Regularization techniques are necessary to compute meaningful solutions to discrete ill-posed inverse problems. The well-known 2-norm Tikhonov regularization method equipped with a discretization of the gradient operator as regularization operator penalizes large gradient components of the solution to overcome instabilities. However, this method is homogeneous, i.e., it does not take into account the orientation of the regularized solution and therefore tends to smooth the desired structures, textures and discontinuities, which often contain important information. If the local orientation field of the solution is known, a possible way to overcome this issue is to implement local anisotropic regularization by penalizing weighted directional derivatives. In this paper, considering problems that are inherently two-dimensional, we propose to automatically and simultaneously recover the regularized solution and the local orientation parameters (used to define the anisotropic regularization term) by solving a bilevel optimization problem. Specifically, the lower level problem is Tikhonov regularization equipped with local anisotropic regularization, while the objective function of the upper level problem encodes some natural assumptions about the local orientation parameters and the Tikhonov regularization parameter. Application of the proposed algorithm to a variety of inverse problems in imaging (such as denoising, deblurring, tomography and Dix inversion), with both real and synthetic data, shows its effectiveness and robustness.

翻译：为计算离散不适定反问题的有效解，正则化技术不可或缺。经典的2-范数Tikhonov正则化方法以梯度算子的离散化作为正则算子，通过惩罚解的大梯度分量来克服不稳定性。然而该方法具有齐次性，即未考虑正则化解的方向特性，因此往往会平滑掉包含重要信息的结构、纹理及间断特征。若已知解的局部方向场，可通过惩罚加权方向导数实现局部各向异性正则化以解决此问题。本文针对本质二维问题，提出通过求解双层优化问题来自动同步恢复正则化解与局部方向参数（用于定义各向异性正则项）。具体而言，下层问题采用配备局部各向异性正则化的Tikhonov正则化，而上层问题的目标函数则编码了关于局部方向参数与Tikhonov正则化参数的自然假设。将所提算法应用于成像领域的多种反问题（如去噪、去模糊、断层扫描及Dix反演），在真实与合成数据上的实验验证了其有效性与鲁棒性。