 Ambrosio–Tortorelli approximation

ID: ambrosio-tortorelli-approximation

The Mumford–Shah functional can be approximated by an auxiliary edge field

0 \leq v \leq 1

E_{ε} = \int (u - g)^{2} + α (v^{2} + η_{ε}) ∣\nabla u ∣^{2} + β [ε ∣\nabla v ∣^{2} + \frac{( 1 - v ) ^{2}}{4 ε}], 0 < η_{ε} = o (ε) .

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 ε)}

has unit edge energy per unit length in this normalization. The 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.

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