Vp� p� Voronoi cells P and their facets, edges and vertices, where Vp = � x £ The dual complex, DP, called the Delaun... |

342 | Surface reconstruction by Voronoi filtering, Discrete and Computational Geometry 22
- Amenta, Bern
- 1999
(Show Context)
Citation Context ...rs of the flat tetrahedra called ‘slivers’, can come close to the surface no matter how dense a sample is. In order to alleviate this problem in the context of surface reconstruction, Amenta and B=-=ern [1] ide-=-ntify some Voronoi vertices called ‘poles’ that remain far from the surface. These poles are the farthest Voronoi vertices from the sample points in their Voronoi cells. Boissonnat and Cazals [7] ... |

204 | A simple algorithm for homeomorphic surface reconstruction
- Amenta, Choi, et al.
- 2000
(Show Context)
Citation Context ...y a key role in capturing information about shapes. This observation has led to a number of algorithms for the related problem of surface reconstruction which exploit the structures of these diagrams =-=[1, 3, 7, 14, 15]. The ¢ Vo-=-ronoi diagram VP for a point £ 3 set P is a cell complex consisting of � Vp� p� Voronoi cells P and their facets, edges and vertices, where Vp = � x £ The dual complex, DP, called the Delaun... |

177 | Approximating polyhedra with spheres for time-critical collision detection
- Hubbard
- 1995
(Show Context)
Citation Context ...s which has been used in a number of applications including image processing [22], computer vision [9, 23], solid modeling [19, 20, 28], mesh generation [25, 26], motion planning [18] and many others =-=[21, 29]-=-. The shapes in this paper are surfaces embedded in three dimensions. Application demands have prompted research in the computational as well as the mathematical aspects of the medial axis in recent y... |

173 | The power crust, union of balls and the medial axis transform
- Amenta, Choi, et al.
(Show Context)
Citation Context ...method based on octree subdivisions of space. Another scheme considered by many uses a set of sample points on the shape and then approximates the medial axis with the Voronoi diagram of these points =-=[4, 5, 6, 11, 25, 29]-=-. We follow the Voronoi diagram approach. It is particularly suitable for point cloud data, which are increasingly being used for geometric modeling over a wide range of applications. It is known that... |

118 | Smooth surface reconstruction via natural neighbour interpolation of distance functions
- Boissonnat, Cazals
- 2000
(Show Context)
Citation Context ... [1] identify some Voronoi vertices called ‘poles’ that remain far from the surface. These poles are the farthest Voronoi vertices from the sample points in their Voronoi cells. Boissonnat and Caz=-=als [7]-=- and Amenta, Choi and Kolluri [4] show that the poles indeed lie close to the medial axis and converge to it as the sample density approaches infinity. The convergence result of poles to the medial ax... |

66 |
Continuous skeleton computation by Voronoi diagram
- Brandt, Algazi
- 1992
(Show Context)
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56 |
Detecting undersampling in surface reconstruction
- Dey, Giesen
(Show Context)
Citation Context ...undaries where it is ‘jagged’. The poor approximation at these places is caused by the sharp edges of the surface where it is nonsmooth and therefore the inherent problem of undersampling occurs, =-=see [13]-=-. In the FOOT data we zoom some places of the toes and the heel. The zoomed region in the toe has undersampling and the medial axis near the boundary has some roughness. However, the heel is well samp... |

55 | A probabilistic roadmap planner for flexible objects with a workspace medialaxis based sampling approach
- Guibas, Holleman, et al.
- 1999
(Show Context)
Citation Context ...entation of the shapes which has been used in a number of applications including image processing [22], computer vision [9, 23], solid modeling [19, 20, 28], mesh generation [25, 26], motion planning =-=[18]-=- and many others [21, 29]. The shapes in this paper are surfaces embedded in three dimensions. Application demands have prompted research in the computational as well as the mathematical aspects of th... |

54 |
A formal classification of 3d medial axis points and their local geometry
- Giblin, Kimia
- 2000
(Show Context)
Citation Context ...well as the mathematical aspects of the medial axis in recent years. As a mathematical structure they are instable since a small change in shape can cause a relatively large change in its medial axis =-=[17, 30]-=-. They are hard to compute exactly due to numerical instability associated with their computations. Few algorithms, and only for special classes of shapes, have been designed till date to compute the ... |

