## Curve reconstruction from unorganized points (2000)

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### Other Repositories/Bibliography

Venue: | Computer Aided Geometric Design |

Citations: | 55 - 3 self |

### BibTeX

@ARTICLE{Lee00curvereconstruction,

author = {In-kwon Lee},

title = {Curve reconstruction from unorganized points},

journal = {Computer Aided Geometric Design},

year = {2000},

volume = {17},

pages = {161--177}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present an algorithm to approximate a set of unorganized points with a simple curve without self-intersections. The moving least-squares method has a good ability to reduce a point cloud to a thin curve-like shape which is a near-best approximation of the point set. In this paper, an improved moving least-squares technique is suggested using Euclidean minimum spanning tree, region expansion and refining iteration. After thinning a given point cloud using the improved moving least-squares technique we can easily reconstruct a smooth curve. As an application, a pipe surface reconstruction algorithm is presented.

### Citations

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Citation Context ...es, the medial axis does not represent the shapes of the point set near the end points of the curve. Many approaches have been suggested for the reconstruction of surfaces from unorganized point sets =-=[1, 2, 9, 10, 18]-=-. For curve reconstruction, there are many solutions for cases where the orderingof points is known. However, there has been little research for curve reconstruction from an unorganized point set. Fan... |

470 | Three Dimensional Alpha Shapes
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Citation Context ... (c) Figure 1: Reconstruction of a surface of revolution: (a) data points and estimated normal vectors, (b) projected points and approximatingcurve, and (c) reconstructed surface of revolution. shape =-=[5]-=- which represents the shape of the point set. Then, the medial axis, representing the skeleton of the point set of this polytop is computed. However, the medial axis of a general polytop may have seve... |

372 | A new voronoi-based surface reconstruction algorithm
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Citation Context ...es, the medial axis does not represent the shapes of the point set near the end points of the curve. Many approaches have been suggested for the reconstruction of surfaces from unorganized point sets =-=[1, 2, 9, 10, 18]-=-. For curve reconstruction, there are many solutions for cases where the orderingof points is known. However, there has been little research for curve reconstruction from an unorganized point set. Fan... |

339 |
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Citation Context ...lusion of this work and suggest some future research directions. 2 Background The concept of moving least-squares has been used in many applications such as scattered data interpolation and smoothing =-=[10, 11, 12, 14, 21]-=-. For each data point, a simple curve or surface is computed which fits some neighborhood of the data point using a weighted regression scheme. Then, the data point is moved to a new position on this ... |

326 |
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(Show Context)
Citation Context ... is repeated in both directions until we cannot continue. Figure 11 shows the ordering process of the thin point cloud. After orderingthe points, we can apply conventional curve approximation methods =-=[11]-=- to the ordered points. Figure 12 shows an example of the curve approximation from a thin point cloud with 2000 points generated from the moving least-squares technique withH=1andtwoiterations. Toorde... |

289 | Data structure for soft objects
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Citation Context ...on computed by the inverse-transformation, M −1 ,of(0,c). Instead of usingthe whole point set for each local regression, we can restrict the neighborhood of each point by introducing a cubic function =-=[26]-=-: ⎧ ⎪⎨ 2 wi = ⎪⎩ r3 r2 − 3 +1 ifr<H H3 H2 (5) 0 if r ≥ H, where r = �Pi − P∗� 2 . The weighting function in Equation (5) forces the weights of the points that are outside of the open circle of radius ... |

118 | The approximation power of moving least-squares
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(Show Context)
Citation Context ...pe of the point set. After the original point set is reduced to a sufficiently thin cloud, ordering the points in this thin cloud is not hard as we will show later. The method of moving least-squares =-=[11, 12]-=- is a very powerful tool for the purpose of thinning the point set. The basic idea of moving least-squares is to compute a simple regression curve/surface C i for each data point P i which locally fit... |

