Regularized Perona-Malik diffusion ($u_t=\mathrm{div}(c(|\nabla (G_{\sigma }\ast u)|)\nabla u)$)

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.