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Variational approach in image processing: existence and approximation properties. (Une approche variationnelle en traitement d’images: résultats d’existence et d’approximation.) (French) Zbl 0682.49003
Summary: We give an existence theorem for image segmentations obtained by minimizing the functional $J(u,K)=\int_{\Omega \setminus K}| \nabla u|^ 2 \,dx+\int_{\Omega}(u-g)^ 2 \,dx+{\mathcal H}_ 1(K),$ over closed subsets $$K$$ of an open rectangle $$\Omega$$, with Hausdorff length $${\mathcal H}_ 1(K)$$, and over $$u\in H^ 1(\Omega \setminus K)$$. $$K$$ represents the “contours” of the image $$g$$ and $$u$$ its regularized version in the connected components of $$\Omega\setminus K$$. This theorem is obtained by proving an equivalence between the above formulation and a new one proposed by E. De Giorgi [“Free discontinuity problems in calculus of variations.” Frontiers in pure and applied mathematics, Coll. Pap. Ded. J.-L. Lions Occas. 60th Birthday, 55–62 (1991; Zbl 0758.49002); “A new type of functional in the calculus of variations” (Italian), Atti Accad. Naz. Lincei, VIII. Ser., Rend Cl. Sci. Fis. Mat. Nat. 82, No. 2, 199–210 (1988; Zbl 0715.49014)] and his school under the name of “free discontinuity problems”.
We also introduce a more computational version of the problem, where $$K$$ is imposed to be the union of $$m$$ curves, and show that this last formulation is asymptotically equivalent to both preceding formulations as $$m$$ tends to infinity.

##### MSC:
 49J10 Existence theories for free problems in two or more independent variables 49J45 Methods involving semicontinuity and convergence; relaxation 68U10 Computing methodologies for image processing 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
##### Keywords:
image processing; image segmentations