Fast iterative image reconstruction for three-dimensional PET and its extension to time-of-flight PET

Positron emission tomography (PET) is a functional biomedical imaging technique that provides in vivo information about physiological processes within the body by reconstructing the 3D distribution of a radiolabelled tracer. It is challenging to estimate the tracer distribution accurately since photon-limited PET data have high statistical variance. Statistical iterative reconstruction methods have shown superior image quality and better quantitative results compared to analytical reconstruction methods. However, the iterative methods involve huge computational loads, particularly for fully 3D PET, consequently limiting their routine use in practice. Clinical 3D time-of-flight (TOF) PET scanners, which become available due to recent developments of fast scintillators, involve an even higher computational cost for image reconstruction because of extra TOF information. A successful pragmatic approach that reduces computation cost for fully 3D PET is to use Fourier rebinning (FORE), reducing 3D PET data to 2D data, combined with a 2D reconstruction method such as the ordered subsets expectation-maximization (OSEM) algorithm. The combination of FORE and 2D OSEM is now routinely used in both clinical and small animal imaging systems. However, data dimensionality reduction using FORE can result in a loss in resolution or noise performance. In the first part of this dissertation, we develop fast projectors that map a 3D image into 3D PET data by using the exact inverse (Fourier) rebinning operator. The inverse rebinning operator is used for calculating 3D PET data from 2D PET data, and combined with a 2D projector to construct a fast 3D projection operator. The inverse rebinning operator is implemented cost-effectively using the fast Fourier transform (FFT), so that the computation time is reduced by an order of magnitude. We utilize this fast projector in the context of fully 3D PET maximum a posteriori (MAP) reconstruction methods. We reduced the reconstruction time substantially while retaining the resolution recovery properties. In the second part of this dissertation, we develop fast 3D image reconstruction methods for TOF PET. We derive a unified framework based on a generalized projection slice theorem. In this framework, we derive mapping equations between different data sets such as 3D TOF, 2D TOF, 3D non TOF and 2D non TOF PET data. Using the mapping equations, we develop a TOF Fourier rebinning method which rebins 3D TOF PET data to 2D TOF data. We also develop other exact and approximate rebinning methods where 3D TOF PET data are rebinned to 2D or 3D non TOF data. Our simulation studies showed that the reconstructed images after TOF Fourier rebinning give superior resolution-variance properties compared to the non TOF case where the TOF information is ignored.