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
- [math]\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 }[/math]
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:
- [math]\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 }[/math]
where ε > 0 is a (small) parameter and ϕ(z) is a potential function. Two typical choices for ϕ(z) are
- [math]\displaystyle{ \phi _1(z) = (1-z^2)/2 \quad z \in [-1,1]. }[/math] This choice associates the edge set B with the set of points z such that ϕ1(z) ≈ 0
- [math]\displaystyle{ \phi _2(z) = z(1-z) \quad z \in [0,1]. }[/math] 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 [math]\displaystyle{ \epsilon\to 0 }[/math], 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]
Minimization by splitting into one-dimensional problems
The Mumford-Shah functional can be split into coupled one-dimensional subproblems. The subproblems are solved exactly by dynamic programming. [4]
See also
- Bounded variation
- Caccioppoli set
- Digital image processing
- Luigi Ambrosio
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
- 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
- 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
- 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. https://archive.org/details/functionsbounded00ambr_983.
- 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, https://dash.harvard.edu/bitstream/handle/1/3637121/Mumford_OptimalApproxPiece.pdf?sequence=1
- Hohm, Kilian; Storath, Martin; Weinmann, Andreas (2015), "An algorithmic framework for Mumford–Shah regularization of inverse problems in imaging", Inverse Problems 31 (11): 115011, doi:10.1088/0266-5611/31/11/115011, Bibcode: 2015InvPr..31k5011H, http://bigwww.epfl.ch/publications/hohm1501.pdf
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