of image and shape data. Results on connections between multivariate median filters and PDE-based image filters are reviewed. Results from existing work, including some by the author of this paper, are reported. In order to present the overarching principles and ideas, proofs of individual results are generally omitted, although hints at important ideas behind these proofs are given where appropriate.

The remainder of this paper is organised as follows. In Section \[sec-med\], different ways to define univariate medians for discrete sets and continuous distributions are juxtaposed, and some equivariance properties and algorithmic aspects are reviewed. Median filtering of univariate (grey-scale) images is discussed in Section \[sec-medf\], covering also a space-continuous model and its relation to PDEs as well as an extension to adaptive neighbourhoods (so-called morphological amoebas). In Section \[sec-mvmed\], a selection of multivariate median concepts are presented, namely, the $L^1$ median, Oja’s simplex median, a transformation–retransformation $L^1$ median, the half-space median and the convex-hull-stripping median. It will be shown how these concepts are derived from generalising different variants of univariate median definitions. Properties of multivariate median concepts that are relevant to image and shape processing are pointed out, with emphasis on the relation between discrete and continuous models, and 