Mumford–Shah functional
# Mumford–Shah functional

The Mumford–Shah functional is a functional that is used to establish an optimality criterion for segmenting an image into sub-regions. An image is modeled as a piecewise-smooth function. The functional penalizes the distance between the model and the input image, the lack of smoothness of the model within the sub-regions, and the length of the boundaries of the sub-regions. By minimizing the functional one may compute the best image segmentation. The functional was proposed by mathematicians David Mumford and Jayant Shah in 1989.

## Definition of the Mumford–Shah functional

Consider an image I with a domain of definition D, call J the image's model, and call B the boundaries that are associated with the model: the Mumford–Shah functional E[ J,B ] is defined as

    $E[J,B]=\alpha \int _{D}(I({\vec {x}})-J({\vec {x}}))^{2}\,\mathrm {d} {\vec {x}}+\beta \int _{D/B}{\vec {\nabla }}J({\vec {x}})\cdot {\vec {\nabla }}J({\vec {x}})\,\mathrm {d} {\vec {x}}+\gamma \int _{B}ds$

Optimization of the functional may be achieved by approximating it with another functional, as proposed by Ambrosio and Tortorelli.

## 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:

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

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

* $\phi _{1}(z)=(1-z^{2})/2\quad z\in [-1,1].$ This choice associates the edge set B with the set of points z such that $ϕ_1(z)$ ≈ 0
* $\phi _{2}(z)=z(1-z)\quad z\in [0,1].$ This choice associates the edge set B with the set of points z such that $ϕ_2(z)$ ≈ 1/4

The non-trivial step in their deduction is the proof that, as $\epsilon \to 0$, 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.

### 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.