89 |
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Citation Context ...; N; i 6= jg. The MST of G is a tree (thus, having no cycle) connecting all points in S so that the sum of its edge lengths is minimum. MST can be computed by well-known Kruskal's or Prim's algorithm =-=[1]-=- in O(N 2 ) time. Instead of using total connectivity graph, we can use Delaunay triangulation (DT) [20] to compute MST due to the well-known fact that MST is a subgraph of DT. Figure 4 shows an examp... |

54 | Drawing contours from arbitrary data points - Mclain - 1974 |

38 |
Two dimensional interpolation from random data, The Computer Journal
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Citation Context ...he points in this thin cloud is not difficult as we will see later. Levin [14] used a method called moving least-squares to thin a point cloud. The moving least-squares method was developed by McLain =-=[16, 17]-=- and used in many applications such as scattered data interpolation and smoothing[12, 13, 14, 16, 17]. The basic idea of movingleast-squares method is to compute a simple regression curve/surface Ci f... |

37 |
Rotational and helical surface approximation for reverse engineering, Computing 60
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(Show Context)
Citation Context ...from a point set is one of the most important problems in the reverse engineering of geometric models. In some cases curve reconstruction plays an important role in the surface reconstruction problem =-=[3, 20, 21, 22]-=-. In this paper, we focus on the reconstruction of a curve from an unorganized point cloud havingno ordering of the point elements. An attractive feature of our solution is that the algorithm can be 1... |

22 | On surface approximation using developable surfaces
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(Show Context)
Citation Context ...rom a point set is one of the most important problems in reverse engineering of geometric models. In some cases, the curve reconstruction plays an important role in the surface reconstruction problem =-=[3, 17, 18, 19]-=-. In this paper, we focus on the reconstruction of a curve from an unorganized point cloud having no ordering of the point elements. An attractive feature of our solution is that the algorithm can be ... |

18 | Reconstructing surfaces and functions on surfaces from unorganized three-dimensional data. Algorithmica
- Bajaj, Bernardini, et al.
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(Show Context)
Citation Context ...al vectors, (b) projected points and approximating curve, (c) reconstructed surface of revolution Several approaches have been suggested for the reconstruction of surfaces from unorganized point sets =-=[2, 7, 8, 15]-=-. For curve reconstruction, there are many solutions in case the ordering of points is known. However, there have been very little researches for curve reconstruction from an unorganized point set. Fa... |

16 |
A graph-based approach to surface reconstruction. Computer Graphics Forum
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(Show Context)
Citation Context ...es, the medial axis does not represent the shapes of the point set near the end points of the curve. Many approaches have been suggested for the reconstruction of surfaces from unorganized point sets =-=[1, 2, 9, 10, 18]-=-. For curve reconstruction, there are many solutions for cases where the orderingof points is known. However, there has been little research for curve reconstruction from an unorganized point set. Fan... |

12 | Reconstruction of kinematic surfaces from scattered data
- Pottmann, Lee, et al.
- 1998
(Show Context)
Citation Context ...from a point set is one of the most important problems in the reverse engineering of geometric models. In some cases curve reconstruction plays an important role in the surface reconstruction problem =-=[3, 20, 21, 22]-=-. In this paper, we focus on the reconstruction of a curve from an unorganized point cloud havingno ordering of the point elements. An attractive feature of our solution is that the algorithm can be 1... |

10 | Geometric least squares fitting of spheres, cylinders, cones and tori
- Lukács, Marshall, et al.
- 1997
(Show Context)
Citation Context ...by several different ways. A simple method is to collect some points from a point set, and computing a torus which locally fits the region. One can use the torus least-squares fitting (Luk'acs et al. =-=[13]-=-) or surface of revolution reconstruction (Pottmann et al. [19]). Once the radius r is found, each data point P i is translated by a vector rN i , where N i is a unit normal vector of P i . If a given... |

