Comparison of discrete Laplacian-Beltrami operators and their applications in surface metrology

Huodao Huang; Wenhan Zeng; Shan Lou; Jane Jiang; Paul Scott

Comparison of discrete Laplacian-Beltrami operators and their applications in surface metrology

Huodao Huang, Wenhan Zeng, Shan Lou, Jane Jiang, Paul Scott

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

In addition to computer image processing, mesh smoothing can also be used in surface metrology to remove noise from free-form surfaces, which are widely represented by triangular meshes. The diffusion equation, also known as the heat equation, is generalised and applied to smooth meshes. The Laplacian operator in the equation is generalised to the Laplacian-Beltrami operator so the equation can be used not only on scalar functions defined in Euclidean space but also on
functions defined on non-Euclidean manifolds. The triangular meshes are piecewise linear approximations of a free-form surface. Therefore, the LaplacianBeltrami operator needs to be discrete.
There are three kinds of discretisation methods widely used currently: uniform graph Laplacian, distance Laplacian and cotangent formula. Their algorithms will be briefly introduced and their performance will be compared in four aspects: smoothness, shrinkage degree, computation time and edge distortion degree. Tests will be conducted on simulated surfaces and measured surfaces. Then, recommendations will be given regarding the requirements of measurement and
characteristics of samples.

Original language	English
Title of host publication	Laser Metrology and Machine Performance XV
Subtitle of host publication	15th International Conference and Exhibition on Laser Metrology, Machine Tool, CMM & Robotic Performance: LAMDAMAP 2023
Editors	Andrew Longstaff, Nan Yu
Publisher	euspen
Pages	257-260
Number of pages	13
ISBN (Electronic)	9781998999125
Publication status	Published - 15 Mar 2023
Event	15th International Conference and Exhibition on Laser Metrology, Coordinate Measuring Machine and Machine Tool Performance - Edinburgh, United Kingdom Duration: 14 Mar 2023 → 15 Mar 2023 Conference number: 15

Conference

Conference	15th International Conference and Exhibition on Laser Metrology, Coordinate Measuring Machine and Machine Tool Performance
Abbreviated title	Lamdamap 2023
Country/Territory	United Kingdom
City	Edinburgh
Period	14/03/23 → 15/03/23

UN SDGs

This output contributes to the following UN Sustainable Development Goals (SDGs)

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https://www.euspen.eu/knowledge-base/LAM23122.pdfLicence: Unspecified

Cite this

Huang, H., Zeng, W., Lou, S., Jiang, J., & Scott, P. (2023). Comparison of discrete Laplacian-Beltrami operators and their applications in surface metrology. In A. Longstaff, & N. Yu (Eds.), Laser Metrology and Machine Performance XV: 15th International Conference and Exhibition on Laser Metrology, Machine Tool, CMM & Robotic Performance: LAMDAMAP 2023 (pp. 257-260). euspen. https://www.euspen.eu/knowledge-base/LAM23122.pdf

Huang, Huodao ; Zeng, Wenhan ; Lou, Shan et al. / Comparison of discrete Laplacian-Beltrami operators and their applications in surface metrology. Laser Metrology and Machine Performance XV: 15th International Conference and Exhibition on Laser Metrology, Machine Tool, CMM & Robotic Performance: LAMDAMAP 2023. editor / Andrew Longstaff ; Nan Yu. euspen, 2023. pp. 257-260

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title = "Comparison of discrete Laplacian-Beltrami operators and their applications in surface metrology",

abstract = "In addition to computer image processing, mesh smoothing can also be used in surface metrology to remove noise from free-form surfaces, which are widely represented by triangular meshes. The diffusion equation, also known as the heat equation, is generalised and applied to smooth meshes. The Laplacian operator in the equation is generalised to the Laplacian-Beltrami operator so the equation can be used not only on scalar functions defined in Euclidean space but also onfunctions defined on non-Euclidean manifolds. The triangular meshes are piecewise linear approximations of a free-form surface. Therefore, the LaplacianBeltrami operator needs to be discrete.There are three kinds of discretisation methods widely used currently: uniform graph Laplacian, distance Laplacian and cotangent formula. Their algorithms will be briefly introduced and their performance will be compared in four aspects: smoothness, shrinkage degree, computation time and edge distortion degree. Tests will be conducted on simulated surfaces and measured surfaces. Then, recommendations will be given regarding the requirements of measurement andcharacteristics of samples.",

