Abstract
This paper addresses a problem arising in the reverse engineering of solid models from depth-maps. We wish to identify and fit surfaces of known type wherever these are a good fit. This paper presents a set of methods for the least-squares fitting of spheres, cylinders, cones and tori to three-dimensional point data. Least-squares fitting of surfaces other planes, even of simple geometric type, has been little studied.
Our method has the particular advantage of being robust in the sense that as the principal curvatures of the surfaces being fitted decrease (or become more equal), the results which are returned naturally become closer and closer to those surfaces of ‘simpler type’, i.e. planes, cylinders, cones, or spheres which best describe the data, unlike other methods which may diverge as various parameters or their combination become infinite.
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References
R. Bajcsy, F. Solina, and A. Gupta, Segmentation versus object representation—are they separable?, In Analysis and Interpretation of Range Images, Eds. R. Jain and A. K. Jain, Springer-Verlag, New York, 1990.
P. J. Besl, Surfaces in Range Image Understanding, Springer-Verlag, New York, USA, 1988.
P. J. Besl and R. K. Jain, Segmentation Through Variable-Order Surface Fitting, IEEE Transactions on Pattern Analysis and Machine Intelligence, 10 (2), 167–192, 1988.
R. M. Bolle and D. B. Cooper, On optimally combining pieces of information, with application to estimating 3-D complex-object position from range data, IEEE Transactions on Pattern Analysis and Machine Intelligence 8, 5, 619–638, 1986.
R. M. Bolle and B. C. Vemuri, On Three-Dimensional Surface Reconstruction Methods, IEEE Transactions on Pattern Analysis and Machine Intelligence, 13 (1), 1–13, 1991.
å. Björk, Numerical methods for least squares problems, SIAM. Society for Industrial and Applied Mathematics, Philadelphia, 1996.
O. D. Faugeras, M. Hebert, and E. Pauchon, Segmentation of Range Data into Planar and Quadric Patches, Proceedings of Third Computer Vision and Pattern Recognition Conference, Arlington, VA, 8–13, 1983.
A.W. Fitzgibbon, M. Pilu, and R.B. Fisher, Direct least-square fitting of ellipses, 13th International Conference on Pattern Recognition (Washington, Brussels, Tokyo), IAPR, IEEE Computer Society Press, June 1996, Proceedings of the 13th ICPR Conference, Vienna Austria, August 1996.
W. Gander, G.H. Golub, and R. Strebel, Least-squares fitting of circles and ellipses, BIT 34, 558–578, 1994.
M. Hebert and J. Ponce, A new method for segmenting 3-D scenes into primitives, 6th International Conference on Pattern Recognition (Munich), 836–838, IAPR DAGM, IEEE Computer Society Press, October 1982, Proceedings of the 6th ICPR Conference, Munich Germany, Oct. 19–22, 1982.
A. Jaklic, A. Leonardis, and F. Solina. Segmentor: An object-oriented framework for image segmentation. Technical Report LRV-96-2, Computer Vision Laboratory, University of Ljubljana, Faculty of Computer and Information Science, 1996.
A. Leonardis. Image analysis using parametric models: model-recovery and modelselection paradigm, PhD dissertation, University of Ljubljana, Faculty of Electrical Engineering and Computer Science, May 1993.
A. Leonardis, A. Gupta, and R. Bajcsy. Segmentation as the search for the best description of the image in terms of primitives. Proceedings of the Third International Conference of Computer Vision, Osaka, Japan, 1990.
A. Leonardis, A. Gupta, and R. Bajcsy. Segmentation of range images as the search for geometric parametric models. International Journal of Computer Vision, 14, 253–277, 1995.
P. Liong, and J. S. Todhunter, Representation and recognition of surface shapes in range images: a differential geometry approach, Computer Vision, Graphics and Image Processing, 52, 1, 78–109, 1990.
G. Lukács, A. D. Marshall, and R. R. Martin, Geometric least-squares fitting of spheres, cylinders, cones and tori, RECCAD, Deliverable Document 2 and 3, COPERNICUS project, No 1068 (Budapest) (R. R. Martin and T. Várady, eds.), Geometric Modelling Laboratory Studies/1997/5, Computer and Automation Research Institute, Budapest, July 1997.
H. Pottmann and T. Randrup, Rotational and helical surface approximation for reverse engineering, Tech. Report 43, Institut für Geometrie, Technische Universität Wien, A-1040 Wien, Wiedner Hauptstraße 8-10, Austria, June 1997, Submitted to Computing.
V. Pratt, Direct least-squares fitting of algebraic surfaces, COMPUTER GRAPHICS Proceedings, vol. 21, Annual Conference Series, no. 4, ACM, Addison Wesley, July 1987, Proceedings of the SIGGRAPH 87 Conference, 145–152, Anaheim, California, 27–31 July 1987.
P. L. Rosin, A note on the least squares fitting of ellipses, Pattern Recognition Letters 14, 799–808, 1993.
P. L. Rosin, Analysing error of fit functions for ellipses, Pattern Recognition Letters 17, 1461–1470, 1996.
G. Taubin, Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence 13 (11), 1115–1138, 1991.
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Lukács, G., Martin, R., Marshall, D. (1998). Faithful least-squares fitting of spheres, cylinders, cones and tori for reliable segmentation. In: Burkhardt, H., Neumann, B. (eds) Computer Vision — ECCV'98. ECCV 1998. Lecture Notes in Computer Science, vol 1406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055697
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DOI: https://doi.org/10.1007/BFb0055697
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