PCA-Based Generative Regularization for the CT Inverse Problem

Abstract

Computed tomography (CT) reconstruction transforms a set of X-ray measurements into cross-sectional images of an object. This process is an ill-posed inverse problem, meaning that small errors in the data can lead to large errors in the reconstruction. A common approach to address this problem is through regularization. In this thesis, we present an introduction to the CT inverse problem and construct a generative regularizer based on principal component analysis (PCA), which incorporates prior structural information from a set of training CT images. The reconstruction problem is formulated as the minimization of a data-fidelity term, combined with a penalty that encourages the solution to remain close to the PCA subspace. We solve this optimization problem with the sparse LSQR algorithm. Numerical experiments show that the PCA-based regularizer achieves up to a sevenfold reduction in mean squared error compared to unregularized or Tikhonov-regularized methods, though at the cost of increased computation time. While the visual differences between reconstructions are subtle, the PCA-regularized images display smoother homogeneous regions, demonstrating the potential benefits of incorporating data-driven priors in CT reconstruction.