On vector and matrix median computation

  • Authors:

  • S. Setzer;G. Steidl;T. Teuber

  • Affiliations:

  • Saarland University, Department of Mathematics and Computer Science, Campus E1.1, 66041 Saarbrücken, Germany;University of Kaiserslautern, Department of Mathematics, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern, Germany;University of Kaiserslautern, Department of Mathematics, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern, Germany

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2012

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Abstract

The aim of this paper is to gain more insight into vector and matrix medians and to investigate algorithms to compute them. We prove relations between vector and matrix means and medians, particularly regarding the classical structure tensor. Moreover, we examine matrix medians corresponding to different unitarily invariant matrix norms for the case of symmetric 2x2 matrices, which frequently arise in image processing. Our findings are explained and illustrated by numerical examples. To solve the corresponding minimization problems, we propose several algorithms. Existing approaches include Weiszfeld's algorithm for the computation of @?"2 vector medians and semi-definite programming, in particular, second order cone programming, which has been used for matrix median computation. In this paper, we adapt Weiszfeld's algorithm for our setting and show that also two splitting methods, namely the alternating direction method of multipliers and the parallel proximal algorithm, can be applied for generalized vector and matrix median computations. Besides, we compare the performance of these algorithms numerically and apply them within local median filters.