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Robust Segmentation of Primitives from Range Data in the Presence of Geometric Degeneracy
March 2001 (vol. 23 no. 3)
pp. 304-314

Abstract—This paper addresses a common problem in the segmentation of range images. We would like to identify and fit surfaces of known type wherever these are a good fit. This paper presents methods for the least-squares fitting of spheres, cylinders, cones, and tori to 3D point data, and their application within a segmentation framework. Least-squares fitting of surfaces other than planes, even of simple geometric type, has been rarely studied. Our main application areas of this research are reverse engineering of solid models from depth-maps and automated 3D inspection where reliable extraction of these surfaces is essential. Our fitting method has the particular advantage of being robust in the presence of geometric degeneracy, i.e., as the principal curvatures of the surfaces being fitted decrease (or become more equal), the results returned naturally become closer and closer to those surfaces of “simpler type,” i.e., planes, cylinders, cones, or spheres, which best describe the data. Many other methods diverge because, in such cases, various parameters or their combination become infinite.

[1] R. Bajcsy, F. Solina, and A. Gupta, “Segmentation versus Object Representation—Are they Separable?” Analysis and Interpretation of Range Images, R. Jain and A.K. Jain, eds., New York: Springer-Verlag, 1990.
[2] P. Benko, G. Kos, and T Varady, “Detecting Translational and Rotational Symmetries in Reverse Engineering,” Proc. Advanced Workshop Confluence of Computer Vision and Computer Graphics, Aug. 1999.
[3] P.J. Besl, Surfaces in Range Image Understanding. New York: Springer-Verlag 1988.
[4] P.J. Besl and R.C. Jain,“Segmentation through variable-order surface fitting,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 10, no. 2, pp. 167-191, Mar. 1988.
[5] R.M. Bolle and D.B. Cooper,“On optimally combining pieces of information, with application to estimating 3D complex-object position from range data,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, no. 5, pp. 619-638, Sept. 1986.
[6] R.M. Bolle and B.C. Vemuri, "On Three-Dimensional Surface Reconstruction Methods," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 1, Jan. 1991.
[7] Å. Björk, Numerical Methods for Least Squares Problems. Philadelphia: SIAM, 1996.
[8] J. Clark, E. Trucco, and H.F. Cheung, “Using Light Polarization in Laser Scanning,” Image and Vision Computing, vol. 15, pp. 107-117, 1997.
[9] O.D. Faugeras, M. Hebert, and E. Pauchon, “Segmentation of Range Data into Planar and Quadric Patches,” Proc. Third Computer Vision and Pattern Recognition Conf., pp. 8-13, 1983.
[10] A.W. Fitzgibbon, M. Pilu, and R.B. Fisher, “Direct Least-Square Fitting of Ellipses,” Proc. 13th Int'l Conf' Pattern Recognition, Aug. 1996.
[11] W. Gander, G.H. Golub, and R. Strebel, “Least-Squares Fitting of Circles and Ellipses,” BIT, vol. 34, pp. 558-578, 1994.
[12] M. Hebert and J. Ponce, “A New Method for Segmenting 3D Scenes into Primitives,” Proc. Sixth Int'l Conf. Pattern Recognition, pp. 836-838, Oct. 1982.
[13] A. Jaklic, A. Leonardis, and F. Solina, “Segmentor: An Object-Oriented Framework for Image Segmentation,” Technical Report LRV-96-2, Computer Vision Laboratory, Faculty of Computer and Information Science, Univ. of Ljubljana, 1996.
[14] D. Keren and C. Gotsman, “Fitting Curves and Surfaces with Constrained Implicit Polynomials,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 1, pp. 31-41, Jan. 1999.
[15] A. Leonardis,A. Gupta,, and R. Bajcsy,“Segmentation as the search for the best description of the image in terms of primitives,” Proc. Int’l Conf. Computer Vision, pp. 121-125, 1990.
