# Mumford–Shah functional

(Redirected from Mumford-Shah energy functional)
Image approximation with Mumford-Shah functional. (left) The image of an eye. (center-left) areas of high gradient in the original image. (center-right) boundaries in the Mumford-Shah model, (right) piecewise-smooth function approximating the image.

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.[1]

## 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

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

Optimization of the functional may be achieved by approximating it with another functional, as proposed by Ambrosio and Tortorelli.[2]

## Minimization of the functional

### Ambrosio–Tortorelli limit

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:

${\displaystyle E[J,z;\epsilon ]=C\int (I({\vec {x}})-J({\vec {x}}))^{2}d{\vec {x}}+A\int z({\vec {x}})|{\vec {\nabla }}J({\vec {x}})|^{2}d{\vec {x}}+B\int \{\epsilon |{\vec {\nabla }}z({\vec {x}})|^{2}+\epsilon ^{-1}\phi ^{2}(z({\vec {x}}))\}d{\vec {x}}}$

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

• ${\displaystyle \phi _{1}(z)=(1-z)/2\quad z\in [0,1].}$ This choice associates the edge set B with the set of points z such that ϕ1(z) ≈ 0
• ${\displaystyle \phi _{2}(z)=3z(1-z)\quad z\in [0,1].}$ This choice associates the edge set B with the set of points z such that ϕ1(z) ≈ ½

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]