Multivariate Median Filters and Partial Differential Equations

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.