7 | Approximation by profile surfaces
- Pottmann, Chen, et al.
- 1998
(Show Context)
Citation Context ...from a point set is one of the most important problems in the reverse engineering of geometric models. In some cases curve reconstruction plays an important role in the surface reconstruction problem =-=[3, 20, 21, 22]-=-. In this paper, we focus on the reconstruction of a curve from an unorganized point cloud havingno ordering of the point elements. An attractive feature of our solution is that the algorithm can be 1... |

7 |
Implicit simplicial models for adaptive curve reconstruction
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(Show Context)
Citation Context ...ieu et al. [4] presented an algorithm for orderingunorganized points assumingthat all points are on the reconstructed curve; thus, this method is not appropriate for a point cloud. Taubin and Ronfard =-=[25]-=- reconstructed a planar curve from unorganized data points using an implicit simplicial curve, which is defined by a planar triangular mesh and the values at the vertices of the mesh. One simple solut... |

7 | Surface Reconstruction Using Alpha Shapes, Computer Graphics Forum - Guo, Menon, et al. - 1997 |

6 | On surface approximation using developable surfaces, Graphical Models and Image Processing 61 - Chen, Lee, et al. - 1999 |

6 | Geometric least-squares of spheres, cylinders, cones and tori - Lukacs, Marshall, et al. - 1997 |

6 | Approximation by pro surfaces - Pottmann, Chen, et al. - 1998 |

4 |
Fitting 3D curves to unorganized data points using deformable curves
- Fang, Gossard
- 1992
(Show Context)
Citation Context ...ve reconstruction, there are many solutions for cases where the orderingof points is known. However, there has been little research for curve reconstruction from an unorganized point set. Fang et al. =-=[6]-=- used a method based on spring energy minimization to approximate an unorganized point set with a curve, which needs a good initial guess of the solution. Dedieu et al. [4] presented an algorithm for ... |

4 |
Mesh-independent surface interpolation, private communication
- Levin
(Show Context)
Citation Context ...ect to the density and shape of the point set. After the original point set is reduced to a sufficiently thin cloud, orderingthe points in this thin cloud is not difficult as we will see later. Levin =-=[14]-=- used a method called moving least-squares to thin a point cloud. The moving least-squares method was developed by McLain [16, 17] and used in many applications such as scattered data interpolation an... |

3 |
Moving weighted least-squares methods
- Lancaster
- 1979
(Show Context)
Citation Context ...lusion of this work and suggest some future research directions. 2 Background The concept of moving least-squares has been used in many applications such as scattered data interpolation and smoothing =-=[10, 11, 12, 14, 21]-=-. For each data point, a simple curve or surface is computed which fits some neighborhood of the data point using a weighted regression scheme. Then, the data point is moved to a new position on this ... |

2 |
Skeletal reconstruction of branching shapes, Computer Graphics Forum 16(5): 283–293
- Ferley, Cani, et al.
- 1997
(Show Context)
Citation Context ...ents the shape of the point set. Then, the medial axis, representing the skeleton of the point set of this polytop is computed. However, the medial axis of a general polytop may have several branches =-=[7]-=- that are difficult to transform into a single curve. The computation of the medial axis of a polytop would be burdened by the numerical problems, especially when the polytop is very complex such as a... |

1 |
Reconstructingsurfaces and functions on surfaces from unorganized 3D data, Algorithmica 19
- Bajaj, Bernardini, et al.
- 1997
(Show Context)
Citation Context |

1 |
On surface approximation usingdevelopable surfaces, Graphical Models and Image Processing
- Chen, Lee, et al.
(Show Context)
Citation Context |

1 |
Algorithms for ordering unorganized points along parametrized curves, Numerical Algorithms 6
- Favardin
- 1994
(Show Context)
Citation Context ...ized point set. Fang et al. [6] used a method based on spring energy minimization to approximate an unorganized point set with a curve, which needs a good initial guess of the solution. Dedieu et al. =-=[4]-=- presented an algorithm for orderingunorganized points assumingthat all points are on the reconstructed curve; thus, this method is not appropriate for a point cloud. Taubin and Ronfard [25] reconstru... |

