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

Volumn 13, Issue 11, 1991, Pages 1115-1138

  Taubin, Gabriel ^a,b  

^a BROWN UNIVERSITY   (United States)

^b IBM T J WATSON RESEARCH CENTER   (United States)

Author keywords

Algebraic curves and surfaces; approximate dis ; generalized eigenvector fit; implicit curve and surface fitting; invariance; object recognition; segmentation; tances

Indexed keywords

MATHEMATICAL TECHNIQUES--APPLICATIONS; PATTERN RECOGNITION; SURFACES;

IMAGE SEGMENTATION; PLANAR CURVES;

IMAGE PROCESSING;

EID: 0026254852     PISSN: 01628828     EISSN: None     Source Type: Journal    
DOI: 10.1109/34.103273     Document Type: Article

References (76)

1. A. Albano, “Representation of digitized contours in terms of conic arcs and straight-line segments,” Comput. Graphics Image Processing, vol. 3. pp. 23–33, 1974.
2. T. M. Apostol, Mathematical Analysis. Reading, MA: Addison Wesley, 1974.
3. R. Bajcsy and F. Solina, “Three dimensional object representation revisited,” in Proc. First Int. Conf. Comput Vision, June 1987, pp. 231–240
4. A. H. Barr, “Superquadrics and angle-preserving transformations,” IEEE Comput. Graphics Applications, vol. 1, pp. 11–23, 1981.
5. P. J. Besl, “Geometric modeling and computer vision,” Proc. IEEE, vol. 76, no. 8, pp. 936–958, 1988.
6. P. J. Besl, Surfaces in Range Image Understanding. Berlin: Springer-Verlag, 1988.
7. P. J. Besl and R. C. Jain, “Segmentation trough variable-order surfaces fitting,” IEEE Trans. Patt. Anal. Machine Intell., vol. 10, Mar. 1988.
8. R. H. Biggerstaff, “Three variation in dental arch form estimated by a quadratic equation,” J. Dental Res., vol. 51, p. 1509, 1972.
9. R. M. Bolle and D. B. Cooper, “Bayesian recognition of local 3D shape by approximating image intensity functions with quadric polynomials,” IEEE Trans. Patt. Anal. Machine Intell., vol. 6, no. 4, July 1984.
10. R. M. Bolle and D. B. Cooper, “On optimally combining pieces of information, with applications to estimating 3D complex-object position from range data,” IEEE Trans. Patt. Analysis Machine Intell., vol. 8, no. 5, Sept. 1986.
11. R.J. Bolle and B. C. Vemuri, “On 3D surface reconstruction methods,” RC-14557, IBM Res. Div., Apr. 1989.
12. R. C. Bolles and P. Horaud, “3DPO: A three-demensinal part orientation system,” lnt. J. Robotics Res., vol. 5, no. 3, pp. 3–26, 1986.
13. R. C. Bolles, P. Horaud, and M.J. Hannah, “3DPO: A three-dimensinal part orientation sytem,” in Proc. 8th. Int. Joint Conf. Artificial Intell., 1983.
14. F. L. Bookstein, “Fitting conic sections to scattered data,” Comput. Vision Graphics Image Processing, vol. 9, pp. 56–71, 1979.
15. T. E. Boult and A. D. Gross, “Recovery of superquadrics from depth information,” in Proc. AAAI Workshop Spatial Reasoning Multisensor Integration, Oct. 1987.
16. T. E. Boult and A. D. Gross, “On the recovery of superellipsoids,” CUCS 331–88, Comput. Sci. Dept., Columbia Univ., 1988.
17. R.A. Bradley and S.S. Srivastava, “Correlation in polynomial regression,” Amer. Statistician, vol. 33, p. 11, 1979.
18. J. F. Canny, The Complexity of Robot Motion Planning. Cambridge, MA: MIT Press, 1988.
19. B. Cernuschi-Frias, “Orientation and location parameter estimation of quadric surfaces in 3D from a sequence of images,” Ph.D. thesis, Brown Univ., May 1984.
20. D. S. Chen, “A data-driven intermediate level feature extraction algorithm,” IEEE Trans. Patt. Anal. Machine Intell., vol. 11, no. 7, July 1989.
21. D. B. Cooper, Y. P. Hung, and G. Taubinl, “A new model-based stereo approach for 3D surface reconstruction using contours on the surfaces pattern, in Proc. Second Int. Conf. Comput. Vision, Dec. 1988.
