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.[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 _{D}(I({\vec {x}})-J({\vec {x}}))^{2}\,\mathrm {d} {\vec {x}}+A\int _{D/B}{\vec {\nabla }}J({\vec {x}})\cdot {\vec {\nabla }}J({\vec {x}})\,\mathrm {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;\varepsilon ]=C\int (I({\vec {x}})-J({\vec {x}}))^{2}\,\mathrm {d} {\vec {x}}+A\int z({\vec {x}})|{\vec {\nabla }}J({\vec {x}})|^{2}\,\mathrm {d} {\vec {x}}+B\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

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

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, arXiv:1403.3388, 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
Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego (2000). Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. New York: The Clarendon Press, Oxford University Press. pp. 434. ISBN 9780198502456. Zbl 0957.49001.
Mumford, David; Shah, Jayant (1989), "Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems" (PDF), Communications on Pure and Applied Mathematics, XLII (5): 577–685, doi:10.1002/cpa.3160420503, MR 0997568, Zbl 0691.49036

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[MumfordShah89-1] Mumford & Shah (1989).

[AmbrosioTortorelli90-2] See Ambrosio & Tortorelli (1990).

[Ambrosio-Fusco-Hutchinson-3] Ambrosio, Fusco & Hutchinson (2003)