Unform rectifiability of image segmentations obtained by a variational method
The author considers the Mumford-Shah functional \[ I(u,K) = \int_{\Omega \backslash K} \bigl(|\nabla u |^2+ \mu|u-g |^2\bigr) dx+\nu {\mathcal H}^1(K) \] where \(\Omega\subset \mathbb{R}^2\) is a rectangle, \({\mathcal H}^1\) is the one-dimensional Hausdorff measure, \(g\in L^\infty (\Omega)\) \((|g |_{L^\infty}\leq 1)\), \(\mu,\nu\) are positive constants, \(K\) is a closed set and \(u\in C^1(\Omega \backslash K)\). The existence of a minimizing pair for \(J\) was proved (in a more general case) in \textit{E. De Giorgi}, \textit{M. Carriero} and \textit{A. Leaci} [Arch. Ration. Mech. Anal. 108, No. 3, 195-218 (1989; Zbl 0682.49002)] and in \textit{G. Dal Maso}, \textit{J. M. Morel} and \textit{S. Solimini} [Acta Math. 168, No. 1-2, 89-151 (1992; Zbl 0772.49006)]. If \((u,K)\) is a minimizing pair, the author proves a projection property of the closed set \(K\), i.e., there exists \(\alpha>0\) such that for every closed disk \(D_r\) with radius \(r\) such that \(K \cap D_{r/4} \neq \emptyset\), one has \[ \forall \theta \in [0,\pi/2[:\;{\mathcal H}^1 \bigl(p_\theta (K\cap D_r) \bigr)+ {\mathcal H}^1 \bigl(p_{\theta+ (\pi/2)} (K\cap D_r) \bigr) \geq \alpha r \] where \(p_\theta\) is the orthogonal projection on any line of direction \(\theta\).
scientific article; zbMATH DE number 811448