22. D. B. Cooper and N. Yalabik, “On the cost of approximating and recognizing noise-perturbed straight ines and quadratic curve segments in the plane,” NASA-NSG-5035/I, Brown Univ., Mar. 1975.
23. D. B. Cooper and N. Yalabik, “On the computational cost of approximating and recognizing noise-perturbed straight lines and quadratic arcs in the plane,” IEEE Trans. Comput., vol. 25, no. 10, pp. 1020–1032, Oct. 1976.
24. C. P. Cox. A Handbook of Introductory Statistical Methods. New York: Wiley, 1987.
25. P. J. Davis, Interpolation and Approximation. New York: Dover, 1975.
26. C. De Boor, A Practical Guide to Splines. Berlin: Springer-Verlag, 1978.
27. J. E. Dennis and R. B. Shnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, NJ: Prentice-Hall, 1983.
28. A. L. Dixon, “The element of three quantics in two independent variables,” in Proc. London Math. Soc., vol. 7, 1908, series 2.
29. R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis. New York: Wiley, 1973.
30. I. Elliot, “Discussion and implementation description of an experiemen-tal interactive superquadric-based 3D drawing sytem,” Internal Rep. LP 2/IME4, Euro. Comput.-Industry Res. Cent., 1986.
31. R. T. Farouki and V.T. Rajan, “On the numerical condition of algebraic curves and surfaces,” RC-126263, IBM Res. Div., Nov. 1987.
32. R. T. Farouki and V.T. Rajan, “On the numberical condition of Bernstein ploynomicals,” RC-13626, IBM Res. Div., Mar. 1987.
33. O. D. Faugeras and M. Hebert, “A 3D recognition and positioning algorithm using geometrical matching between primitive surfaces,” in Proc. 8th. Int. Joint Conf. Artificial Intell. (UCAl), 1983.
34. O. D. Faugeras, M. Hebert, and E. Pauchon,” Segmentation of range data into planar and quadric patches,” in Proc. IEEE Conf. Comput. Vision Patt. Recog. (CVPR), 1983.
35. Y. Gardan, Numberical Methods for CAD. Cambridge, MA: MIT Press, 1986.
36. B.S. Garbow, J. M. Boyle, J.J. Dongarra, and C. B. Moller, Matrix Eigensystem Routines—EISPACK Guide Extension, volume 51 of Lecture Notes in Computer Science. Berlin: Springer-Verlag, 1977.
37. M. Gardiner, “The superellipse: A curve that lies between the ellipse and the rectangle, Sci. Amer., vol. 213, no. 3, pp. 222–234, Sept. 1965.
38. D. B. Gennery, “Object detection and measurement using stereo vision,” in Proc. DARPA Image Understanding Workshop, Apr. 1980, pp. 161–167
39. A. A. Giordano and F. M. Hsu, Least Squares Estimation with Applications to Digital Signal Processing. New York: Wiley, 1985.
40. R. Gnanadesikan, Methods for Statistical Data Analysis of Multivariate Observations. New York: Wiley, 1977.
41. R. N. Goldman and T. W. Sederberg, “Some applications of resultants to problems in computational geometry,” Visual Comput., vol. 1, pp. 101–107, 1985.
42. G. Golub and C. F. Van Loan, Matrix Computaitons. Baltimore: John Hopkins University Press, 1983.
43. G. H. Golub and R. Underwood, “Stationary values of the ratio of quadratic forms subject to linear constraints,” Z. Angew. Math. Phys., vol. 21, pp. 318–326, 1970.
44. A. D. Gross and T. E. Boult, “Error of fit measures for recovering parametric solids,” in Proc. Second Int. Conf. Comput. Vision, Dec. 1988.
45. E. L. Hall, J. B. K. Tio, C. A. McPherson, and F. A. Sadjadi, “Measuring curved surfaces for robot vision, IEEE Comput., Dec. 1982, pp. 42–54.
46. H. Hotelling, “Analysis of a complex of statistical variables into principal components,” J. Educ. Psychol, vol. 24, pp. 417—444, 498–520, 1933.
47. E. Isaacson and H.B. Keller, Analysis of Numberical Methods. New York: Wiley, 1966.
48. I.T. Jollife, Principal Component Analysis. New York: Springer-Verlag, 1986.
