Mumford–Shah functional

Ambrosio and Tortorelli[2] 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:

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

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

• ${\displaystyle \phi _{1}(z)=(1-z^{2})/2\quad z\in [-1,1].}$ This choice associates the edge set B with the set of points z such that ϕ₁(z) ≈ 0
• ${\displaystyle \phi _{2}(z)=z(1-z)\quad z\in [0,1].}$ 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 ${\displaystyle \epsilon \to 0}$, 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.[3]

The Mumford-Shah functional can be split into coupled one-dimensional subproblems. The subproblems are solved exactly by dynamic programming. [4]