  Multivariate median filters have been proposed as generalisations of the
well-established median filter for grey-value images to multi-channel images.
As multivariate median, most of the recent approaches use the $L^1$ median,
i.e.\ the minimiser of an objective function that is the sum of distances to
all input points. Many properties of univariate median filters generalise to
such a filter. However, the famous result by Guichard and Morel about
approximation of the mean curvature motion PDE by median filtering does not
have a comparably simple counterpart for $L^1$ multivariate median filtering.
We discuss the affine equivariant Oja median and the affine equivariant
transformation--retransformation $L^1$ median as alternatives to $L^1$ median
filtering. We analyse multivariate median filters in a space-continuous
setting, including the formulation of a space-continuous version of the
transformation--retransformation $L^1$ median, and derive PDEs approximated by
these filters in the cases of bivariate planar images, three-channel volume
images and three-channel planar images. The PDEs for the affine equivariant
filters can be interpreted geometrically as combinations of a diffusion and a
principal-component-wise curvature motion contribution with a cross-effect term
based on torsions of principal components. Numerical experiments are presented
that demonstrate the validity of the approximation results.
