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
E[J,B]=\alpha\intD(I(\vecx)-J(\vecx))2d\vecx+\beta \intD/B\vec\nablaJ(\vecx) ⋅ \vec\nablaJ(\vecx)d\vecx+\gamma\intB 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:
E[J,z;\varepsilon]=\alpha\int(I(\vecx)-J(\vecx))2d\vecx+ \beta\intz(\vecx)|\vec\nablaJ(\vecx)|2d\vecx+\gamma\int \{\varepsilon|\vec\nablaz(\vecx)|2+\varepsilon-1\phi2(z(\vec x))\}d\vecx
where ε > 0 is a (small) parameter and ϕ(z) is a potential function. Two typical choices for ϕ(z) are
\phi1(z)=(1-z2)/2 z\in[-1,1].
\phi2(z)=z(1-z) z\in[0,1].
The non-trivial step in their deduction is the proof that, as
\epsilon\to0
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.
The Mumford-Shah functional can be split into coupled one-dimensional subproblems.The subproblems are solved exactly by dynamic programming.