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 \(p \rightarrow 0\), but for \(p \rightarrow -1\). 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.

本地M-平滑器是有趣且重要的信号和图像处理技术，与其他方法有很多联系。在本文中，我们推导出家庭偏微分方程（PDE的），其结果在一个，两个，和三个维度为限制从M-平滑器工艺是基于本地命令- p装置滚珠其半径趋于内零。阶数p可以取任何非零值\（>-1 \），也可以为负值。与文献结果相反，我们在空间连续的情况下表明，对于\（p \ rightarrow 0 \）不会发生模式滤波，而对于\（p \ rightarrow -1 \）会发生模式滤波。将我们的过滤器类扩展为小于的p值\（-1 \）允许包含例如Gabor的经典图像锐化流程。我们在1D，2D和3D中得出的PDE显示出很大的结构相似性。由于我们的PDE类是高度各向异性的，并且可能包含反向抛物线算符，因此设计适当的数值方法很困难。我们提出一个\（L ^ \ infty \） -稳定的显式有限差分方案，该方案满足离散的最大-最小原理，具有出色的旋转不变性，并采用分成四个小数步的方式来允许更大的时间步长。尽管它近似于抛物线型PDE，但其受益于双曲线型PDE数值的稳定概念。我们的2D实验表明\（p <1 \）的PDE 特别感兴趣：它们的后抛物线术语可产生有利的锐化特性，而它们似乎保持了平均曲率运动的强烈形状简化特性。