Mumford–Shah Functional
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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 David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded th ...
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 : E ,B= \alpha \int_D (I(\vec x) - J(\vec x))^2 \,\mathrm\vec x + \beta \int _ \vec \nabla J(\vec x) \cdot \vec \nabla J(\vec x) \,\mathrm \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.See .


Minimization of the functional


Ambrosio–Tortorelli limit

Ambrosio and Tortorelli showed that the 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 ,z;\varepsilon= \alpha \int (I(\vec x) - J(\vec x))^2 \,\mathrm \vec x + \beta \int z(\vec x) , \vec \nabla J(\vec x), ^2 \,\mathrm \vec x + \gamma \int \ \,\mathrm \vec x where ε > 0 is a (small) parameter and ''ϕ''(''z'') is a potential function. Two typical choices for ''ϕ''(''z'') are * \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 * \phi _2(z) = z(1-z) \quad z \in ,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 \epsilon\to 0, the last two terms of the energy function (i.e. the last
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
term of the energy functional) converge to the edge set integral ∫Bd''s''. 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 In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...
of the singular set of minimizers of the Mumford-Shah energy.


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.


See also

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Bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
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Caccioppoli set In mathematics, a Caccioppoli set is a subset of \R^n whose boundary is (in a suitable sense) measurable and has (at least locally) a ''finite measure''. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli set if its ...
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Digital image processing Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allo ...
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Luigi Ambrosio Luigi Ambrosio (born 27 January 1963) is a professor at Scuola Normale Superiore in Pisa, Italy. His main fields of research are the calculus of variations and geometric measure theory. Biography Ambrosio entered the Scuola Normale Superiore d ...


Notes


References

* * * * * * {{DEFAULTSORT:Mumford-Shah functional Image segmentation