## E.: Butterworth filtering and implicit fairing of irregular meshes

Venue: | In Proceedings of Pacific Graphics (2003 |

Citations: | 5 - 2 self |

### BibTeX

@INPROCEEDINGS{Zhang_e.:butterworth,

author = {Hao Zhang},

title = {E.: Butterworth filtering and implicit fairing of irregular meshes},

booktitle = {In Proceedings of Pacific Graphics (2003},

year = {},

pages = {502--506}

}

### OpenURL

### Abstract

In this paper, we propose efficient numerical techniques for Butterworth filtering and implicit fairing of large irregular triangle meshes, where the corresponding filters are rational polynomials and the resulting large linear systems need to be solved iteratively. We show that significant speed-up can be achieved for Butterworth filtering by factorizing the linear system in the complex domain. As for implicit fairing, with our estimate of the optimal extrapolation parameter ω, successive overrelaxation (SOR) offers great improvements, both in speed and space usage, over the more familiar conjugate gradient type solvers. 1.

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Citation Context ...iciently, for rendering, modeling, and visualization, while preserving the basic shape and/or features of the original model. A signal processing approach to mesh fairing was first proposed by Taubin =-=[8]-=-, where the mesh geometry is represented as a 3D signal defined over the vertices of the underlying mesh graph. Compared to traditional techniques relying on nonlinear geometric optimization [5], Taub... |

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Citation Context ...not require the computation of inner products, which are communication-intensive [1]. The success of SOR depends critically on the choice of the relaxation parameter ω. In the classical work of Young =-=[11]-=-, the optimal ω, one which achieves the fastest convergence, for the so-called consistently ordered systems is derived. Although the IF system (3) is not consistently ordered, we can show [12] that it... |

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Citation Context ...aubin [8], where the mesh geometry is represented as a 3D signal defined over the vertices of the underlying mesh graph. Compared to traditional techniques relying on nonlinear geometric optimization =-=[5]-=-, Taubin’s application of polynomial low-pass filters [8, 9] is known for its simplicity and efficiency. Such an approach builds upon the premise that the geometric irregularities over a mesh have an ... |

118 |
Laplacian Smoothing and Delaunay Triangulations
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Citation Context ... typically use either polynomials or rational polynomials to approximate the ideal filter. Eugene Fiume Department of Computer Science University of Toronto, Canada The well-known Laplacian smoothing =-=[3]-=- applies a polynomial filter which attenuates all but the zero frequency, causing severe shrinkage and shape distortion. Taubin [8] remedies this with the λ-µ filter, where pass-band frequencies are a... |

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et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition
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Citation Context ... integration. We have observed however that in terms of execution time, the improvement would hardly be obvious since one BCG iteration is about three times as costly as one explicit integration step =-=[1]-=-. Desbrun et al. [2] also suggests that filtering results may be improved by increasing the degree of the denominator of the IIR filter for implicit fairing (we call it the IF filter in this paper) wi... |

43 |
Optimal surface smoothing as filter design
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Citation Context ... signal defined over the vertices of the underlying mesh graph. Compared to traditional techniques relying on nonlinear geometric optimization [5], Taubin’s application of polynomial low-pass filters =-=[8, 9]-=- is known for its simplicity and efficiency. Such an approach builds upon the premise that the geometric irregularities over a mesh have an intuitive frequency-domain characterization. For the ideal l... |

17 |
Matrix iterative Analysis, Second Edition
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Citation Context ...much smoother convergence and achieves about twofold speed-up over BCG quite consistently. SOR for IF filtering: The SOR method uses an extrapolation parameter ω to accelerate Gauss-Seidel iterations =-=[10]-=-. It is trivial to implement SOR so that its per-iteration cost is only about 1/3 and memory requirement about 1/2 (assuming that the average mesh vertex degree is about six) of that of the BCG type s... |

7 |
filtering and implicit fairing of irregular meshes
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- 2003
(Show Context)
Citation Context ...e comparable. In terms of iteration counts however, Butterworth filtering should definitely be preferred. Due to lack of space, we report these findings in detail in an extended version of this paper =-=[12]-=-. Notations: We represent an irregular triangle mesh M with n vertices by its coordinate signal x ∈ R n×3 , where the k-th row of x gives the 3D coordinates of vertex k. The centroid matrix C of M is ... |

3 |
Implicit Fairing of Irregular Meshes using
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Citation Context ...55 7.34 16.02 SOR 0.87 2.14 2.67 3.71 3.15 6.34 Gain% 61% 42% 62% 61% 57% 60% Table 1. Execution time in seconds and gain for SOR. Top block: λ = 10. Bottom: λ = 100. As pointed out by Desbrun et al. =-=[2]-=-, ⌊λ⌋ steps of Laplacian smoothing (explicit integration) produces about the same smoothness result as implicit fairing using λ as the time step. For small values of λ, e.g., λ ≤ 15, ⌊λ⌋ steps of Lapl... |