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Fitting Curves and Surfaces With Constrained Implicit Polynomials
January 1999 (vol. 21 no. 1)
pp. 31-41

Abstract—A problem which often arises while fitting implicit polynomials to 2D and 3D data sets is the following: Although the data set is simple, the fit exhibits undesired phenomena, such as loops, holes, extraneous components, etc. Previous work tackled these problems by optimizing heuristic cost functions, which penalize some of these topological problems in the fit. This paper suggests a different approach—to design parameterized families of polynomials whose zero-sets are guaranteed to satisfy certain topological properties. Namely, we construct families of polynomials with star-shaped zero-sets, as well as polynomials whose zero-sets are guaranteed not to intersect an ellipse circumscribing the data or to be entirely contained in such an ellipse. This is more rigorous than using heuristics which may fail and result in pathological zero-sets. The ability to parameterize these families depends heavily on the ability to parameterize positive polynomials. To achieve this, we use some powerful recent results from real algebraic geometry.

[1] C.L. Bajaj, J. Chen, and G. Xu, "Modeling with Cubic A-Patches," ACM Trans. Graphics, vol. 14, no. 2, pp. 103-133, 1995.
[2] 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.
[3] M.D. Choi, T.Y. Lam, and B. Reznick, "Sums of Squares of Real Polynomials," Proc. Symposia Pure Mathematics, vol. 58.2, pp. 103-126, 1995.
[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] D. Hilbert, "Uber die darstellung definiter formen als summe von formen-quadsraten," Math. Ann., vol. 32, pp. 342-350, 1888.
[7] D. Keren, "Some New Invariants in Computer Vision." IEEE Trans. On Pattern Analysis and Machine Intelligence, pp. 1,143-1,149, Nov. 1994.
[8] 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.
[9] D.J. Kriegman and J. Ponce, "On Recognizing and Positioning Curve 3-D Objects From Image Contours," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, pp. 1,127-1,137, Dec. 1990.
[10] 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.
[11] D. Levin and E. Nadler, "Convexity Preserving Interpolation by Algebraic Curves and Surfaces," Numerical Algorithms, vol. 9, pp. 113-139, 1995.
[12] P.D Sampson, "Fitting Conic Sections to Very Scattered Data: An Iterative Improvement of the Bookstein Algorithm," Computer Vision, Graphics, and Image Processing, vol. 18, pp. 97-108, 1982.
[13] 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.
[14] S. Sullivan, L. Sandford, and J. Ponce, "Using Geometric Distance Fits for 3D Object Modeling and Recognition," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 12, pp. 1,183-1,196, 1994.
[15] 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.
[16] 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.

Index Terms:
Implicit polynomials, fitting, free-form shapes, topological integrity, starshaped curves and surfaces, positive polynomials.
Citation:
Daniel Keren, Craig Gotsman, "Fitting Curves and Surfaces With Constrained Implicit Polynomials," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 21, no. 1, pp. 31-41, Jan. 1999, doi:10.1109/34.745731
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