We propose a bilevel optimization approach for the estimation of parameters in nonlocal image denoising models. The parameters we consider are both the fidelity weight and weights within the kernel of the nonlocal operator. In both cases, we investigate the differentiability of the solution operator in function spaces and derive a first-order optimality system that characterizes local minima. For the numerical solution of the problems, we use a second-order trust-region algorithm in combination with a finite element discretization of the nonlocal denoising models and introduce a computational strategy for the solution of the resulting dense linear systems. Several experiments illustrate the applicability and effectiveness of our approach.

我们提出了一种用于非局部图像去噪模型中参数估计的双层优化方法。我们考虑的参数是保真度权重和非局部运算符内核内的权重。在这两种情况下，我们研究函数空间中解算子的可微性，并得出表征局部极小值的一阶最优系统。对于问题的数值解，我们结合非局部降噪模型的有限元离散化使用了二阶信任域算法，并引入了一种计算策略来解决由此产生的密集线性系统。几个实验说明了我们方法的适用性和有效性。