 Mumford–Shah functional

The energy ${\int }_{\Omega \setminus K}(u-g{)}^2+\alpha {\int }_{\Omega \setminus K}|\nabla u{|}^2+\beta {\mathcal{H}}^1(K)$ balances data fidelity, within-region smoothness and image edge length. In its relaxed SBV space formulation the image edge set is $J_u$. Clipping to the bounded data range, the SBV compactness theorem and lower semicontinuity prove existence; the full segmentation problem is not strictly convex.