Orlicz空间松弛的全变分去噪

中文摘要 AI 辅助

我们考虑具有空间依赖的Orlicz正则化的变分图像去噪，并引入基于缩放$L\log L$型密度的ROF模型的Orlicz-Sobolev近似。对于满足$\Delta_2$条件的一般均匀超线性被积函数类，我们建立了极小元的存在性和唯一性，并导出了具有对偶可达性和逐点最优性条件的Fenchel对偶问题。然后，我们将该框架专门化到对数密度，其校正在规定的局部梯度尺度之上被激活。对于该模型，我们获得了Fenchel对偶和最优性条件的显式表达式，以及涉及Lambert $W$函数的对偶近端映射的逐点径向公式。当对数参数趋于零时，我们证明了等强制性和对ROF泛函的$\Gamma$-收敛性，以及相应极小元的收敛性。数值示例表明，对数校正可以减少漫反射过渡中的阶梯效应，同时在尖锐界面和自然图像上保持与ROF相当的行为。

英文摘要

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.

发表机构

• University of Graz（格拉茨大学）
• TU Wien（维也纳工业大学）

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