  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.
