Mumford–Shah functional

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Image approximation with Mumford-Shah functional. (left) The image of an eye. (center-left) areas of high gradient in the original image. (center-right) boundaries in the Mumford-Shah model, (right) piecewise-smooth function approximating the image.

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[edit]

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.[2]

Minimization of the functional[edit]

Ambrosio–Tortorelli limit[edit]

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:

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.

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]

See also[edit]