over a region of $\mathbb{R}^2$, which should be compact and connected. For a deeper analysis, several function spaces have been proposed, such as functions of bounded variation (BV) or sums of the BV space (to represent piecewise smooth images) and Sobolev spaces (to represent noise and/or texture components), see e.g. [@Aujol-IJCV05; @Aujol-JMIV05; @Osher-MMS03; @Yin-vlsm05] and the numerous references therein. In this work, we will for simplicity assume that images are smooth functions.

\[def-img\] Let $\varOmega\subset\mathbb{R}^2$ be compact and equal to the closure of its interior. Let $u:\varOmega\to\mathbb{R}$ be a bounded smooth function on $\varOmega$. We call $u$ *smooth image with domain $\varOmega$*.

We denote by $\boldsymbol{\nabla}u$ the gradient of $u$, and by $\mathrm{D}^2u$ its Hessian, i.e., the symmetric $2\times2$-tensor of its second-order derivatives. We call $\boldsymbol{x}\in\varOmega$ *regular point of $u$* if $\boldsymbol{\nabla}u(\boldsymbol{x})\ne\boldsymbol{0}$.

We will now give some definitions to formalise the two steps of a local filter in a space-continuous setting. We start with the selection step.

\[def-sel\] Let $u$ be a smooth image with domain $\varOmega\subset\mathbb{R}^2$. Let $S$ be a map that assigns to each location $\boldsymbol{x}\in\varOmega$ a closed neighbourhood $S(\boldsymbol{x})\subset\mathbb{R}^2$. Then $S$ is called *selector* for $u$.

Whereas Definition \[def-sel\] actually allows some more generality (with regard to 