A new proof of regularity for two-shaded image segmentations (Q1392985)

The paper deals with the regularity of minimizers of the Mumford-Shah problem under the constraint that \(u\) takes only two values. Under this constraint the functional reduces to \[ \alpha\int_\Omega (u-g)^2 dx+\beta{\mathcal H}^{n-1}(S_u) \] and the regularity theory for quasi minimizers of the area functional provides the \(C^{1,\alpha}\) partial regularity of the jump set of the solution, with full regularity up to \(n=7\). The paper contains however an independent, and more direct, proof of the regularity in the two dimensional case. The proof relies on explicit estimates, different from the ones usually obtained with indirect arguments in the theory of minimal surfaces.