Mumford-Shah 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 which 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

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

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

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:

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

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

• $\phi _1(z) = (1-z)/2 \quad z \in [0,1].$ This choice associates the edge set B with the set of points z such that ϕ₁(z) ≈ 0

• $\phi _2(z) = 3 z(1-z) \quad z \in [0,1].$ 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 $\epsilon\to 0$, 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.

See also

• Bounded variation
• Caccioppoli set
• Digital image processing
• Luigi Ambrosio

References

• Ambrosio, Luigi; Tortorelli, Vincenzo Maria (1990), "Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence", Communications on Pure and Applied Mathematics 43 (8): 999–1036, doi:10.1002/cpa.3160430805, MR 1075076, Zbl 0722.49020 
• Mumford, David; Shah, Jayant (1989), "Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems", Communications on Pure and Applied Mathematics XLII (5): 577–685, doi:10.1002/cpa.3160420503, MR 0997568, Zbl 0691.49036