On Learned Operator Correction in Inverse Problems

Area of Science:

• Applied Mathematics
• Image Reconstruction
• Computational Imaging

Background:

• Inverse problems are central to many scientific and engineering fields.
• Variational methods are widely used for regularized solutions in inverse problems.
• Explicitly learning model errors offers a path to improved reconstruction accuracy.

Purpose of the Study:

• To investigate the feasibility of learning data-driven explicit model corrections for inverse problems.
• To develop a variational framework incorporating learned model corrections for regularized reconstructions.
• To analyze the convergence properties of solutions obtained with learned corrections.

Main Methods:

• A novel forward-adjoint correction is proposed, acting in both data and solution spaces.
• Conditions for convergence of variational solutions with learned corrections to true solutions are derived.
• The method is applied to limited view photoacoustic tomography.

Main Results:

• The proposed forward-adjoint correction effectively addresses model deficiencies in inverse problems.
• Convergence of the learned correction approach to solutions with the correct operator is demonstrated under specific conditions.
• The method shows competitive performance compared to the Bayesian approximation error method in photoacoustic tomography.

Conclusions:

• Learning data-driven model corrections is a viable strategy for enhancing inverse problem solutions.
• The proposed forward-adjoint correction offers a robust framework for regularized reconstructions.
• This work advances the state-of-the-art in model-based iterative reconstruction techniques.