Non-linear least squares fitting of Bézier surfaces to unstructured point clouds

Joseph Lifton; Tong Liu; John McBride; Joseph Lifton; Tong Liu; John McBride

doi:10.3934/math.2021190

AIMS Mathematics

2021, Volume 6, Issue 4: 3142-3159. doi: 10.3934/math.2021190

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Research article

Non-linear least squares fitting of Bézier surfaces to unstructured point clouds

1.
Intelligent Product Verification Group, Advanced Remanufacturing and Technology Centre, 637143, Singapore
2.
Precision Measurements Group, Singapore Institute of Manufacturing Technology, 637662, Singapore
3.
Mechatronics Engineering Group, Mechanical Engineering Department, Faculty of Engineering and Physical Sciences, University of Southampton, SO17 1BJ, UK

Received: 01 October 2020 Accepted: 29 December 2020 Published: 14 January 2021
MSC : 65D10

Algorithms for linear and non-linear least squares fitting of Bézier surfaces to unstructured point clouds are derived from first principles. The presented derivation includes the analytical form of the partial derivatives that are required for minimising the objective functions, these have been computed numerically in previous work concerning Bézier curve fitting, not surface fitting. Results of fitting fourth degree Bézier surfaces to complex simulated and measured surfaces are presented, a quantitative comparison is made between fitting Bézier surfaces and fitting polynomial surfaces. The developed fitting algorithm is used to remove the geometric form of a complex engineered surface such that the surface roughness can be evaluated.
- Bézier surfaces,
- least squares fitting,
- surface metrology
Citation: Joseph Lifton, Tong Liu, John McBride. Non-linear least squares fitting of Bézier surfaces to unstructured point clouds[J]. AIMS Mathematics, 2021, 6(4): 3142-3159. doi: 10.3934/math.2021190

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Abstract

Algorithms for linear and non-linear least squares fitting of Bézier surfaces to unstructured point clouds are derived from first principles. The presented derivation includes the analytical form of the partial derivatives that are required for minimising the objective functions, these have been computed numerically in previous work concerning Bézier curve fitting, not surface fitting. Results of fitting fourth degree Bézier surfaces to complex simulated and measured surfaces are presented, a quantitative comparison is made between fitting Bézier surfaces and fitting polynomial surfaces. The developed fitting algorithm is used to remove the geometric form of a complex engineered surface such that the surface roughness can be evaluated.

References

[1]	A. Sóbester, A. I. J. Forrester, Aircraft aerodynamic design: geometry and optimization, John Wiley & Sons, West Sussex, 2015.
[2]	L. Shao, H. Zhou, Curve fitting with Bézier cubics, Graphical Models and Image Processing, 58 (1996), 223–232. doi: 10.1006/gmip.1996.0019
[3]	T. Várady, P. Salvi, M. Vaitkus, Á. Sipos, Multi-sided Bézier surfaces over curved, multi-connected domains, Computer Aided Geometric Design, 78 (2020), 101828.
[4]	J. N. Petzing, J. M. Coupland, R. K. Leach, The measurement of rough surface topography using coherence scanning interferometry, National Physical Laboratory, UK, 2010
[5]	T. A. Pastva, Bézier curve fitting, Thesis, Naval Postgraduate School, Monterey, California, 1998.
[6]	C. F. Borges, T. Pastva, Total least squares fitting of Bézier and B-spline curves to ordered data, Computer Aided Geometric Design, 19 (2002), 275–289. doi: 10.1016/S0167-8396(02)00088-2
[7]	P. Kovacs, A. M. Fekete, Nonlinear least-squares spline fitting with variable knots, Appl. Math. Comput., 354 (2019), 490–501.
[8]	A. Gálvez, A. Iglesias, A. Cobo, J. Puig-Pey, J. Espinola, Bézier curve and surface fitting of 3D point clouds through genetic algorithms, functional networks and least-squares approximation, Computational Science and Its Applications, 4706 (2007), 680–693.
[9]	K. S. Reddy, A. Mandal, K. K. Verma, G. Rajamohan, Fitting of Bézier surfaces using the fireworks algorithm, International Journal of Advances in Engineering & Technology, 9 (2016), 396–403.
[10]	A. Gálvez, A. Iglesias, Firefly Algorithm for Polynomial Bézier Surface Parameterization, J. Appl. Math., 2013 (2013).
[11]	L. Piegl, W. Tiller, The NURBS book, 2 Eds., Springer-Verlag, Berlin, Heidelberg, New York, 1995.
[12]	J. Arvo, Graphics Gems Ⅱ, Academic Press Professional, San Diego, CA, United States, 1991.
[13]	T. Varady and R. Martin, Handbook of Computer Aided Geometric Design, North-Holland, Amsterdam, The Netherlands, 2002.
[14]	A. B. Forbes, Least-squares best-fit geometric elements, NPL Report DITC 140/89, 1991.
[15]	Y. Cao, D. M. Yan, P. Wonka, Patch layout generation by detecting feature networks, Computers & Graphics, 46 (2015), 275–282.
[16]	H. Lin, W. Chen, H. Bao, Adaptive patch-based mesh fitting for reverse engineering, Computer-Aided Design, 39 (2007), 1134–1142. doi: 10.1016/j.cad.2007.10.002
[17]	I. Kovács, T. Várady, Constrained fitting with free-form curves and surfaces, Computer-Aided Design, 122 (2020), 102816. doi: 10.1016/j.cad.2020.102816

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