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 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

E[J,B] = C \int (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:

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 \phi(\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

$\phi _1(z) = (1-z)/2 \quad z \in [0,1].$ 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.

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]

Notes

References

Camillo, De Lellis; Focardi, Matteo; Ruffini, Berardo (October 2013), "A note on the Hausdorff dimension of the singular set for minimizers of the Mumford–Shah energy", Advances in Calculus of Variations 7 (4): 539–545, doi:10.1515/acv-2013-0107, ISSN 1864-8258, Zbl 1304.49091
Ambrosio, Luigi; Fusco, Nicola; Hutchinson, John E. (2003), "Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford-Shah functional", Calculus of Variations and Partial Differential Equations 16 (2): 187–215, doi:10.1007/s005260100148, Zbl 1047.49015
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

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