in iterated filtering the reduced filtering effect of a single step can be compensated by increasing the number of iterations by a suitable scale. In the limit $\varrho\to0$, the number of filtering steps goes to infinity whereas the effect of each step goes to zero. Instead of a sequence of progressively filtered images, an image evolution with a continuous time parameter results.

Let $\varOmega\subset\mathbb{R}^2$ be as in Definition \[def-img\], and $[0,T]\subset\mathbb{R}$ an interval. The coordinates of $\Omega$ will be called *spatial coordinates,* the additional coordinate from $[0,T]$ *time coordinate.* Let $u:\varOmega\times[0,T]\to\mathbb{R}$ be a bounded smooth function. We call $u$ an *image evolution with spatial domain $\varOmega$.*

We denote by $\boldsymbol{\nabla}u$ the spatial gradient of $u$, i.e., the vector of its first-order derivatives w.r.t. the spatial coordinates, and by $\mathrm{D}^2u$ its spatial Hessian, i.e., the $2\times2$-tensor of its second-order derivatives w.r.t. the spatial coordinates.

By $u(t^*)$ for $t^*\in[0,T]$ we denote the smooth image with $u(t^*)(\boldsymbol{x}):=u(\boldsymbol{x},t^*)$ for all $\boldsymbol{x}\in\Omega$.

Time-continuous image evolutions can be described by partial differential equations (PDEs). To capture the limit transition $\varrho\to0$ of an iterated local filter, including the appropriate scaling of the iteration count, we use the following definition.

\[def-regpdelimit\] Let $\varSigma$ be a scaled selector in 