Freeform Surface Filtering Using the Diffusion Equation

Research output: Contribution to journalArticle

15 Citations (Scopus)

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 languageEnglish
Pages (from-to)841-859
Number of pages19
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume467
Issue number2127
DOIs
Publication statusPublished - 8 Mar 2011

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Free-form Surface
Diffusion equation
Filtering
Euclidean
Gaussian Filter
textures
Surface Texture
Textures
Least Squares Approximation
surface distortion
Laplace-Beltrami Operator
Gaussian Function
Least squares approximations
pulse detonation engines
Surface measurement
Standard deviation
Texture
Convolution
approximation
convolution integrals

Cite this

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

In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 467, No. 2127, 08.03.2011, p. 841-859.

Research output: Contribution to journalArticle

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