= Ambrosio–Tortorelli approximation
{c}
{title2=$E_\varepsilon(u,v)$}
The <Mumford–Shah functional> can be approximated by an auxiliary edge field $0\le v\le1$ and energy
$$
E_\varepsilon=\int(u-g)^2+\alpha(v^2+\eta_\varepsilon)|\nabla u|^2+\beta\left[\varepsilon|\nabla v|^2+\frac{(1-v)^2}{4\varepsilon}\right],\qquad0<\eta_\varepsilon=o(\varepsilon).
$$
The field is near one inside regions and near zero across an <image edge>, weakening smoothing there. The profile $v(s)=1-e^{-|s|/(2\varepsilon)}$ has unit edge energy per unit length in this normalization. The https://onlinelibrary.wiley.com/doi/10.1002/cpa.3160430805[1990 construction] is a <Gamma-convergence> approximation; its global-minimizer <limit> does not guarantee global optimality of an alternating numerical solve. Each fixed-field subproblem is quadratic, but the joint energy is nonconvex.
Back to article page
