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Zbl 0884.49024
Tamanini, Italo; Congedo, Giuseppe
Optimal segmentation of unbounded functions. (English)
[J] Rend. Semin. Mat. Univ. Padova 95, 153-174 (1996). ISSN 0041-8994

We denote by $n$ an integer $\ge 2$, by $p$ and $\lambda$ two real numbers with $p\ge 1$ and $\lambda>0$, by $\Omega$ an open subset of the Euclidean $n$-dimensional space $\bbfR^n$, by ${\cal H}^{n-1}$ the $(n-1)$-dimensional Hausdorff measure in $\bbfR^n$, and by $g$ a measurable, real-valued function defined on $\Omega$.\par The authors prove that the functional $$F(C, u)=\lambda\int_{\Omega\backslash C}|u-g|^p dx+{\cal H}^{n-1}(C\cap\Omega)$$ achieves its minimum on pairs $(C,u)$ with $C$ closed and $u$ constant on each connected component of $\Omega\backslash C$. Moreover, they show that the family of connected components of $\Omega\backslash K$ is locally finite in $\Omega$, for any minimizer $(K, w)$ of $F$.\par Variational problems of the preceding type are encountered for instance in computer vision theory, where a basic problem is to obtain ``optimal segmentations'' of a given image.
[C.Udrişte (Bucureşti)]

MSC 2000:: *49Q20 Variational problems in geometric measure-theoretic setting
68U10 Image processing

Keywords: optimal segmentation; variational problems

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