Automatic nonstationary anisotropic Tikhonov regularization through bilevel optimization

To compute meaningful solutions to ill-posed inverse problems, one should employ some regularization techniques. 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 nonstationary anisotropic regularization by penalizing weighted directional derivatives. In this paper, considering linear 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 nonstationary 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, tomography and Dix velocity inversion) shows its effectiveness and robustness.