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PDE Evolutions for M-Smoothers in One Two and Three Dimensions

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Document pages: 46 pages

Abstract: Local M-smoothers are interesting and important signal and image processingtechniques with many connections to other methods. In our paper we derive afamily of partial differential equations (PDEs) that result in one, two, andthree dimensions as limiting processes from M-smoothers which are based onlocal order-$p$ means within a ball the radius of which tends to zero. Theorder $p$ may take any nonzero value $>-1$, allowing also negative values. Incontrast to results from the literature, we show in the space-continuous casethat mode filtering does not arise for $p to 0$, but for $p to -1$. Extendingour filter class to $p$-values smaller than $-1$ allows to include e.g. theclassical image sharpening flow of Gabor. The PDEs we derive in 1D, 2D, and 3Dshow large structural similarities. Since our PDE class is highly anisotropicand may contain backward parabolic operators, designing adequate numericalmethods is difficult. We present an $L^ infty$-stable explicit finitedifference scheme that satisfies a discrete maximum--minimum principle, offersexcellent rotation invariance, and employs a splitting into four fractionalsteps to allow larger time step sizes. Although it approximates parabolic PDEs,it consequently benefits from stabilisation concepts from the numerics ofhyperbolic PDEs. Our 2D experiments show that the PDEs for $p<1$ are ofspecific interest: Their backward parabolic term creates favourable sharpeningproperties, while they appear to maintain the strong shape simplificationproperties of mean curvature motion.

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