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Stable Fitting of 2D Curves and 3D Surfaces by Implicit Polynomials
October 2004 (vol. 26 no. 10)
pp. 1283-1294
This work deals with fitting 2D and 3D implicit polynomials (IPs) to 2D curves and 3D surfaces, respectively. The zero-set of the polynomial is determined by the IP coefficients and describes the data. The polynomial fitting algorithms proposed in this paper aim at reducing the sensitivity of the polynomial to coefficient errors. Errors in coefficient values may be the result of numerical calculations, when solving the fitting problem or due to coefficient quantization. It is demonstrated that the effect of reducing this sensitivity also improves the fitting tightness and stability of the proposed two algorithms in fitting noisy data, as compared to existing algorithms like the well-known 3L and gradient-one algorithms. The development of the proposed algorithms is based on an analysis of the sensitivity of the zero-set to small coefficient changes and on minimizing a bound on the maximal error for one algorithm and minimizing the error variance for the second. Simulation results show that the proposed algorithms provide a significant reduction in fitting errors, particularly when fitting noisy data of complex shapes with high order polynomials, as compared to the performance obtained by the abovementioned existing algorithms.

[1] T.W. Sederberg and D.C. Anderson, Implicit Representation of Parametric Curves and Surfaces Computer Vision, Graphics, and Image Processing, vol. 28, no. 1, pp. 72-84, 1984.
[2] 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.
[3] 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.
[4] D. Forsyth, J.L. Mundy, A. Zisserman, C. Coelho, A. Heller, and C. Rothwell, "Invariant descriptors for 3-D object recognition and pose," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 10, pp. 971-991, 1991.
[5] D.A. Forsyth, Recognizing Algebraic Surfaces from Their Outlines Proc. Int'l Conf. Computer Vision, pp. 476-480, May 1993.
[6] M. Barzohar, D. Keren, and D. Cooper, Recognizing Groups of Curves Based on New Affine Mutual Geometric Invariants, with Applications to Recognizing Intersecting Roads in Aerial Images Proc. IAPR Int'l Conf. Pattern Recognition, vol. 1, pp. 205-209, Oct. 1994.
[7] J.-P. Trael and D.B. Cooper, The Complex Representation of Algebraic Curves and Its Simple Exploitation for Pose Estimation and Invariant Recognition IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 7, pp. 663-674, July 2000.
[8] A. Helzer, M. Bar-Zohar, and D. Malah, Using Implicit Polynomials for Image Compression Proc. 21st IEEE Convention of the Electrical and Electronic Eng. in Israel, pp. 384-388, Apr. 2000.
[9] T. Tasdizen and D.B. Cooper, Boundary Estimation from Intensity/Color Images with Algebraic Curve Models Int'l Conf. Pattern Recognition, pp. 1225-1228, 2000.
[10] C. Bajaj, I. Ihm, and J. Warren, Higher-Order Interpolation and Least-Squares Approximation Using Implicit Algebraic Surfaces ACM Trans. Graphics, vol. 12, no. 4, pp. 327-347, 1993.
[11] J. Subrahmonia, D.B. Cooper, and D. Keren, “Practical, Reliable, Bayesian Recognition of 2D and 3D Objects Using Implicit Polynomials and Algebraic Invariants,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 5, pp. 505-519, May 1996.
[12] J.-P. Tarel, W.A. Wolovich, and D.B. Cooper, Covariant Conics Decomposition of Quartics for 2-D Object Recognition and Affine Alignment Proc. Int'l Conf. Image Processing, Oct. 1998.
[13] M.M. Blane, Z. Lei, H. Civil, and D.B. Cooper, The 3L Algorithm for Fitting Implicit Polynomials Curves and Surface to Data IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 3, Mar. 2000.
[14] Z. Lei and D.B. Cooper, New, Faster, More Controlled Fitting of Implicit Polynomial 2D Curves and 3D Surfaces to Data Proc. IEEE Conf. Computer Vision and Pattern Recognition, June 1996.
[15] Z. Lei and D.B. Cooper, Linear Programming Fitting of Implicit Polynomials IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 2, pp. 212-217, Feb. 1998.
[16] 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.
[17] T. Tasdizen, J.-P. Tarel, and D.B. Cooper, Improving the Stability of Algebraic Curves for Applications IEEE Trans. Image Processing, vol. 9, no. 3, pp. 405-416, Mar. 2000.
[18] A. Helzer, M. Bar-Zohar, and D. Malah, Robust Fitting of Implicit Polynomials with Quantized Coefficients to 2D Data Proc. 15th Int'l Conf. Pattern Recognition, pp. 290-293, Sept. 2000.
[19] A. Helzer, Using Implicit Polynomials for Contour Coding MSc Thesis, Technion-Israel Inst. of Technology, Haifa, Israel, Dec. 2000.
[20] C. Oden, A. Ercil, V.T. Yildiz, H. Kirmizita, and B. Buke, Hand Recognition Using Implicit Polynomials and Geometric Features Proc. Third Int'l Conf. Audio-and-Video-Based Biometric Person Authentication, AVBPA-2001, June 2001.

Index Terms:
Implicit polynomials, zero-set sensitivity, curve and surface fitting, stable fitting.
Citation:
Amir Helzer, Meir Barzohar, David Malah, "Stable Fitting of 2D Curves and 3D Surfaces by Implicit Polynomials," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 10, pp. 1283-1294, Oct. 2004, doi:10.1109/TPAMI.2004.91
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