Perona–Malik equation
Here is an implementation of the Perona–Malik equation,
\[\pd{u}{t}=\vnabla \cdot \left (\mathrm{e}^{-D |\vnabla u|^2}\vnabla u\right),\]which is used for image denoising. In particular, the nonlinear anisotropic diffusion causes sharp gradients to sharpen, and smooths out more shallow noisy regions.
- Load the interactive simulation: it starts with some text from a quote by Bernt Øksendal, with some noise added on top of it.
- Press to cause the initial condition to sharpen, making the text clearer.
- You can pause the simulation and use the brush to add more noise to the image.
- Play with the parameter $D$ to see how it influences the ability to denoise the text.
You can change the image to one of a noisy aperiodic tiling by modifying the initial conditions to use $I_S$ rather than $I_T$, and then restarting the simulation.
Other images will also work, but these may need some fine-tuning to have this algorithm improve their quality. In particular, finer meshes may be needed to preserve small edges.