[16] A. Leonardis, “Image Analysis Using Parametric Models: Model-Recovery and Model-Selection Paradigm,” PhD dissertation, Faculty of Electrical Eng., and Computer Science, Univ. of Ljubljana, May 1993.
[17] A. Leonardis, A. Gupta, and R. Bajcsy, “Segmentation of Range Images as the Search for Geometric Parametric Models,” Int'l J. Computer Vision, vol. 14, no. 3, pp. 253-277, 1995.
[18] 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, vol. 52, no. 1, pp. 78-109, 1990.
[19] 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. 1,068 (Budapest) Geometric Modelling Laboratory Studies/1997/5, Computer and Automation Research Institute, Budapest, R.R. Martin and T. Várady, eds., July 1997. http://www.cs.cf.ac.uk/Dave/3DVGFITTING.
[20] G. Lukács, A.D. Marshall, and R.R. Martin, “Faithful Least-Squares Fitting of Spheres, Cylinders, Cones, and Tori for Reliable Segmentation,” Proc. Fifth European Conf. Computer Vision (ECCV '98), H. Burkhardt and B. Neumann eds., vol. I, pp. 671-686, 1998.
[21] H. Pottmann and T. Randrup, “Rotational and Helical Surface Approximation for Reverse Engineering,” Computing, vol. 60, pp. 307-322, 1998.
[22] V. Pratt, “Direct Least-Squares Fitting of Algebraic Surfaces,” Proc. Ann. Conf. Series Computer Graphics, vol. 21, no. 4, pp. 145-152, July 1987.
[23] P.L. Rosin, “A Note on the Least Square Fitting of Ellipses,” Pattern Recognition Letters, vol. 14, no. 10, pp. 799-808, 1993.
[24] P.L. Rosin, “Analysing Error of Fit Functions for Ellipses,” Pattern Recognition Letters, vol. 17, pp. 1461-1470, 1996.
[25] G. Taubin,“Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 11, pp. 1115-1137, Nov. 1991.
[26] W. Thompson, J. Owen, J. de St. Germain, S. Stark, and T. Henderson, “Feature-Based Reverse Engineering of Mechanical Parts,” IEEE Trans. Robotics and Automation, vol. 15, no. 1, pp. 1-9, 1999.
[27] E. Trucco, R.B. Fisher, A.W. Fitzgibbon, and D.K. Naidu, “Calibration, Data Consistency, and Model Acquisition with a 3D Laser Striper,” Int'l J. Computer Integrated Manufacturing, vol. 11, no. 4, pp. 293-320, 1998.
[28] N. Werghi, R.B. Fisher, A. Ashbrook, and C. Robertson, “Improving Model Shape Acquisition by Incorporating Geometric Constraints,” Proc. British Machine Vision Conf. (BMVC '97), A. Clark, ed., pp. 520-529, 1997.
[29] N. Werghi, R.B. Fisher, C. Robertson, and A. Ashbrook, “Modelling Object Having Quadric Surfaces Incorporating Geometric Constraints,” Proc. Fifth European Conf. Computer Vision (ECCV '98), H. Burkhardt and B. Neumann, eds., vol II, pp. 185-201, 1998.
[30] N. Werghi, R.B. Fisher, C. Robertson, and A. Ashbrook, “Towards Object Modelling by Incorporating Geometric Constraints,” Proc. IEEE Workshop Model-Based 3D Image Analysis, E. Cuchet and G. Subsol, eds., pp. 45-53, 1998.
[31] N. Werghi, R.B. Fisher, A. Ashbrook, and C. Robertson, “Faithful Recovering of Quadric Surface from 3D Range Data,” Proc. Second Int'l Conf. 3D Digital Imaging and Modelling, M. Rioux, P. Boulanger, and D. Laurendeau, eds., pp. 45-53, 1999.
[32] N. Werghi, R.B. Fisher, A. Ashbrook, and C. Robertson, “Object Reconstruction by Incorporating Geometric Constraints in Reverse Engineering,” Computer Aided Design, vol. 31, pp. 363-399, 1999.
[33] T. Varady, R.R. Martin, and J. Cox, “Reverse Engineering of Geometric Models—An Introduction,” Computer Aided Design, vol. 29, no. 4, pp. 255-268, 1997.

Index Terms:
Nonlinear least squares, geometric distance, cylinder, cone, sphere, torus, surface fitting, segmentation.
Citation:
David Marshall, Gabor Lukacs, Ralph Martin, "Robust Segmentation of Primitives from Range Data in the Presence of Geometric Degeneracy," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23, no. 3, pp. 304-314, March 2001, doi:10.1109/34.910883
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