Orlicz space relaxation of total variation denoising
We consider variational image denoising with spatially dependent Orlicz regularization and introduce an Orlicz--Sobolev approximation of the ROF model based on a scaled $L\log L$-type density. For a general class of uniformly superlinear integrands satisfying a $Δ_2$-condition, we establish existence and uniqueness of minimizers and derive a Fenchel dual problem with dual attainment and pointwise optimality conditions. We then specialize the framework to a logarithmic density whose correction is activated above a prescribed local gradient scale. For this model, we obtain explicit expressions of the Fenchel dual and optimality conditions as well as a pointwise radial formula for the dual proximal map involving the Lambert $W$-function. As the logarithmic parameter tends to zero, we prove equicoercivity and $Γ$-convergence to the ROF functional, together with convergence of the corresponding minimizers. Numerical illustrations indicate that the logarithmic correction can reduce staircasing for diffuse transitions, while retaining behavior comparable to ROF for sharp interfaces and a natural image.