Ballester, C.; Caselles, V.; González, M. Lower semicontinuity of the affine total variation used in image segmentation. (English) Zbl 0939.49025 Rev. Unión Mat. Argent. 41, No. 1, 41-60 (1998). In the last years, several affine invariant models for edge detection and image filtering have been studied; in this framework, affine invariance is thought as an approximation of the – more correct – projective invariance. The paper, which is the continuation of a previous work by the same authors, deals with an affine invariant simplification of the Mumford-Shah functional, obtained by dropping the Dirichlet integral term (thus thinking the image as a piecewise constant function) and replacing the Hausdorff \(1\)-dimensional measure by an affine invariant energy. Specifically, the energy has the form \[ E_{af}(u,B):=\int_\Omega |u-g|^2 dx+\lambda\text{ATV}(B) \] where \(\text{ATV}(B)\), the Affine Total Variation of \(B\), is defined as \[ \sum_{i,j}\int_{\Gamma_i}\int_{\Gamma_j}|\tau_i\wedge\tau_j|d{\mathcal H}^1\times d{\mathcal H}^1 \] whenever \(B\) is the union of finitely many Jordan curves \(J_i\) with disjoint interiors and tangents \(\tau_i\). The main result is the extension of the lower semicontinuity results of the previous paper to a more general setting, where \(B\) is any continuum with finite \({\mathcal H}^1\)-measure. Reviewer: L.Ambrosio (Pisa) MSC: 49Q20 Variational problems in a geometric measure-theoretic setting 68U10 Computing methodologies for image processing 49J45 Methods involving semicontinuity and convergence; relaxation Keywords:image segmentation; affine invariance; variational methods; continuum; Hausdorff measure; Mumford-Shah functional PDFBibTeX XMLCite \textit{C. Ballester} et al., Rev. Unión Mat. Argent. 41, No. 1, 41--60 (1998; Zbl 0939.49025)