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The 3L Algorithm for Fitting Implicit Polynomial Curves and Surfaces to Data
March 2000 (vol. 22 no. 3)
pp. 298-313

Abstract—In this paper, we introduce a completely new approach to fitting implicit polynomial geometric shape models to data and to studying these polynomials. The power of these models is in their ability to represent nonstar complex shapes in two- (2D) and three-dimensional (3D) data to permit fast, repeatable fitting to unorganized data which may not be uniformly sampled and which may contain gaps, to permit position-invariant shape recognition based on new complete sets of Euclidean and affine invariants and to permit fast, stable single-computation pose estimation. The algorithm represents a significant advancement of implicit polynomial technology for four important reasons. First, it is orders of magnitude faster than existing fitting methods for implicit polynomial 2D curves and 3D surfaces, and the algorithms for 2D and 3D are essentially the same. Second, it has significantly better repeatability, numerical stability, and robustness than current methods in dealing with noisy, deformed, or missing data. Third, it can easily fit polynomials of high, such as 14th or 16th, degree. Fourth, additional linear constraints can be easily incorporated into the fitting process, and general linear vector space concepts apply.

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Index Terms:
Least-squares fitting, implicit polynomial representations, curve and surface fitting, fitting with constraints, algebraic curves and surfaces.
Michael M. Blane, Zhibin Lei, Hakan Çivi, David B. Cooper, "The 3L Algorithm for Fitting Implicit Polynomial Curves and Surfaces to Data," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 3, pp. 298-313, March 2000, doi:10.1109/34.841760
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