Perona–Malik equation

Art | Parabolic

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.

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