Quadratic regularization design for 3D cylindrical PET

Statistical methods for tomographic image reconstruction lead to improved spatial resolution and noise properties in PET. Penalized-likelihood (PL) image reconstruction methods involve maximizing an objective function that is based on the log-likelihood of the sinogram measurements and on a roughness penalty function to control noise. In emission tomography, PL methods (and MAP methods) based on conventional quadratic regularization functions lead to nonuniform and anisotropic spatial resolution, even for idealized shift-invariant imaging systems. We have previously addressed this problem for parallel-beam 2D emission tomography, and for fan-beam 2D transmission tomography by designing data-dependent, shift-variant regularizers that improve resolution uniformity and isotropy, even for idealized shift-invariant imaging systems. This paper extends those methods to 3D cylindrical PET, using an analytical design approach that is numerically efficient.