Freeform surface filtering using the diffusion equation

Jiang, Xiang, Cooper, Philip and Scott, Paul J. (2011) Freeform surface filtering using the diffusion equation. Proceedings of the Royal Society, A, 467 (2127). pp. 841-859. ISSN 1364-5021

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 scale-limited 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 Graphic. 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.

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