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The Determination of Implicit Polynomial Canonical Curves
October 1998 (vol. 20 no. 10)
pp. 1080-1090

Abstract—A new method is presented for identifying and comparing closed, bounded, free-form curves that are defined by even implicit polynomial (IP) equations in the Cartesian coordinates x and y. The method provides a new expression for an IP involving a product of conic factors with unique conic factor centers. The critical points for an IP curve also are defined. The conic factor centers and the critical points are shown to be useful related points that directly map to one another under affine transformations. In particular, the explicit determination of such points implies both a canonical form for the curves and the transformation matrix which relates affine equivalent curves.

[1] D.H. Ballard and C.M. Brown, Computer Vision.Englewood Cliffs, N.J.: Prentice Hall, 1982.
[2] J. Bloomenthal, Introduction to Implicit Surfaces.San Mateo, Calif.: Morgan Kaufmann Publishers, 1997.
[3] K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. New York: Academic Press, 1990.
[4] R.M. Haralick, H. Joo, C.-N. Lee, X. Zhuang, and M.B. Kim, “Pose Estimation from Corresponding Point Data,” IEEE Trans. Systems, Man, and Cybernetics, vol. 19, no. 6, p. 1426, 1989.
[5] D. Keren,D. Cooper,, and J. Subrahmonia,“Describing complicated objects by implicit polynomials,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 1, pp. 38-53, 1994.
[6] Z. Lei, D. Keren, and D.B. Cooper, “Computationally Fast Bayesian Recognition of Complex Objects Based on Mutual Algebraic Invariants,” Proc. IEEE Int'l Conf. Image Processing, Oct. 1995.
[7] Z. Lei, M.M. Blane, and D.B. Cooper, “3L Fitting of Higher Degree Implicit Polynomials,” Proc. Third IEEE Workshop Applications of Computer Vision, pp. 148-153, Dec. 1996.
[8] Z. Lei, "Implicit Polynomial Shape Modeling and Recognition, and Application to Image/Video Databases," PhD dissertation, Division of Eng., Brown Univ., Providence, R.I., May, 1997.
[9] The MATLAB User's Guide. Math Works, Inc., 21 Eliot St., South Natick, MA 01760.
[10] J.L. Mundy and A. Zisserman, Geometric Invariants in Computer Vision.Cambridge, Mass.: MIT Press, 1992.
[11] S.M. Selby, CRC Standard Mathematical Tables, 30th ed. Chemical Rubber Company, 1996.
[12] G. Taubin and D.B. Cooper, "2D and 3D Object Recognition and Positioning With Algebraic Invariants and Covariants," Chapter 6 of Symbolic and Numerical Computation for Artificial Intelligence.New York: Academic Press, 1992.
[13] G. Taubin, F. Cukierman, S. Sullivan, J. Ponce, and D.J. Kriegman, “Parameterized Families of Polynomials for Bounded Algebraic Curve and Surface Fitting,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 3, pp. 287-303, Mar. 1994.
[14] M. Unel and W.A. Wolovich, "A Unique Decomposition of Algebraic Curves," Technical Note LEMS-166, 1997.
[15] B. Vijayakumar, D. Kriegman, and J. Ponce, "Invariant-Based Recognition of Complex 3D Curved Objects From Image Contours," Proc. Fifth Int'l Conf. Computer Vision, pp. 508-514,Boston, 1995.
[16] W.A. Wolovich and M. Unel, "Vision-Based System Identification and State Estimation," The Confluence of Vision and Control, Lecture Notes in Control and Information Systems. New York: Springer-Verlag, 1998.

Index Terms:
Implicit polynomials, affine transformations, object recognition, pose estimation, canonical curves.
William A. Wolovich, Mustafa Unel, "The Determination of Implicit Polynomial Canonical Curves," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20, no. 10, pp. 1080-1090, Oct. 1998, doi:10.1109/34.722620
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