On the existence of solutions to a problem in multidimensional segmentation
The authors give an existence result for a minimum problem related to image segmentation in computer vision. In the framework of the calculus of variations this minimum problem pertains to the class of ``Minimal boundary problems with free discontinuity'' recently studied by De Giorgi-Carriero and Leaci. The main result is the following: Let \(n\geq 2\), M open \(\subset R^ n\), \(0<\lambda <+\infty\), \(1\leq p<+\infty\), \(g\in L^ p(M)\cap L^{\infty}(M)\). Then there exists at least one pair (K,u) minimizing \[ F(K,u)=\lambda \int_{M\setminus K}| u-g|^ p dx+H^{n-1}(K\cap M), \] with k closed \(\subset R^ n\), \(u\in C'(M\setminus K)\) such that \(\nabla u\equiv 0\) in \(M\setminus K\).
Minimal boundary problems with free discontinuity
scientific article; zbMATH DE number 4205225
scientific article