Sparse-View X-Ray CT Reconstruction Using $\ell_1$ Prior with Learned Transform

A major challenge in X-ray computed tomography (CT) is reducing radiation dose while maintaining high quality of reconstructed images. To reduce the radiation dose, one can reduce the number of projection views (sparse-view CT); however, it becomes difficult to achieve high-quality image reconstruction as the number of projection views decreases. Researchers have applied the concept of learning sparse representations from (high-quality) CT image dataset to the sparse-view CT reconstruction. We propose a new statistical CT reconstruction model that combines penalized weighted-least squares (PWLS) and $\ell_1$ regularization with learned sparsifying transform (PWLS-ST-$\ell_1$), and a corresponding efficient algorithm based on Alternating Direction Method of Multipliers (ADMM). To moderate the difficulty of tuning ADMM parameters, we propose a new ADMM parameter selection scheme based on approximated condition numbers. We interpret the proposed model by analyzing the minimum mean square error of its ($\ell_2$-norm relaxed) image update estimator. Numerical experiments with the extended cardiac-torso (XCAT) phantom show that, for sparse-view 2D fan-beam CT and 3D axial cone-beam CT, PWLS-ST-$\ell_1$ significantly improves the quality of reconstructed images compared to the CT reconstruction methods using edge-preserving regularizer and $\ell_2$ regularization with learned ST. These experiments also show that, for sparse-view 2D fan-beam CT, PWLS-ST-$\ell_1$ outperforms PWLS-DL using a learned overcomplete dictionary by providing both significantly better image quality and much shorter runtime. Numerical experiments with clinical data show that, PWLS-ST-$\ell_1$ using the unsupervised learned regularizer generalizes better than a state-of-the-art deep regression neural network that does not use a physical imaging model.