\({L_q}\)-Closest-Point to Affine Subspaces Using the Generalized Weiszfeld Algorithm

摘要

This paper presents a method for finding an \(L_q\)-closest-point to a set of affine subspaces, that is a point for which the sum of the q-th power of orthogonal distances to all the subspaces is minimized, where \(1 \le q < 2\). We give a theoretical proof for the convergence of the proposed algorithm to a unique \(L_q\) minimum. The proposed method is motivated by the \(L_q\) Weiszfeld algorithm, an extremely simple and rapid averaging algorithm, that finds the \(L_q\) mean of a set of given points in a Euclidean space. The proposed algorithm is applied to the triangulation problem in computer vision by finding the \(L_q\)-closest-point to a set of lines in 3D. Our experimental results for the triangulation problem confirm that the \(L_q\)-closest-point method, for \(1 \le q < 2\), is more robust to outliers than the \(L_2\)-closest-point method.