| Recent advances in medical imaging have led to the ability to create detailed cross-sectional images of the inside of the human body using x-rays, gamma rays, or ultrasound. Common to these methods is the mathematical reconstruction of a two-dimensional image from series of line integral measurements (so-called projections). In most applications of cross-sectional imaging, images are reconstructed from a complete set of symmetrically arranged projections, and the mathematical theory and methods of image reconstruction are well established. However, in certain applications of cross-sectional imaging, such as rapid scanning of the heart, images are reconstructed from an incomplete set of arbitrarily arranged projections. For these so-called limited angle applications, the mathematical theory is incomplete, and suitable methods for image reconstruction are not available.; In this dissertation, a new theory of image reconstruction is developed, based on the Minimum Variance Estimator, which establishes the interrelation between several existing reconstruction methods, and enables the derivation and implementation of a powerful new class of reconstruction methods for limited angle. In this theory, optimal reconstruction methods are derived, called Fast Estimators, that are as easy to compute as Convolution Back-Projection. In the Fast Estimators, the computational requirements precluding previous implementation of the Minimum Variance Estimator are avoided, because for each the measurement covariance matrix is carefully chosen such that it can be factored into a product of matrices that is easy to invert.; The unification of several existing reconstruction methods under one mathematical framework has been made by generalizing the derivation of methods based on the Minimum Variance Estimator to include a priori measurement covariance information. Existing deterministic methods are shown to have implicit measurement covariance assumptions which poorly approximate the true measurement covariances occurring in practical implementation. In the new theory, a prescription is provided to derive and implement reconstruction methods which incorporate as much a priori measurement covariance information as required to obtain a desired image quality. The theory optimizes the incorporation of a priori covariance information for a given required speed of execution. |
