= Mumford–Shah functional
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{title2=$\int(u-g)^2+\alpha\int|\nabla u|^2+\beta\mathcal H^1(J_u)$}
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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>.
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