local level line, one can write the same equation as [@Sapiro-Book01 (2.26)] $$\begin{aligned}
u_t &= u_{\boldsymbol{\xi\xi}}\;,\end{aligned}$$ i.e., a diffusion process in level line direction, which gives reason to the names *geometric diffusion* or *geometric heat flow* that are also often used.

As an application of the result in Proposition \[prop-gm\], it has been proposed that median filtering may be used as a discretisation of mean curvature flow, see e.g. [@Merriman-tr92] which anticipated the relation between the two processes, and [@Jalba-ismm09].

Combination with Adaptive Structuring Elements
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An interesting modification of median filtering is obtained by altering the selection step to use image-adaptive neighbourhoods. Instead of a sliding window that is shifted to each pixel, the window shape is determined at each pixel individually. One approach to do this are the *morphological amoebas* introduced by Lerallut et al.[@Lerallut-ismm05; @Lerallut-IVC07]. Here, the distance between neighbouring pixels is measured by an *amoeba distance* combining standard Euclidean distance with contrast information, i.e., $$\begin{aligned}
d_\beta\bigl((i,j),(i',j')\bigr)&:=1+\beta\,\lvert u_{i,j}-u_{i',j'}\rvert
\label{damoeba-lerallut}\end{aligned}$$ for two pixels $(i,j)$ and $(i',j')$ in the image $u$ which are either horizontal or vertical neighbours, i.e., $\lvert i-i'\rvert+\lvert j-j'\rvert\le1$. This distance measure is extended to non-adjacent pixels via shortest paths, and for each pixel $(i,j)$ the 