Shock filter for image enhancement ($u_t=-\mathrm{sign}(\Delta u)|\nabla u|$)

The formal shock-filter equation $u_t=-\mathrm{sign}(\Delta u)|\nabla u|$ steepens transitions around inflection boundaries. It is a Hamilton–Jacobi-type transport mechanism rather than positive diffusion. Combining it with controlled forward image smoothing limits noise amplification.