In many applications, e.g., medical or satellite imaging, the edges are key features and thus must be preserved sharp and undistorted in smoothing/denoising.
Edge-preserving filters are designed to automatically limit the smoothing at “edges” in images measured, e.g., by high gradient magnitudes.
For example, the motivation for anisotropic diffusion (also called nonuniform or variable conductance diffusion) is that a Gaussian smoothed image is a single time slice of the solution to the heat equation, that has the original image as its initial conditions.
A common default choice for the parameters of the filter is aimed for natural images and results in strong denoising at the cost of some smoothing of the edges.
While not-physical for the heat equation, this effect results in sharpening corners of one-dimensional signals, when used in graph-based smoothing filters, as shown in reference [3] that also provides an alternative physical interpretation using the wave equation describing mechanical vibrations of a mass-spring system with some repulsive springs.