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Lower semicontinuity of the affine total variation used in image segmentation. (English) Zbl 0939.49025

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
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