## Estimation of Euler Characteristic from Point Data

Citations: | 4 - 0 self |

### BibTeX

@MISC{Klain_estimationof,

author = {Daniel A. Klain and Konstantin Rybnikov and Karen Daniels and Bradford Jones and Cristina Neacsu},

title = {Estimation of Euler Characteristic from Point Data },

year = {}

}

### OpenURL

### Abstract

Determination of the geometry and topology of a 3-dimensional body is an important problem

### Citations

449 |
Elements of Algebraic Topology
- Munkres
- 1984
(Show Context)
Citation Context ...ase fi stands for the numbers of simplices (cells) of dimension i. If one chooses this route, the proof of invariance with respect to the choice of triangulation (cell-partition) can be found in e.g. =-=[Mun84]-=-). In analogy to the Fubini theorem (or Cavalieri principle) for Euclidean volume, the Euler characteristic is also multiplicative over direct products: If K ∈ A in Rk , L ∈ A in Rl , then the Euler c... |

237 | Integral Geometry and Geometric Probability - Santaló - 1976 |

201 |
Algebraic Geometry
- Bochnak, Coste, et al.
- 1998
(Show Context)
Citation Context ...or any k ∈ Z≥0. Of course, it is not clear from the above definition that χ exists. The proof of existence of χ for OpenConv can be found in [KR97], the proof of existence for SemiAlg can be found in =-=[JB98]-=-, and that for SubAn in [Loj93]. Since AC has a generating system of disjoint cells, then by defining χ on cells we define it on all of A – the Euler characteristic of any K ∈ A, which is not a cell, ... |

97 |
Introduction to Geometric Probability
- Klain, Rota
- 1997
(Show Context)
Citation Context ...ive measure, which is defined on a k-cell C ∈ C as (−1) k for any k ∈ Z≥0. Of course, it is not clear from the above definition that χ exists. The proof of existence of χ for OpenConv can be found in =-=[KR97]-=-, the proof of existence for SemiAlg can be found in [JB98], and that for SubAn in [Loj93]. Since AC has a generating system of disjoint cells, then by defining χ on cells we define it on all of A – t... |

89 |
Sur les points singuliers d’une forme de Pfaff completement integrable ou d’une fonction numerique, Comptes Rendus Acad. Sciences Paris 222:847–849
- Reeb
- 1946
(Show Context)
Citation Context ...algorithm based on the idea of construction of a Reeb graph. Let K ⊂ R2 ,lethd : K −→ R be the height function hd(x) =d · x for the direction d ∈ R2 ,andletG(hd) be the graph of hd in K × R. The Reeb =-=[Ree46]-=- complex of K for hd, which we denote by Reebd(K), is G(hd) factored by the following equivalence relation: (x,hd(x)) ∼ (x1,hd(x1)) iff hd(x) =hd(x1) andbothxand x1 lie in the same connected component... |

71 | Removing excess topology from isosurfaces - Wood, Hoppe, et al. - 2004 |

54 | Some integral calculus based on Euler characteristic in: Topology and GeometryRohlin - Viro - 1988 |

52 | Shape segmentation and matching with flow discretization
- Dey, Giesen, et al.
- 2003
(Show Context)
Citation Context ...least 4 times the length of the minimal vector of the lattice we can find the exact value of χ(K) in a mere 7 seconds on an average 32-bit 2.8GHz Xeon workstation. To compare, Dey, Giesen and Goswami =-=[DGG03]-=- report that, on a 755MHz Pentium III with 512MB of RAM, reconstruction of a shape from a sampling of 2 20 points takes 53 minutes. The fastest code that we know of (as of July 2005) for construction ... |

52 | Delaunay based shape reconstruction from large data
- Dey, Giesen, et al.
- 2005
(Show Context)
Citation Context ...l algorithms attempting to reconstruct a body from a point sampling, on faster hardware takes at least 1 minute (Dey, Giesen and Hudson et al. report 70.5 sec for 120,000 points and 53 sec for 95,000 =-=[DGH01]-=-). While Section 1 provides a brief theoretical description of the Euler characteristic, more detailed treatments can be found in [KR97, Mun84, Vir88]. Section 1.0.2 presents a recursive technique for... |

48 |
Deformable smooth surface design
- Edelsbrunner
- 1999
(Show Context)
Citation Context ...nd many reconstruction methods have been suggested – e.g., methods based on Fourier analysis, methods based on weighted Delaunay or Voronoi complexes, such as e.g. the α-shapes method of Edelsbrunner =-=[Ede99]-=-, etc. However, if the number of sample points is very large, a piecewise-linear reconstruction is not always possible – for example, at present it is impossible to use any methods based on Delaunay c... |

24 | Time-varying Reeb graphs for continuous space-time data - Edelsbrunner, Harer, et al. |

20 | Compact representations of simplicial meshes in two and three dimensions
- Blandford, Blelloch, et al.
- 2003
(Show Context)
Citation Context ... of Delaunay triangulations finds Delaunay triangulation for a uniformly distributed set of 2 20 points in a unit cube in about 200 seconds on hardware and software practically identical to ours (see =-=[BBCK03]-=-). 1sFor uniform and Poisson data our methods require more runs, but still, for a 100,000 uniform point sampling, with density 100 or larger, from a 3D body with local feature size 1 our algorithm pro... |

13 |
la géométrie semi- et sous-analytique
- Lojasiewicz, Sur
- 1993
(Show Context)
Citation Context ...is not clear from the above definition that χ exists. The proof of existence of χ for OpenConv can be found in [KR97], the proof of existence for SemiAlg can be found in [JB98], and that for SubAn in =-=[Loj93]-=-. Since AC has a generating system of disjoint cells, then by defining χ on cells we define it on all of A – the Euler characteristic of any K ∈ A, which is not a cell, can be computed by the inclusio... |

12 | Asymptotic moments of near neighbour distance distributions
- Evans, Jones, et al.
- 2002
(Show Context)
Citation Context ...n the expectation of the distance to the k-th nearest neighbor EN,k is given by (6) V −1/n Γ(k +1/n) n Γ(k)N 1/n � ϕ(x) P 1−1/n 1 dx + O( ), N 2/n−ρ where Vn is the volume of the unit ball in Rn (see =-=[EJS02]-=-). It is sufficient if the following requirements are met: 1) ϕ is continuous, 2) partial derivatives of ϕ exist and are bounded in absolute value, 3) ϕ>0. For the precise conditions on the body P see... |

8 | Scaling universalities of kth nearest neighbor distances on closed manifolds
- Percus, Martin
- 1998
(Show Context)
Citation Context ...ies and all bodies considered in our experiments satisfy them. If P is a cube and the distances are measured under periodic boundary conditions, the second 1 term can be improved to O( N 1+1/n ) (see =-=[PM98]-=-). For data coming from applications EN,k should be estimated by taking the median of the distance to the k-th nearest neighbor. In practice, the median estimate agrees with the theoretical estimate m... |