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Point-wise behavior of the Geman-McClure and the Hebert-Leahy image restoration models. (English) Zbl 1359.94034

Summary: We present new continuous variants of the Geman-McClure model and the Hebert-Leahy model for image restoration, where the energy is given by the nonconvex function \(x \mapsto x^2/(1+x^2)\) or \(x \mapsto \log(1+x^2)\), respectively. In addition to studying these models’ \(\Gamma\)-convergence, we consider their point-wise behaviour when the scale of convolution tends to zero. In both cases the limit is the Mumford-Shah functional.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
49J45 Methods involving semicontinuity and convergence; relaxation
49N45 Inverse problems in optimal control
68U10 Computing methodologies for image processing
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