L q-closest-point to affine subspaces using the generalized Weiszfeld Algorithm

Abstract

This paper presents a method for finding an Lq - 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 ≤ q < 2. We give a theoretical proof for the convergence of the proposed algorithm to a unique Lq minimum. The proposed method is motivated by the Lq Weiszfeld algorithm, an extremely simple and rapid averaging algorithm, that finds the Lq 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 Lq -closest-point to a set of lines in 3D. Our experimental results for the triangulation problem confirm that the Lq -closest-point method, for 1 ≤ q < 2, is more robust to outliers than the L2-closest-point method Show more