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
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:
where ε > 0 is a (small) parameter and ϕ(z) is a potential function. Two typical choices for ϕ(z) are
- This choice associates the edge set B with the set of points z such that ϕ1(z) ≈ 0
- 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 , 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
- 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