### Abstract

The measurement of texture for geometric surfaces is well established for surfaces that are of a planar (Euclidean) nature. Gaussian filtering is the fundamental base for scalelimited surfaces used in surface texture, but cannot be applied to non-Euclidean surfaces without distortion of the results. A link exists between Gaussian filtering and solutions of the PDE that models linear isotropic diffusion. In particular, an analytical solution of this diffusion equation over a planar region at a time t is given by the continuous convolution of the initial distribution of the diffused quantity with a Gaussian function of standard deviation δ =√2t. A practical implementation of the standard Gaussian filter on sampled data can be viewed as a discretization of this process. On a non- Euclidean surface, the diffusion equation is formulated by using the Laplace-Beltrami operator. Using this generalization, a method of Gaussian filtering for freeform surface data is proposed by solving the diffusion equation for approximation residuals defined on a freeform least-squares approximation of the measurement surface data. Results of the application of these methods to simulated and experimental data are presented. This journal is

Original language | English |
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Pages (from-to) | 841-859 |

Number of pages | 19 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 467 |

Issue number | 2127 |

DOIs | |

Publication status | Published - 8 Mar 2011 |

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*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 467, no. 2127, pp. 841-859. https://doi.org/10.1098/rspa.2010.0307

**Freeform Surface Filtering Using the Diffusion Equation.** / Jiang, X.; Scott, P. J.; Cooper, P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Freeform Surface Filtering Using the Diffusion Equation

AU - Jiang, X.

AU - Scott, P. J.

AU - Cooper, P

PY - 2011/3/8

Y1 - 2011/3/8

N2 - The measurement of texture for geometric surfaces is well established for surfaces that are of a planar (Euclidean) nature. Gaussian filtering is the fundamental base for scalelimited surfaces used in surface texture, but cannot be applied to non-Euclidean surfaces without distortion of the results. A link exists between Gaussian filtering and solutions of the PDE that models linear isotropic diffusion. In particular, an analytical solution of this diffusion equation over a planar region at a time t is given by the continuous convolution of the initial distribution of the diffused quantity with a Gaussian function of standard deviation δ =√2t. A practical implementation of the standard Gaussian filter on sampled data can be viewed as a discretization of this process. On a non- Euclidean surface, the diffusion equation is formulated by using the Laplace-Beltrami operator. Using this generalization, a method of Gaussian filtering for freeform surface data is proposed by solving the diffusion equation for approximation residuals defined on a freeform least-squares approximation of the measurement surface data. Results of the application of these methods to simulated and experimental data are presented. This journal is

AB - The measurement of texture for geometric surfaces is well established for surfaces that are of a planar (Euclidean) nature. Gaussian filtering is the fundamental base for scalelimited surfaces used in surface texture, but cannot be applied to non-Euclidean surfaces without distortion of the results. A link exists between Gaussian filtering and solutions of the PDE that models linear isotropic diffusion. In particular, an analytical solution of this diffusion equation over a planar region at a time t is given by the continuous convolution of the initial distribution of the diffused quantity with a Gaussian function of standard deviation δ =√2t. A practical implementation of the standard Gaussian filter on sampled data can be viewed as a discretization of this process. On a non- Euclidean surface, the diffusion equation is formulated by using the Laplace-Beltrami operator. Using this generalization, a method of Gaussian filtering for freeform surface data is proposed by solving the diffusion equation for approximation residuals defined on a freeform least-squares approximation of the measurement surface data. Results of the application of these methods to simulated and experimental data are presented. This journal is

KW - Diffusion Filtering

KW - Freeform Surfaces

KW - Gaussian Scale Space

KW - Surface Metrology

UR - http://www.scopus.com/inward/record.url?scp=79952328801&partnerID=8YFLogxK

U2 - 10.1098/rspa.2010.0307

DO - 10.1098/rspa.2010.0307

M3 - Article

VL - 467

SP - 841

EP - 859

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

IS - 2127

ER -