Fast regularization design for tomographic image reconstruction for uniform and isotropic spatial resolution

Statistical methods for tomographic image reconstruction have improved noise and spatial resolution properties that may improve image quality in X-ray CT and PET. Final converged solutions from maximum likelihood (ML) and weighted least squares (WLS) reconstruction are often extremely noisy due to the ill conditioned nature of the system. One can stop the iterative algorithm before convergence and before images become too noisy, however this results in non-uniform and anisotropic spatial resolution because resolution uniformity and isotropy improve with successive iterations. Alternatively, one can run the iterative algorithm to completion and post-filter the resulting noise, however, this often requires a large number of iterations. Instead we use penalized likelihood (PL) and penalized weighted least squares (PWLS) with a roughness penalty to regularize the problem which filters out noise, and leads to faster convergence. Unfortunately, interactions between the weightings, which are implicit in PL methods and explicit in PWLS methods, and conventional quadratic regularization lead to nonuniform and anisotropic spatial resolution. Previous work focuses on matrix algebra methods to design data-dependent, shift variant regularizers that improve resolution uniformity. This thesis develops fast analytical regularization design methods for 2D fan-beam X-ray CT imaging, for which parallel-beam tomography is a special case. This thesis uses continuous space analogs to greatly simplify the regularization design problem which yields a mostly analytical solution for efficient computation. This thesis extends regularization design to 3D systems using a computationally efficient iterative approach. Finally, this thesis explores using 2D regularization with z-dimension post-reconstruction denoising. This is an attempt to combine the improved XY isotropy associated with 2D regularization design, and the computational efficiency of the mostly analytical solution and use it for 3D geometries. The spatial resolution and noise properties of this method is analyzed for quadratic regularizers. Simulation results have also been performed using non-quadratic edge-preserving regularizers which show that, though this method has potential, more work needs to be done to ensure that the spatial resolution and noise properties of this method are desirable.