### 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 |