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  • 1991
  • Issue No. 11 - November
  • Abstract - Estimation of Planar Curves, Surfaces, and Nonplanar Space Curves Defined by Implicit Equations with Applications to Edge and Range Image Segmentation
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Estimation of Planar Curves, Surfaces, and Nonplanar Space Curves Defined by Implicit Equations with Applications to Edge and Range Image Segmentation
November 1991 (vol. 13 no. 11)
pp. 1115-1138

The author addresses the problem of parametric representation and estimation of complex planar curves in 2-D surfaces in 3-D, and nonplanar space curves in 3-D. Curves and surfaces can be defined either parametrically or implicitly, with the latter representation used here. A planar curve is the set of zeros of a smooth function of two variables x-y, a surface is the set of zeros of a smooth function of three variables x-y-z, and a space curve is the intersection of two surfaces, which are the set of zeros of two linearly independent smooth functions of three variables x-y-z For example, the surface of a complex object in 3-D can be represented as a subset of a single implicit surface, with similar results for planar and space curves. It is shown how this unified representation can be used for object recognition, object position estimation, and segmentation of objects into meaningful subobjects, that is, the detection of 'interest regions' that are more complex than high curvature regions and, hence, more useful as features for object recognition.

[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," inProc. 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, Aug. 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 application to estimating 3-D complex-object position from range data,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 619-638, 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 dimensional part orientation svstem,"Int. J. Robotics Res., vol. 5, no. 3, Fall 1986, pp. 3-26.
[13] R. C. Bolles, P. Horaud, and M. J. Hannah, "3DPO: A three-dimensinal part orientation sytem," inProc. 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," inProc. 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, inProc. 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-5035/1, 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," inProc. 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 experiemental 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," inProc. 8th. Int. Joint Conf. Artificial Intell. (IJCAI), 1983.
[34] O.D. Faugeras, M. Hebert, and E. Pauchon," Segmentation of range data into planar and quadric patches," inProc. 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 ofLecture Note 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," inProc. 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," inProc. 2nd Int. Conf. Comput. vision, Dec. 1988, pp. 690-694.
[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," inProc. 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," inProc. IEEE Workshop on Interpretation of 3D Scenes, Austin, TX, Nov. 1989, pp. 61-67.
[60] V. Pratt, "Direct least squares fitting 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. T. Schwartz and M. Sharir, "Identification of objects in two and three dimensions by matching noisy characteristic curves,"Int. J. Robotics Res., vol. 6, no. 2, pp. 29-44, 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. Bovle, J.J. Dongarra, B.S. Garbow, Y. Ikebe, V.C. Klema, and C.B. Moller,Matrix Eigensystem Routines-EISFACK Guide, volume 6 ofLecture Notes in Computer Science. New York: Springer-Verlag, 1976.
[69] F. Solina, "Shape recovery and segmentation with deformable part models," Ph.D. dissertation, Univ. Pennsylvania, 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," inProc. 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. dissertation, Univ. of Texas at Austin, Dept. of Electrical and Computer Engineering, Austin, 1987.
[74] B.C. Vemuri, A. Mitiche, and J.K. Aggarwal, "Curvature-based representation of objects from range data,"Image and 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.

Index Terms:
pattern recognition; algebra; optimisation; edge segmentation; planar curves; surfaces; nonplanar space curves; implicit equations; range image segmentation; parametric representation; 2-D; 3-D; zeros; object recognition; object position estimation; algebra; curve fitting; optimisation; pattern recognition
Citation:
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, vol. 13, no. 11, pp. 1115-1138, Nov. 1991, doi:10.1109/34.103273
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