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Geometric Fitting of Parametric Curves and Surfaces

  • Ahn, Sung-Joon (Sch. of Inf. and Comm. Eng., Sungkyunkwan University)
  • 발행 : 2008.12.31

초록

This paper deals with the geometric fitting algorithms for parametric curves and surfaces in 2-D/3-D space, which estimate the curve/surface parameters by minimizing the square sum of the shortest distances between the curve/surface and the given points. We identify three algorithmic approaches for solving the nonlinear problem of geometric fitting. As their general implementation we describe a new algorithm for geometric fitting of parametric curves and surfaces. The curve/surface parameters are estimated in terms of form, position, and rotation parameters. We test and evaluate the performances of the algorithms with fitting examples.

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참고문헌

  1. R.J. Adcock, “Note on the method of least squares, ” The Analyst, Des Moines, Iowa, vol.4, pp.183-184, 1877 https://doi.org/10.2307/2635777
  2. S.J. Ahn, W. Rauh, H.S. Cho, H.-J. Warnecke, “Orthogonal Distance Fitting of Implicit Curves and Surfaces,” IEEE Trans. Pattern Analy. Mach. Intell, vol.24, no.5, pp.620-638, 2002 https://doi.org/10.1109/34.1000237
  3. S.J. Ahn, E. Westkamper, W. Rauh, “Orthogonal Distance Fitting of Parametric Curves and Surfaces, ” Proc. 4th Int'l Symp. Algorithms for Approximation, J. Levesley et al. (eds.), U.K.: Univ. of Huddersfield, pp.122-129, 2002
  4. P.T. Boggs, R.H. Byrd, R.B. Schnabel, “A stable and efficient algorithm for nonlinear orthogonal distance regression,” SIAM J. Sci. Stat. Compt., vol.8, pp. 1052-1078, 1987 https://doi.org/10.1137/0908085
  5. P.T. Boggs, J.R. Donaldson, R.H. Byrd, R.B. Schnabel, “Algorithm 676 – ODRPACK: Software for Weighted Orthogonal Distance Regression,” ACM Trans. Math. Softw., vol.15, no.4, pp.348-364, 1989 https://doi.org/10.1145/76909.76913
  6. F.L. Bookstein, “Fitting conic sections to scattered data,” Comput. Graph. Image Process., vol.9, no.1, pp.56-71, 1979 https://doi.org/10.1016/0146-664X(79)90082-0
  7. B.P. Butler, A.B. Forbes, P.M. Harris, “Algorithms for Geometric Tolerance Assessment,” Report no. DITC 228/94, Teddington, U.K.: NPL, 1994
  8. X. Cao, N. Shrikhande, G. Hu, “Approximate orthogonal distance regression method for fitting quadric surfaces to range data,” Pattern Recogn. Lett., vol.15, no.8, pp. 781-796, 1994 https://doi.org/10.1016/0167-8655(94)90006-X
  9. A. Fitzgibbon, M. Pilu, R.B. Fisher, “Direct Least Square Fitting of Ellipses,” IEEE Trans. Pattern Analy. Mach. Intell, vol.21, no.5, pp.476-480, 1999 https://doi.org/10.1109/34.765658
  10. H.-P. Helfrich and D. Zwick, “A trust region method for implicit orthogonal distance regression,” Numer. Algorithms, vol.5, pp.535-545, 1993 https://doi.org/10.1007/BF02108668
  11. H.-P. Helfrich and D. Zwick, “A trust region algorithm for parametric curve and surface fitting,” J. Comput. Appl. Math., vol.73, pp.119-134, 1996 https://doi.org/10.1016/0377-0427(96)00039-8
  12. J. Hoschek, F.-J. Schneider, P. Wassum, “Optimal approximate conversion of spline surfaces”, Comput. Aided Geom. Design, vol.6, pp.293-306, 1989 https://doi.org/10.1016/0167-8396(89)90030-7
  13. ISO 10360-6, “Geometrical Product Specifications (GPS) – Acceptance and reverification test for coordinate measuring machines (CMM) – Part 6: Estimation of errors in computing Gaussian associated features,” Geneva, Switzerland: ISO, Dec. 2001
  14. D. Marshall, G. Lukacs, R. Martin, “Robust Segmentation of Primitives from Range Data in the Presence of Geometric Degeneracy,” IEEE Trans. Pattern Analy. Mach. Intell, vol.23, no.3, pp.304-314, 2001 https://doi.org/10.1109/34.910883
  15. K. Pearson, “On Lines and Planes of Closest Fit to Systems of Points in Space,” The Philos. Mag., ser.6, vol.2, no.11, pp.559-572, 1901 https://doi.org/10.1080/14786440109462720
  16. L. Piegl and W. Tiller, “The NURBS Book,” 2nd Ed., Berlin: Springer, 1997
  17. F. Solina and R. Bajcsy, “Recovery of Parametric Models from Range Images: The Case for Superquadrics with Global Deformations,” IEEE Trans. Pattern Analy. Mach. Intell, vol.12, no.2, pp.131-147, 1990 https://doi.org/10.1109/34.44401
  18. D. Sourlier, “Three Dimensional Feature Independent Bestfit in Coordinate Metrology,” Ph.D. Thesis, no. 11319, Zurich, Switzerland: ETH Zurich, 1995
  19. S. Sullivan, L. Sandford, J. Ponce, “Using Geometric Distance Fits for 3-D Object Modeling and Recognition,” IEEE Trans. Pattern Analy. Mach. Intell, vol.16, no.12, pp.1183-1196, 1994 https://doi.org/10.1109/34.387489
  20. G. Taubin, “Estimation of Planar Curves, Surfaces, Nonplanar Space Curves Defined by Implicit Equations with Applications to Edge and Range Image Segmentation,” IEEE Trans. Pattern Analy. Mach. Intell, vol.13, no. 11, pp.1115-1138, 1991 https://doi.org/10.1109/34.103273
  21. D.A. Turner, “The approximation of Cartesian coordinate data by parametric orthogonal distance regression,” Ph.D. Thesis, U.K.: Univ. of Huddersfield, 1999

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