Perona-Malik equation ($u_t=\mathrm{div}(c(|\nabla u|)\nabla u)$)

The Perona-Malik equation uses decreasing diffusivity to smooth low-gradient image regions while reducing transport across edges. For $r=|\nabla u|>0$, its principal diffusion coefficients are $c(r)$ tangent to a level set and $c(r)+r{c}^{\prime }(r)$ normal to it. A negative normal coefficient allows formal sharpening but causes forward-backward ill-posedness. Positivity of $c$ alone is insufficient for parabolicity.

For one-dimensional flux $F(p)=p|p|e^{-p^2/(2{\lambda }^2)}$, the nonlinear diffusion equation is $u_t=F^{\prime }(u_x)u_{xx}$. The sole one-dimensional member of the principal diffusion coefficients is positive for $0<|u_x|<\sqrt{2}\lambda$, negative above $\sqrt{2}\lambda$, and zero at the two endpoints. Negative diffusion gives formal sharpening with ill-posedness through high-frequency growth. The extra factor $|p|$ matters: the unweighted exponential conductance has threshold $\lambda$ instead.

Computing conductance from a smoothed gradient gives $u_t=\mathrm{div}(c(|\nabla (G_{\sigma }\ast u)|)\nabla u)$. The image edge detector is nonlocal, while the local highest-order diffusion has positive coefficient when conductance is positive. A smooth kernel, lower conductance bound and appropriate regularity/boundary hypotheses support a well-posed regularized filter.