Automatic nonstationary anisotropic Tikhonov regularization through bilevel optimization

TL;DR

The paper tackles ill-posed 2D inverse problems by introducing a bilevel framework that jointly recovers the unknown model $\mathbf{m}$ and a spatially varying local orientation field $\boldsymbol\theta$ for anisotropic regularization. The lower level solves a Tikhonov problem with a nonuniform, rotation-enabled directional penalty, while the upper level enforces a discrepancy-principle-inspired fit, smooth orientation fields, and orientation alignment, solved with L-BFGS-B using stabilized gradient estimates. Without requiring training data, the approach demonstrates robustness across imaging tasks—denoising, deblurring, tomography, and Dix velocity inversion—achieving sharper, more structure-preserving reconstructions compared to isotropic regularization. The work provides a practical, unsupervised route to nonstationary anisotropy in regularization, with clear paths for scalability and future extensions.

Abstract

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.

Figures (13)

• Figure 1: The 2D Hilbert transform filters.
• Figure 2: Denoising test problem. (a) exact image; (b) available data; (c) image recovered by the (isotropic) Tikhonov regularization method; (d) image recovered by the new anisotropic Tikhonov regularization method.
• Figure 3: Denoising test problem. (a) pixel-wise orientation parameters recovered solving the bilevel optimization method \ref{['eq:bilevel']}; (b) 2-norm of the directional derivatives along $x'$ and $z'$ versus L-BFGS-B iterations.
• Figure 4: Denoising test problem: (a) upper level objective function values versus L-BFGS-B iterations; (b) relative reconstruction errors versus L-BFGS-B iterations; (c) Tikhonov regularization parameter $\mu$ versus L-BFGS-B iterations.
• Figure 5: Deblurring test problem. (a) exact image; (b) available data; (c) image recovered by the (isotropic) Tikhonov regularization method; (d) image recovered by the new anisotropic Tikhonov regularization method.