Mumford–Shah functional

Minimization of the functional

Ambrosio–Tortorelli limit

Ambrosio and Tortorelli showed that Mumford–Shah functional E[ J,B ] can be obtained as the limit of a family of energy functionals E[ J,z,ε ] where the boundary B is replaced by continuous function z whose magnitude indicates the presence of a boundary. Their analysis show that the Mumford–Shah functional has a well-defined minimum. It also yields an algorithm for estimating the minimum.

The functionals they define have the following form:

${\displaystyle E[J,z;\epsilon ]=\alpha \int (I(\overrightarrow{x})-J(\overrightarrow{x}){)}^2\mathrm{d}\overrightarrow{x}+\beta \int z(\overrightarrow{x})|\overrightarrow{\mathrm{\nabla }}J(\overrightarrow{x}){|}^2\mathrm{d}\overrightarrow{x}+\gamma \int \{\epsilon |\overrightarrow{\mathrm{\nabla }}z(\overrightarrow{x}){|}^2+{\epsilon }^{-1}{\varphi }^2(z(\overrightarrow{x}))\}\mathrm{d}\overrightarrow{x}}$

where ε > 0 is a (small) parameter and ϕ(z) is a potential function. Two typical choices for ϕ(z) are

• ${\displaystyle {\varphi }_1(z)=(1-z^2)/2z\in [-1,1].}$ This choice associates the edge set B with the set of points z such that ϕ₁(z) ≈ 0
• ${\displaystyle {\varphi }_2(z)=z(1-z)z\in [0,1].}$ This choice associates the edge set B with the set of points z such that ϕ₂(z) ≈ 1/4

The non-trivial step in their deduction is the proof that, as ${\displaystyle ϵ\to 0}$, the last two terms of the energy function (i.e. the last integral term of the energy functional) converge to the edge set integral ∫_Bds.

The energy functional E[ J,z,ε ] can be minimized by gradient descent methods, assuring the convergence to a local minimum.

Ambrosio, Fusco, and Hutchinson, established a result to give an optimal estimate of the Hausdorff dimension of the singular set of minimizers of the Mumford-Shah energy.

Minimization by splitting into one-dimensional problems

The Mumford-Shah functional can be split into coupled one-dimensional subproblems. The subproblems are solved exactly by dynamic programming.