9623077 Mair The proposers investigate a refinement of the Shepp-Vardi probabilistic model for positron emission tomography (PET), and the corresponding expectation-maximization maximum-likelihood (EM-ML) reconstruction algorithm. This model replaces the usual finite linear system with an integral equation in which the unknown is a Borel measure. The research includes an analysis of the convergence properties of this algorithm, numerical methods of implementation by spline and wavelet bases, and an analysis of the smoothing effects of these bases. The convergence results provide mathematical reasons in terms of the regularity properties of Borel measures, for the seeming divergence of the numerical implementations of the finite dimensional EM-ML algorithm. Also, since the EM-ML algorithm has been recently extended to linear integral equations in which the kernel and the unknown function are nonnegative, these results provide important insight into the open problem of convergence of this algorithm. In addition, the research introduces, analyzes, and develops novel algorithms for dealing with data errors due to accidental coincidences and attenuation. The proposers also investigate the mathematical properties of the probability functions which link the emissions to the detector geometry in both PET and single photon emission computed tomography (SPECT). Preliminary work for SPECT links these functions to the classical Poisson kernel. This study is important for developing accurate, efficient, fast, alternatives to the EM-ML reconstruction algorithm. %%% The mathematical realization of many problems in science, engineering, and medicine, give rise to inverse problems which are ill-posed in the sense of Hadamard, and in which positivity plays an important role. This proposal deals mainly with the particular example of such a problem occuring in the nuclear imaging procedure of PET. In this procedure, a patient is given a radiopharmaceutical which is absorbed disprop ortionately by various regions of the organ of interest and emits positrons according to the amount absorbed. These emissions are collected by PET scanners which surround the region of interest and then used in reconstruction algorithms which generate images containing important information about the metabolism of an organ or region of interest. This information is extremely useful for blood flow and metabolic activity studies. For instance, it is a valuable tool in the diagnosis of tumors, in determining their rates of growth; in psychological studies for mapping activation areas of the brain to cognitive tasks, in determining the effects of various drugs on the brain, and in quantitative measures of the health of the human heart. The proposers develop a refined mathematical model for the PET process and introduce novel mathematical and numerical methods for the reconstruction of PET images. These reconstruction algorithms are sufficiently flexible to deal with significant errors in the PET data, caused by anatomical obstructions which prevent some of the emitted photons from being registered in the appropriate detector. An important component of this work is the development and evaluation of efficient, reliable, accurate algorithms for reconstructing PET images. Although primarily directed to PET studies, the results obtained in this reserach are also applicable to other medical and engineering problems, such as those which occur in liver biopsies, nondestructive evaluation of manufactured items, and the restoration of blurred images from outer space. ***
