International Journal of Computer Vision

, Volume 114, Issue 1, pp 1–15 | Cite as

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

Article

Abstract

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.

Keywords

\(L_q\) Weiszfeld algorithm Affine subspaces Triangulation \(L_q\) mean 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.College of Engineering and Computer ScienceAustralian National University and National ICT AustraliaCanberraAustralia
  2. 2.Research School of EngineeringAustralian National UniversityCanberraAustralia

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