title: Mumford–Shah functional
text: # 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.
  
  ## 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
  
      $E[J,B]=\alpha \int _{D}(I({\vec {x}})-J({\vec {x}}))^{2}\,\mathrm {d} {\vec {x}}+\beta \int _{D/B}{\vec {\nabla }}J({\vec {x}})\cdot {\vec {\nabla }}J({\vec {x}})\,\mathrm {d} {\vec {x}}+\gamma \int _{B}ds$
  
  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:
  
      $E[J,z;\varepsilon ]=\alpha \int (I({\vec {x}})-J({\vec {x}}))^{2}\,\mathrm {d} {\vec {x}}+\beta \int z({\vec {x}})|{\vec {\nabla }}J({\vec {x}})|^{2}\,\mathrm {d} {\vec {x}}+\gamma \int \{\varepsilon |{\vec {\nabla }}z({\vec {x}})|^{2}+\varepsilon ^{-1}\phi ^{2}(z({\vec {x}}))\}\,\mathrm {d} {\vec {x}}$
  
  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 $ϕ_1(z)$ ≈ 0
  * $\phi _{2}(z)=z(1-z)\quad z\in [0,1].$ This choice associates the edge set B with the set of points z such that $ϕ_2(z)$ ≈ 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 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.
  
  ### 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.
  
  