混合高斯和椒盐噪声去除的变分模型分析与优化
Analysis and optimisation of a variational model for mixed Gaussian and Salt & Pepper noise removal
摘要：
我们分析了最近在[14]中提出的用于混合噪声去除的变分正则化问题。该模型的数据差异项是由$L^1$项和$L^2$项组成的下确界卷积，适用于高斯噪声和椒盐噪声的联合去除。在这项工作中，我们进行了更精细的分析模型，强调的平衡效果的两个参数出现在差异项。也就是说，我们研究了这些参数的大值和小值的模型的渐近行为，并将其与相应的变分模型与$L^1$和$L^2$数据保真度进行比较。此外，我们计算精确解的简单数据函数的全变差作为正则化。使用这些理论结果，然后，我们分析研究的双层优化策略，自动选择模型的参数，通过一个训练集。最后，我们报告了一些数值结果的最佳噪声模型的选择，通过这种策略，证实了我们的分析的有效性和使用流行的数据模型的情况下，“盲目”的模型选择。
英文摘要：
We analyse a variational regularisation problem for mixed noise removal that was recently proposed in [14]. The data discrepancy term of the model combines $L^1$ and $L^2$ terms in an infimal convolution fashion and it is appropriate for the joint removal of Gaussian and Salt & Pepper noise. In this work we perform a finer analysis of the model which emphasises on the balancing effect of the two parameters appearing in the discrepancy term. Namely, we study the asymptotic behaviour of the model for large and small values of these parameters and we compare it to the corresponding variational models with $L^1$ and $L^2$ data fidelity. Furthermore, we compute exact solutions for simple data functions taking the total variation as regulariser. Using these theoretical results, we then analytically study a bilevel optimisation strategy for automatically selecting the parameters of the model by means of a training set. Finally, we report some numerical results on the selection of the optimal noise model via such strategy which confirm the validity of our analysis and the use of popular data models in the case of "blind" model selection.
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