1 |
Surface reconstruction usingalpha shapes, Computer Graphics Forum 16(4
- Guo, Menon, et al.
- 1997
(Show Context)
Citation Context ... Note that the α-shape of Edelsbrunner et al. [5] is another subset of DT which can represent the appropriate connectivity of the point elements though the parameter α is usually chosen interactively =-=[1, 8]-=-. Now, in the local regression for P∗, we use the EMST to collect the neighboring points. Let A denote a set of neighboring points of P∗ with respect to a distance H. The recursive algorithm in Algori... |

1 |
Mesh optimization, Computer Graphics 27
- Hoppe, DeRose, et al.
- 1993
(Show Context)
Citation Context |

1 |
Movingweighted least-squares methods
- Lancaster
- 1979
(Show Context)
Citation Context ...hod called moving least-squares to thin a point cloud. The moving least-squares method was developed by McLain [16, 17] and used in many applications such as scattered data interpolation and smoothing=-=[12, 13, 14, 16, 17]-=-. The basic idea of movingleast-squares method is to compute a simple regression curve/surface Ci for each data point Pi which locally fits a certain neighborhood of Pi using a weighted regression sch... |

1 |
The approximation power of movingleast-squares
- Levin
- 1998
(Show Context)
Citation Context ...hod called moving least-squares to thin a point cloud. The moving least-squares method was developed by McLain [16, 17] and used in many applications such as scattered data interpolation and smoothing=-=[12, 13, 14, 16, 17]-=-. The basic idea of movingleast-squares method is to compute a simple regression curve/surface Ci for each data point Pi which locally fits a certain neighborhood of Pi using a weighted regression sch... |

1 |
Geometric least-squares fittingof spheres, cylinders, cones, and tori
- Lukács, Marshall, et al.
- 1997
(Show Context)
Citation Context ... several different ways. A simple method is to collect some points from a point set, and then computinga torus which locally fits the region. One can use the torus least-squares fitting(Lukács et al. =-=[15]-=-) or surface of revolution reconstruction (Pottmann et al. [22]). Once the radius r is found, each data point Pi is translated by a vector rNi, whereNiis a unit normal vector of Pi. If a given point s... |

1 |
Drawingcontours from arbitrary data points
- McLain
- 1974
(Show Context)
Citation Context ...he points in this thin cloud is not difficult as we will see later. Levin [14] used a method called moving least-squares to thin a point cloud. The moving least-squares method was developed by McLain =-=[16, 17]-=- and used in many applications such as scattered data interpolation and smoothing[12, 13, 14, 16, 17]. The basic idea of movingleast-squares method is to compute a simple regression curve/surface Ci f... |

1 |
Probability (Springer-Verlag
- Pitman
- 1992
(Show Context)
Citation Context ...movingleast-squares by comparing two results with and without the EMST structure. The size of H must be determined to reflect the thickness of the point cloud. We introduce the concept of correlation =-=[19]-=- developed in probability theory, which can be used to compute the size of an appropriate H. LetXand Y be two random variables. The covariance of X and Y is defined by Cov(X, Y )=E[(X − E(X))(Y − E(Y ... |

1 |
Algorithms: 2nd ed (Addison{Wesley
- Sedgewick
- 1988
(Show Context)
Citation Context ... j}. The EMST of G is a tree (thus, havingno cycle) connectingall points in S so that the sum of its edge lengths is minimum. EMST of G can be computed by the well-known Kruskal’s or Prim’s algorithm =-=[24]-=-. Instead of using a total connectivity graph, we can also use Delaunay triangulation (DT) to compute EMST due to the well-known fact that EMST is a subgraph of DT [23]. Figure 4 shows an example of D... |