Exploring non-linear diffusion: The diffusion echo

The Gaussian serves as Green's function for the linear diffusion equation and as a source for intuitive understanding of the linear diffusion process. In general, non-linear diffusion equations have no known closed form solutions and thereby no equally simple description. This article introduces a simple, intuitive description of these processes in terms of the Diffusion Echo. The Diffusion Echo offers intuitive visualisations for non-linear diffusion processes. In addition, the Diffusion Echo has potential for offering simple formulations for grouping problems. Furthermore, the Diffusion Echo can be considered a deep structure summary and thereby offers an alternative to multi-scale linking and flooding techniques.