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Parameterized Families of Polynomials for Bounded Algebraic Curve and Surface Fitting
March 1994 (vol. 16 no. 3)
pp. 287-303

Interest in algebraic curves and surfaces of high degree as geometric models or shape descriptors for different model-based computer vision tasks has increased in recent years, and although their properties make them a natural choice for object recognition and positioning applications, algebraic curve and surface fitting algorithms often suffer from instability problems. One of the main reasons for these problems is that, while the data sets are always bounded, the resulting algebraic curves or surfaces are, in most cases, unbounded. In this paper, the authors propose to constrain the polynomials to a family with bounded zero sets, and use only members of this family in the fitting process. For every even number d the authors introduce a new parameterized family of polynomials of degree d whose level sets are always bounded, in particular, its zero sets. This family has the same number of degrees of freedom as a general polynomial of the same degree. Three methods for fitting members of this polynomial family to measured data points are introduced. Experimental results of fitting curves to sets of points in R/sup 2/ and surfaces to sets of points in R/sup 3/ are presented.

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Index Terms:
computer vision; matrix algebra; polynomials; surface fitting; curve fitting; polynomials; bounded algebraic curve fitting; surface fitting; geometric models; shape descriptors; model-based computer vision tasks; data sets; bounded zero sets
G. Taubin, F. Cukierman, S. Sullivan, J. Ponce, D.J. Kriegman, "Parameterized Families of Polynomials for Bounded Algebraic Curve and Surface Fitting," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 16, no. 3, pp. 287-303, March 1994, doi:10.1109/34.276128
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