On vector and matrix median computation

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 2 × 2 matrices which frequently arise in image processing. Our findings are explained and illustrated by numerical examples. To solve the corresponding minimization problems we propose different algorithms. Weiszfeld’s algorithm was developed for the l2 vector median computation and semi-definite programming, in particular second order cone programming, was used for the matrix median computation in the literature. We show that also two splitting methods, the alternating direction method of multipliers and the parallel proximal algorithm, can be applied for generalized vector and matrix median computations. Furthermore, we adapt Weiszfeld’s algorithm for our setting. We compare these algorithms numerically and apply them within local median filters.