Freeform Surface Filtering Using the Diffusion Equation

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22 Citations (Scopus)


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 languageEnglish
Pages (from-to)841-859
Number of pages19
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2127
Publication statusPublished - 8 Mar 2011


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