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The multi-grid expectation maximization algorithm (MGEM), an extension of the maximum likelihood (ML) algorithm, has been applied to the problem of reconstruction in positron emission tomography (PET). The MGEM algorithm implemented the expectation maximization (EM) algorithm on different size image grids. The algorithm is based on the idea that the low frequency image components can be recovered faster than the high frequency components. The algorithm begins with a coarse grid, where the low frequency components are recovered. After the low frequency components have been recovered, the solution is projected onto the next finer grid. On the next finder grid, the high frequency components from the previous grid become the low frequency components. The algorithm continues iterating and switching levels until the finest grid has been reached. This method provides faster convergence than the single grid EM algorithm. An important issue concerning the MGEM algorithm is when to stop iterating at a particular grid and project to the next grid, or stop at the finest grid. The convergence rate of the MGEM algorithm was used as a grid level transition criterion. A grid level transition criterion which used the co-occurrence matrix statistics of the reconstructed image is presented. The spatial distribution and dependence among gray levels in a local area of the reconstructed image can be studied by gray level dependent co-occurrence statistics. The second order histogram represents the probability of occurrence of a pair of gray levels separated by a given displacement vector. The features computed from the second-order histogram were entropy, contrast, angular second moment, inverse difference moment, correlation, mean, and deviation. The difference histogram is derived from the second order histogram and represents the probability of occurrence of a difference in gray levels of two pixels separated by a displacement vector. The features extracted from the difference histogram were entropy, contrast, and angular second moment. The behavior of each of the statistics was compared to the root mean squared (RMS) error, to evaluate the potential for use as a transition criterion. An appropriate window was used to average the variance of each statistic. The indication of switching to the next finer grid level, or of stopping at the finest grid level was provided by a decline in the variance of a statistic. These statistics were computed on an image reconstructed from simulated PET data. Each of the statistics was evaluated versus the number of iterations at each grid level. The transitions for each level occurred when the variance of one of the statistics decreased sharply. For comparison, the single grid EM algorithm was allowed to run for the same amount of CPU time as the MGEM algorithm. The RMS error indicates that the MGEM method, using the co- occurrence statistics as a transition criterion, produced a better reconstruction than the single grid EM algorithm.
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