We present a new approach for nonlocal image denoising, based around the application of an unnormalized extended Gaussian ANOVA kernel within a bilevel optimization algorithm. A critical bottleneck when solving such problems for finely-resolved images is the solution of huge-scale, dense linear systems arising from the minimization of an energy term. We tackle this using a Krylov subspace approach, with a Nonequispaced Fast Fourier Transform utilized to approximate matrix-vector products in a matrix-free manner. We accelerate the algorithm using a novel change of basis approach to account for the (known) smallest eigenvalue-eigenvector pair of the matrices involved, coupled with a simple but frequently very effective diagonal preconditioning approach. We present a number of theoretical results concerning the eigenvalues and predicted convergence behavior, and a range of numerical experiments which validate our solvers and use them to tackle parameter learning problems. These demonstrate that very large problems may be effectively and rapidly denoised with very low storage requirements on a computer.

翻译：我们提出了一种新的非局部图像去噪方法，其核心在于将未归一化的扩展高斯ANOVA核应用于双层优化算法中。当针对高分辨率图像求解此类问题时，一个关键瓶颈源于能量项最小化所产生的大规模稠密线性系统的求解。我们采用Krylov子空间方法应对这一挑战，并利用非等间距快速傅里叶变换以无矩阵方式近似矩阵-向量乘积。我们通过一种新颖的基变换方法来加速算法，该方法结合了所涉及矩阵（已知的）最小特征值-特征向量对，并辅以一种简单但通常非常有效的对角预处理策略。我们给出了关于特征值及预测收敛行为的若干理论结果，并通过一系列数值实验验证了求解器的有效性，同时将其应用于参数学习问题。实验表明，该方法能够以极低的存储需求在计算机上高效、快速地完成大规模图像的去噪处理。