Image reconstruction algorithms for achieving high-resolution positron emission tomography images

In this dissertation, we present four algorithms for reconstructing high-resolution images in PET. The first algorithm, referred to as the penalized maximum likelihood (PML) algorithm, iteratively minimizes a PML objective function. At each iteration, the PML algorithm generates a function, called a surrogate function, that satisfies certain conditions. The next iterate is defined to be the nonnegative minimizer of the surrogate function. The PML algorithm utilizes standard de-coupled surrogate functions for the maximum likelihood objective function of the data and de-coupled surrogate functions for a certain class of penalty functions. As desired, the PML algorithm guarantees nonnegative estimates and monotonically decreases the PML objective function with increasing iterations. For the case where the PML objective function is strictly convex, which is true for the class of penalty functions under consideration, the PML algorithm has been shown to converge to the minimizer of the PML objective function.; The drawback of the PML algorithm is that it converges slowly. Thus, a "fast" version of the PML algorithm, referred to as the accelerated PML (APML) algorithm, was developed where an additional search step, called a pattern search step, is performed after each standard PML iteration. In the pattern search step, an accelerated iterate, which has lower cost than the standard PML iterate, is found by solving a certain constrained optimization problem that arises at each pattern search step. The APML algorithm retains the nice properties of the PML algorithm.; The third algorithm, referred to as the quadratic edge preserving (QEP) algorithm, aims to preserve edges in the reconstructed images so that fine details, such as small tumors, are more resolvable. The QEP algorithm is based on an iteration dependent, de-coupled penalty function that introduces smoothing while preserving edges. The penalty function was developed by modifying the surrogate functions of the, penalty function for the PHIL method.; In PET, there are several errors that have the net effect of introducing blur into the reconstructed images. We propose a method that aims to reduce blur in PET images. The method is based on the assumption that the "true" probability matrix for the observed emission data is a product of an unknown nonnegative matrix, called a scatter matrix, and a "conventional" probability matrix. Under the suggested framework, the problem is to jointly estimate the scatter matrix and emission means. We propose an alternating minimization algorithm to estimate them by minimizing a certain distance.; The algorithms are qualitatively and quantitatively assessed using synthetic phantom data, and real phantom data.