42 | Skeleton-Space: A Multiscale Shape Description combining Region and Boundary Information
- Ogniewicz
- 1994
(Show Context)
Citation Context ...e than one closest point on the shape. The medial axis provides a compact representation of the shapes which has been used in a number of applications including image processing [22], computer vision =-=[9, 23]-=-, solid modeling [19, 20, 28], mesh generation [25, 26], motion planning [18] and many others [21, 29]. The shapes in this paper are surfaces embedded in three dimensions. Application demands have pro... |

41 | Computing the medial axis of a polyhedron
- Culver
- 2000
(Show Context)
Citation Context ... compute exactly due to numerical instability associated with their computations. Few algorithms, and only for special classes of shapes, have been designed till date to compute the exact medial axis =-=[12, 20]-=-. Consequently, efforts have been made to approximate the medial axis. For polyhedral input Etzion and Rappoport [16] suggest an approximation method based on octree subdivisions of space. Another sch... |

38 | An algorithm for the medial axis transform of 3d polyhedral solids - Sherbrooke, Patrikalakis, et al. - 1996 |

37 |
Computing and simplifying 2d and 3d continuous skeletons
- Attali, Montanvert
- 1997
(Show Context)
Citation Context ...method based on octree subdivisions of space. Another scheme considered by many uses a set of sample points on the shape and then approximates the medial axis with the Voronoi diagram of these points =-=[4, 5, 6, 11, 25, 29]-=-. We follow the Voronoi diagram approach. It is particularly suitable for point cloud data, which are increasingly being used for geometric modeling over a wide range of applications. It is known that... |

35 | Cut locus and medial axis in global shape interrogation and representation. Memorandum 92-2
- Wolter
- 1992
(Show Context)
Citation Context ...well as the mathematical aspects of the medial axis in recent years. As a mathematical structure they are instable since a small change in shape can cause a relatively large change in its medial axis =-=[17, 30]-=-. They are hard to compute exactly due to numerical instability associated with their computations. Few algorithms, and only for special classes of shapes, have been designed till date to compute the ... |

33 | Divergence-based medial surfaces
- Bouix, Siddiqi
- 2000
(Show Context)
Citation Context ...e than one closest point on the shape. The medial axis provides a compact representation of the shapes which has been used in a number of applications including image processing [22], computer vision =-=[9, 23]-=-, solid modeling [19, 20, 28], mesh generation [25, 26], motion planning [18] and many others [21, 29]. The shapes in this paper are surfaces embedded in three dimensions. Application demands have pro... |

30 |
Skeletonbased modeling operations on solids
- Storti, Turkiyyah, et al.
- 1997
(Show Context)
Citation Context ... on the shape. The medial axis provides a compact representation of the shapes which has been used in a number of applications including image processing [22], computer vision [9, 23], solid modeling =-=[19, 20, 28]-=-, mesh generation [25, 26], motion planning [18] and many others [21, 29]. The shapes in this paper are surfaces embedded in three dimensions. Application demands have prompted research in the computa... |

29 | Assisted articulation of closed polygonal models
- Teichmann, Teller
- 1998
(Show Context)
Citation Context ...s which has been used in a number of applications including image processing [22], computer vision [9, 23], solid modeling [19, 20, 28], mesh generation [25, 26], motion planning [18] and many others =-=[21, 29]-=-. The shapes in this paper are surfaces embedded in three dimensions. Application demands have prompted research in the computational as well as the mathematical aspects of the medial axis in recent y... |

27 | Hexahedral mesh generation using the embedded voronoi graph
- Sheffer, Etzion, et al.
- 1999
(Show Context)
Citation Context ... provides a compact representation of the shapes which has been used in a number of applications including image processing [22], computer vision [9, 23], solid modeling [19, 20, 28], mesh generation =-=[25, 26]-=-, motion planning [18] and many others [21, 29]. The shapes in this paper are surfaces embedded in three dimensions. Application demands have prompted research in the computational as well as the math... |

26 |
Shape reconstruction with Delaunay complex
- Edelsbrunner
- 1998
(Show Context)
Citation Context ...y a key role in capturing information about shapes. This observation has led to a number of algorithms for the related problem of surface reconstruction which exploit the structures of these diagrams =-=[1, 3, 7, 14, 15]. The ¢ Vo-=-ronoi diagram VP for a point £ 3 set P is a cell complex consisting of � Vp� p� Voronoi cells P and their facets, edges and vertices, where Vp = � x £ The dual complex, DP, called the Delaun... |

