and their variants such as Gaussian smoothing are useful for denoising e.g. additive Gaussian noise, they break down in the presence of noise featuring more outliers such as impulsive noise. In contrast, the median filter performs surprisingly well in removing such noise; at the same time, it is capable of preserving sharp signal or image boundaries. Moreover, an interesting connection to partial differential equations (PDEs) could be proven [@Guichard-sana97].

The success of univariate median filtering has inspired researchers to investigate similar procedures for multivariate images. In several works on this subject [@Astola-PIEEE90; @Spence-icip07; @Welk-dagm03], the minimisation of an objective function composed of distances of the median point to the input values was considered, building on or partly rediscovering concepts that had been around long before in the statistical community [@Gini-Met29] and are nowadays often referred to as $L^1$ medians. In the statistical literature, however, shortcomings of this concept have been discussed since the 1940s [@Haldane-Biomet48], and various alternative ideas have been developed since then [@Small-ISR90]. Recently, attempts have been made to introduce some of these alternative concepts to image processing [@Welk-JMIV16; @Welk-ssvm19].

The ability of the median to reduce a data set to a single value representing its position, and 