The 3L algorithm for fitting implicit polynomial curves and surfaces to data

In this paper, we introduce a completely new approach to fitting implicit p olynomial 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, repeat able fitting to unorganized data which may not be uniformly sampled and whi ch may contain gaps, to permit position-invariant shape recognition based o n new complete sets of Euclidean and affine invariants and to permit fast, stable single-computation pose estimation. The algorithm represents a signi ficant advancement of implicit polynomial technology for four important rea sons. 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 rep eatability, numerical stability. and robustness than current methods in dea ling with noisy, deformed. or missing data. Third, it can easily fit polyno mials of high, such as 14th or 16th, degree. Fourth, additional linear cons traints can be easily incorporated into the fitting process, and general li near vector space concepts apply.