Quantitative comparisons of statistical methods in image reconstruction.

Statistical methods for approaching image reconstruction and restoration problems have generated much interest among statisticians in the past decade. In this dissertation, we examine in detail various statistical methods of image reconstruction through the simulation of a multiple-pinhole coded-aperture imaging system for use in emission tomography. We reconstruct each object from a class of 64 total objects, obtaining a reconstruction for each of the 64 originals by several different methods. Among the methods that we use to obtain these reconstructions are maximum likelihood techniques, where we make use of both the popular expectation-maximization (EM) algorithm and a Monte Carlo search routine. We also examine methods that include, in some form, various kinds of prior information. One way of using prior information is through the specification of a prior probability density on the object (or class of objects) to be reconstructed. We investigate the use of Markov random field (MRF) models as a means of specifying the prior densities that will be used to obtain reconstructions. Once given a prior density, these reconstructions are taken to be approximations to the maximum a posteriori (MAP) estimate of the original object. We also investigate reconstructions obtained through other prior densities plus reconstructions obtained by introducing prior information in alternate ways. Once all the reconstructions are obtained, we attempt to answer the important question, "which reconstruction method is 'best'?" We define "best" in this context to be the method that allows a human observer to perform a specified task the most accurately. The task to be performed is to determine whether or not a small protrusion exists on an elliptical object. (This task is motivated by the desire to detect wall-motion abnormalities in the left ventricle of the heart.) We generate 32 objects with protrusions (abnormal objects) and 32 objects without protrusions (normal objects). These objects constitute our class of 64 originals which are reconstructed by the various methods. The reconstruction methods are then analyzed through receiver operating characteristic (ROC) analysis, and a performance index, the area under the curve (AUC), is obtained for each method. Statistical tests are then performed on certain pairs of methods so that the hypothesis that no difference between the AUC's exists can be tested. We found that the reconstruction methods that used the largest amount of (accurate) prior information were generally superior to other methods considered. We also compute calculable figures of merit (FOM) associated with each reconstruction method with the hope that these FOM's will predict the performance of the human observer. Unfortunately, our results indicate that the FOM's that we considered do not correlate well with the performance of the human