steps are combined in the following definition.

\[def-localfilter-2-1\] Let $u$ be a smooth image with domain $\varOmega$, and $S$ a selector for $u$. Let $u{\upharpoonright} S$ be the restriction of $u$ to $\varOmega\cap S$, i.e., $u{\upharpoonright} S:\varOmega\cap S\to\mathbb{R}$, $(u{\upharpoonright} S)(\boldsymbol{x})
=u(\boldsymbol{x})$ for $\boldsymbol{x}\in\varOmega\cap S$.

Let $A$ be a $\varGamma$-aggregator on $\mathbb{R}$. If for each $\boldsymbol{x}\in\varOmega$, the density $\gamma(u{\upharpoonright} (S(\boldsymbol{x})):
\mathbb{R}\to\mathbb{R}^+_0$ of the values of $u{\upharpoonright}\bigl(S(\boldsymbol{x})\bigr)$ belongs to $\varGamma$, we define $A(u,S):\varOmega\to\mathbb{R}^n$ by $A(u,S)(\boldsymbol{x}):=
A\bigl(\gamma(u{\upharpoonright} (\boldsymbol{x}+S))\bigr)$ for all $\boldsymbol{x}\in\varOmega$ and call it *$(A,S)$-filtering of $u$.*

If $\varSigma$ is a scaled selector for $u$, and the $(A,\varSigma_{u,\varrho})$-filtering of $u$ exists for all $\varrho>0$, we call the family of $(A,\varSigma_{u,\varrho})$-filterings the *$(A,\varSigma)$-filtering of $u$.*

Continuous median filtering can then be described as $(\operatorname{med},S)$-filtering with a suitable selector. For example, one step of $(\operatorname{med},D_\varrho)$-filtering assigns to each location $\boldsymbol{x}$ as filtered value the median of the grey-value density within a disc of radius $\varrho$ around this location.

In the following we will be interested in the family of such filters, i.e., the (scaled) $(\operatorname{med},\mathcal{D})$-filtering.

Infinitesimal Analysis of Median Filtering {#ssec-meddf-pde}
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When the neighbourhood radius $\varrho$ of a $(\operatorname{med},D_\varrho)$-filtering is reduced, the filtering result $\operatorname{med}(u,D_\varrho)$ will differ less and less from $u$ itself. However, 