  Local M-smoothers are interesting and important signal and image processing
techniques with many connections to other methods. In our paper we derive a
family of partial differential equations (PDEs) that result in one, two, and
three dimensions as limiting processes from M-smoothers which are based on
local order-$p$ means within a ball the radius of which tends to zero. The
order $p$ may take any nonzero value $>-1$, allowing also negative values. In
contrast to results from the literature, we show in the space-continuous case
that mode filtering does not arise for $p \to 0$, but for $p \to -1$. Extending
our filter class to $p$-values smaller than $-1$ allows to include e.g. the
classical image sharpening flow of Gabor. The PDEs we derive in 1D, 2D, and 3D
show large structural similarities. Since our PDE class is highly anisotropic
and may contain backward parabolic operators, designing adequate numerical
methods is difficult. We present an $L^\infty$-stable explicit finite
difference scheme that satisfies a discrete maximum--minimum principle, offers
excellent rotation invariance, and employs a splitting into four fractional
steps to allow larger time step sizes. Although it approximates parabolic PDEs,
it consequently benefits from stabilisation concepts from the numerics of
hyperbolic PDEs. Our 2D experiments show that the PDEs for $p<1$ are of
specific interest: Their backward parabolic term creates favourable sharpening
properties, while they appear to maintain the strong shape simplification
properties of mean curvature motion.
