$\mathbb{R}^2$, and $A$ a $\varGamma$-aggregator on $\mathbb{R}$.

The second-order PDE $u_t=F(\boldsymbol{\nabla}u,\mathrm{D}^2u)$ is the *regular PDE limit of $(A,\varSigma)$-filtering* with *time scale $\tau(\varrho)$* if for each smooth image $u$ the $(A,\varSigma)$-filtering exists, and if there exists an $\varepsilon>0$ such that for each regular point $\boldsymbol{x}$ of $u$ one has for $\varrho\to0$ $$\begin{aligned}
\frac{A(u,\varSigma_{u,\varrho})
(\boldsymbol{x})
-u(\boldsymbol{x})}{\tau(\varrho)}
- F\bigl(\boldsymbol{\nabla}u(\boldsymbol{x}),
\mathrm{D}^2u(\boldsymbol{x})\bigr) =
\mathcal{O}(\varrho^{\varepsilon}) \;.
\label{regpdelimit}\end{aligned}$$

**PDE limit of median filtering.**

Guichard and Morel [@Guichard-sana97] proved that space-continuous median filtering approximates for $\varrho\to0$ a well-known image filtering PDE evolution, namely *(mean) curvature flow* [@Alvarez-SINUM92].

With our definitions above, we can state their result as follows.

\[prop-gm\] For a smooth image, $(\operatorname{med},\mathcal{D})$-filtering with the median as aggregator and the family of discs $\mathcal{D}$ as scaled selector has the regular PDE limit with time scale $\tau(\varrho)=\varrho^2/6$ given by $$\begin{aligned}
u_t &= \lVert\boldsymbol{\nabla}u\rVert\,\operatorname{div}\,
\left(\frac{\boldsymbol{\nabla}u}{\lVert\boldsymbol{\nabla}u\rVert}\right)
\;.
\label{mcm}\end{aligned}$$

The PDE can be written in various forms. We mention that $\operatorname{div}\bigl(\boldsymbol{\nabla}u/\lVert\boldsymbol{\nabla}u\rVert
\bigr)$ gives at any regular image location $\boldsymbol{x}$ the curvature $\kappa(\boldsymbol{x})$ of the level line passing through $\boldsymbol{x}$. Therefore can be written as [@Sapiro-Book01 (2.25)] $$\begin{aligned}
u_t &= \kappa\,\lVert\boldsymbol{\nabla}u\rVert\;.\end{aligned}$$ Moreover, if $\boldsymbol{\xi}$ denotes at each image location a unit vector perpendicular to the gradient, thus, in the direction of the 