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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.

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