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]

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

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

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

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:

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

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

• ${\displaystyle \phi _{1}(z)=(1-z^{2})/2\quad z\in [-1,1].}$ This choice associates the edge set B with the set of points z such that ϕ₁(z) ≈ 0
• ${\displaystyle \phi _{2}(z)=z(1-z)\quad z\in [0,1].}$ This choice associates the edge set B with the set of points z such that ϕ₂(z) ≈ 1/4

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]

The Mumford-Shah functional can be split into coupled one-dimensional subproblems. The subproblems are solved exactly by dynamic programming. ^[4]

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

• 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, S2CID 2040612, 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, S2CID 55078333, 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

• Hohm, Kilian; Storath, Martin; Weinmann, Andreas (2015), "An algorithmic framework for Mumford–Shah regularization of inverse problems in imaging" (PDF), Inverse Problems, 31 (11): 115011, Bibcode:2015InvPr..31k5011H, doi:10.1088/0266-5611/31/11/115011, S2CID 15365352