Linear Smoothing Morphological Sharpening + Kuwahara-Nagao Operator

This report sheds some new light on an old and trusted tool in the image processing toolbox. The Kuwahara-Nagao operator is known as an edge preserving smoothing operator. This report shows that we don’t need to trust on our intuition to verify this claim but that it follows from basic principles. To that end we will first cast the classical Kuwahara-Nagao operator into a more modern framework using Gaussian convolutions to calculate the mean and variance of local image patches and using morphological flow fields to select the neighborhood of minimal variance. The main result of this report is the formulation of the PDE whose solution is equivalent to the iteration of a Kuwahara-Nagao operator using infinitesimally small neighborhoods. The resulting PDE is a combination of linear diffusion and morphological sharpening.