Automatic nonstationary anisotropic Tikhonov regularization through bilevel optimization

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.