Stable Border Bases for Ideals of Points
This work provides a robust algebraic tool for geometric modeling and computer vision applications where point data is noisy.
The paper addresses the problem of computing a basis for the vanishing ideal of a set of points with uncertain coordinates, and presents a method to obtain a basis that is structurally stable under small perturbations. The result is a polynomial basis that remains valid for nearby point sets.
Let $X$ be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal $I(X)$ independent of the data uncertainty. We present a method to compute a polynomial basis $B$ of $I(X)$ which exhibits structural stability, that is, if $\widetilde X$ is any set of points differing only slightly from $X$, there exists a polynomial set $\widetilde B$ structurally similar to $B$, which is a basis of the perturbed ideal $ I(\widetilde X)$.