The functional was proposed by mathematicians David Mumford and Jayant Shah in 1989.
Two typical choices for ϕ(z) are The non-trivial step in their deduction is the proof that, as
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] The Mumford-Shah functional can be split into coupled one-dimensional subproblems.