24 | The crust and the /~-skeleton: combinatorial curve reconstruction - Amenta, Bern, et al. - 1998 |

24 |
Generating skeletons and centerlines from the distance transform
- Niblack, Gibbons, et al.
- 1992
(Show Context)
Citation Context ...l points that have more than one closest point on the shape. The medial axis provides a compact representation of the shapes which has been used in a number of applications including image processing =-=[22]-=-, computer vision [9, 23], solid modeling [19, 20, 28], mesh generation [25, 26], motion planning [18] and many others [21, 29]. The shapes in this paper are surfaces embedded in three dimensions. App... |

20 |
Convergence and continuity criteria for discrete approximations of the continuous planar skeleton
- BRANDT
- 1994
(Show Context)
Citation Context ... increasingly being used for geometric modeling over a wide range of applications. It is known that the Voronoi vertices approximate the medial axis of a curve in 2D. In fact, Schmitt [24] and Brandt =-=[10]-=- show that if the sample density approaches infinity, the Voronoi vertices in this case converge to the medial axis. Unfortunately, the same is not true in three dimensions. Amenta, Bern and Eppstein ... |

18 | Computing Voronoi skeletons of a 3-d polyhedron by space subdivision
- Etzion, Rappoport
(Show Context)
Citation Context ...asses of shapes, have been designed till date to compute the exact medial axis [12, 20]. Consequently, efforts have been made to approximate the medial axis. For polyhedral input Etzion and Rappoport =-=[16]-=- suggest an approximation method based on octree subdivisions of space. Another scheme considered by many uses a set of sample points on the shape and then approximates the medial axis with the Vorono... |

17 | Delaunay conforming iso-surface; skeleton extraction and noise removal, Computational Geometry: Theory and Applications
- Attali, Lachaud
(Show Context)
Citation Context |

14 |
Some examples of algorithms analysis in computational geometry by means of mathematical morphological techniques
- Schmitt
- 1989
(Show Context)
Citation Context ... data, which are increasingly being used for geometric modeling over a wide range of applications. It is known that the Voronoi vertices approximate the medial axis of a curve in 2D. In fact, Schmitt =-=[24]-=- and Brandt [10] show that if the sample density approaches infinity, the Voronoi vertices in this case converge to the medial axis. Unfortunately, the same is not true in three dimensions. Amenta, Be... |

14 |
Shape description by medial axis construction
- SHEEHY, ARMSTRONG, et al.
- 1996
(Show Context)
Citation Context ... provides a compact representation of the shapes which has been used in a number of applications including image processing [22], computer vision [9, 23], solid modeling [19, 20, 28], mesh generation =-=[25, 26]-=-, motion planning [18] and many others [21, 29]. The shapes in this paper are surfaces embedded in three dimensions. Application demands have prompted research in the computational as well as the math... |

12 |
Natural neighbor coordinates of points on a surface. Computational Geometry: Theory and Applications
- Boissonnat, Cazals
- 2001
(Show Context)
Citation Context ...hat for a long Delaunay edge pq there must exists a point w £ Dual pq which cannot be too far from a medial axis point. This lemma is extracted from a result (Proposition 18) of Boissonnat and Cazals=-= [8]. Altho-=-ugh we use slightly different version with different constants and exponents, the proof remains same. ¦ § q¦ ¥ ¡s¡ ¡sLemma 8 Let pq be a Delaunay edge with p , where is the radius of a medial b... |

8 |
Automated interrogation and adaptive subdivision of shape using medial axis transform
- Gursoy, Patrikalakis
- 1991
(Show Context)
Citation Context ... on the shape. The medial axis provides a compact representation of the shapes which has been used in a number of applications including image processing [22], computer vision [9, 23], solid modeling =-=[19, 20, 28]-=-, mesh generation [25, 26], motion planning [18] and many others [21, 29]. The shapes in this paper are surfaces embedded in three dimensions. Application demands have prompted research in the computa... |

7 |
How to Construct the Skeleton of CSG Objects
- Hoffman
- 1990
(Show Context)
Citation Context ... on the shape. The medial axis provides a compact representation of the shapes which has been used in a number of applications including image processing [22], computer vision [9, 23], solid modeling =-=[19, 20, 28]-=-, mesh generation [25, 26], motion planning [18] and many others [21, 29]. The shapes in this paper are surfaces embedded in three dimensions. Application demands have prompted research in the computa... |