Point-wise behavior of the Geman--McClure and the Hebert--Leahy image restoration models

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.