PARAMETERIZED FAMILIES OF POLYNOMIALS FOR BOUNDED ALGEBRAIC CURVE ANDSURFACE FITTING

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 mak e them a natural choice for object recognition and positioning applica tions, algebraic curve and surface fitting algorithms often suffer fro m 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, we pr opose 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 we introduce a new parameterized family of polynomials of degree d whose level sets are always bounded, in particular, its ze ro sets. This family has the same number of degrees of freedom as a ge neral polynomial of the same degree. Three methods for fitting members of this polynomial family to measured data points are introduced. Exp erimental results of fitting curves to sets or points in R2 and surfac es to sets of points in R3 are presented.