keywords = "Laplacian-Beltrami operators, surface metrology, computer image processing, mesh smoothing, free-form surfaces",

author = "Huodao Huang and Wenhan Zeng and Shan Lou and Jane Jiang and Paul Scott",

note = "Funding Information: The authors gratefully acknowledge the UK{\textquoteright}s Engineering and Physical Sciences Research Council (EPSRC) funding via the Future Advanced Metrology Hub (EP/P006930/1), EPSRC Fellowship in Manufacturing (EP/R024162/1), and new investigator award (EP/S000453/1) for supporting this work. Publisher Copyright: {\textcopyright} euspen Headquarters March 2023.; 15th International Conference and Exhibition on Laser Metrology, Coordinate Measuring Machine and Machine Tool Performance, Lamdamap 2023 ; Conference date: 14-03-2023 Through 15-03-2023",

year = "2023",

month = mar,

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Huang, H, Zeng, W , Lou, S , Jiang, J & Scott, P 2023, Comparison of discrete Laplacian-Beltrami operators and their applications in surface metrology. in A Longstaff & N Yu (eds), Laser Metrology and Machine Performance XV: 15th International Conference and Exhibition on Laser Metrology, Machine Tool, CMM & Robotic Performance: LAMDAMAP 2023. euspen, pp. 257-260, 15th International Conference and Exhibition on Laser Metrology, Coordinate Measuring Machine and Machine Tool Performance, Edinburgh, United Kingdom, 14/03/23. <https://www.euspen.eu/knowledge-base/LAM23122.pdf>

Comparison of discrete Laplacian-Beltrami operators and their applications in surface metrology. / Huang, Huodao; Zeng, Wenhan ; Lou, Shan et al.

Laser Metrology and Machine Performance XV: 15th International Conference and Exhibition on Laser Metrology, Machine Tool, CMM & Robotic Performance: LAMDAMAP 2023. ed. / Andrew Longstaff; Nan Yu. euspen, 2023. p. 257-260.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Lou, Shan

AU - Jiang, Jane

AU - Scott, Paul

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N2 - In addition to computer image processing, mesh smoothing can also be used in surface metrology to remove noise from free-form surfaces, which are widely represented by triangular meshes. The diffusion equation, also known as the heat equation, is generalised and applied to smooth meshes. The Laplacian operator in the equation is generalised to the Laplacian-Beltrami operator so the equation can be used not only on scalar functions defined in Euclidean space but also onfunctions defined on non-Euclidean manifolds. The triangular meshes are piecewise linear approximations of a free-form surface. Therefore, the LaplacianBeltrami operator needs to be discrete.There are three kinds of discretisation methods widely used currently: uniform graph Laplacian, distance Laplacian and cotangent formula. Their algorithms will be briefly introduced and their performance will be compared in four aspects: smoothness, shrinkage degree, computation time and edge distortion degree. Tests will be conducted on simulated surfaces and measured surfaces. Then, recommendations will be given regarding the requirements of measurement andcharacteristics of samples.

AB - In addition to computer image processing, mesh smoothing can also be used in surface metrology to remove noise from free-form surfaces, which are widely represented by triangular meshes. The diffusion equation, also known as the heat equation, is generalised and applied to smooth meshes. The Laplacian operator in the equation is generalised to the Laplacian-Beltrami operator so the equation can be used not only on scalar functions defined in Euclidean space but also onfunctions defined on non-Euclidean manifolds. The triangular meshes are piecewise linear approximations of a free-form surface. Therefore, the LaplacianBeltrami operator needs to be discrete.There are three kinds of discretisation methods widely used currently: uniform graph Laplacian, distance Laplacian and cotangent formula. Their algorithms will be briefly introduced and their performance will be compared in four aspects: smoothness, shrinkage degree, computation time and edge distortion degree. Tests will be conducted on simulated surfaces and measured surfaces. Then, recommendations will be given regarding the requirements of measurement andcharacteristics of samples.

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A2 - Longstaff, Andrew

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ER -

Comparison of discrete Laplacian-Beltrami operators and their applications in surface metrology

Abstract

Conference

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