Lq-Closest-Point to Affine Subspaces Using the Generalized Weiszfeld Algorithm

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.