49. K. Levenberg, “A method for the solution of certain problems in least squares, Quar. Appl. Math., vol. 2, pp. 164—168, 1944.
50. E. A. Lord and C. B. Wilson, The Mathematical Description of Shape and Form. Chichester: Ellis Horwood, 1984.
51. F. S. Macaulay. The Algebraic Theory of Modular Systems. Cambridge, UK: Cambridge University Press, 1916.
52. D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math, vol. 11, pp. 431–441, 1963.
53. J.J. More, B. S. Garbow, and K.E. Hillstrom, “User guide for minpack-1,” ANL-080-74, Argone Nat. Lab., 1980.
54. K. A. Paton, “Conic section in automatic chromosome analysis,” Machine Intell., vol. 5, p. 411, 1970.
55. K. A. Paton, “Conic sections in chromosome analysis,” Patt. Recogn., vol. 2, p. 39, 1970.
56. C. Pearson, Numerical Methods in Engineering and Science. New York: Van Nostrand Reinhold, 1986.
57. K. Pearson, “On lines and planes of closest fit to systems fo points in space,” Philos. Mag., vol. 2, p. 559, 1901, series 2.
58. A. P. Pentland, “Recognition by parts,” in Proc. First Int. Conf. Comput. Vision, June 1987, pp. 612–620.
59. J. Ponce and D. J. Kriegman, “On recognizing and positioning curved 3D objects from image contours,” in Proc. IEEE Workskhop Interpretation 3D Scenes, Nov. 1989.
60. V. Pratt, “Direct least squares fititng of algebraic surfaces, Comput. Graphics, vol. 21, no. 4, pp. 145–152, July 1987.
61. J.O. Ramsay, “A comparative study of sevseral robust estimates of slope, intercept, and scale in linear regression,” J Amer. Stat. Assoc., vol. 72, pp. 608–615, 1977.
62. M. Rioux and L. Cournoyer, “The NRCC three-dimensional image data files,” CNRC 29077, Nat. Res. Council Canada, June 1988.
63. G. Salmon, Modern Higher Algebra. Dublin: Hodges, Smith and Co., 1866.
64. P. D. Sampson, “Fitting conic sections to very scattered data: An iterative refinement of the Bookstein algorithm, Comput. Vision Graphics Image Processing, vol. 18, pp. 97–108, 1982.
65. J. Schwartz and M. Sharir, “Identificatin of partially obscured objects in two and three dimensions by matching noisy characteristic curves,” Int. J. Robotics Res., vol. 6, no. 2, 1987.
66. T. W. Sederberg, D. C. Anderson, and R. N. Goldman, “Implicit representation of parametric curves and surfaces,” Comput. Vision Graphics Image Processing, vol. 28, pp. 72–84, 1984.
67. J. F. Silverman and D. B. Cooper, “Bayesian clustering for unsupervised estimation of surface and texture models,” IEEE Trans. Patt. Anal. Machine Intell, vol. 10, July 1988.
68. B.T. Smith, J.M. Boyle, J.J. Dongarra, B.S. Garbow, Y. Ikebe, V.C. Klema, and C. B. Moller. Matrix Eigensystem Routines-EISPACK Guide, volume 6 of Lecture Notes in Computer Science. New York: Springer-Verlag, 1976.
69. F. Solina, “Shape recovery and segmentation with deformable part models,” Ph.D. thesis, Univ. Pennsylvania, Dec. 1987.
70. G. Taubin, “Algebraic nonplanar curve and surface estimation in 3-space with applications to positoin estimation,” Tech. Rep. LEMS-43, Brown Univ., Feb. 1988.
71. G. Taubin, “Nonplanar curve and surface estimation in 3-space,” in Proc. IEEE Conf. Robotics Automation, May 1988.
72. K. Turner, “Computer perception of curved objects using a television camera,” Ph.D. thesis, Univ. Edinburgh, Nov. 1974.
73. B. C. Vemuri, “Representation and recognition of objects from dense range maps,” Ph.D. thesis, Univ. Texas at Austin, Dept. Elect, Eng., 1987.
74. B. C. Vemuri, A. Mitiche, and J. K. Aggarwal, “Curvature-based representations of objects from range data,” Image Vision Comput., vol. 4, no. 2, pp. 107–114, May 1986.
75. R. Walker, Algebraic Curves. Princeton, NJ: Princeton University Press, 1950.
76. H. Weyl, The Classical Groups. Princeton, NJ: Princeton University Press, 1939.