Efficient MVE image reconstruction for arbitrary measurement geometries

The minimum variance estimator (MVE) is applied to three, dimensional image reconstruction and efficient implementations are explored for selected geometries. For two-dimensional image reconstruction using fan beam or parallel beam geometries with complete rotation about the object of interest, the inversion of a block circulent measurement covariance matrix is reduced from O(n³) to O(n²) operations. Similar savings can be achieved for less than complete rotations if the angular increment is constant. In this paper, these result are extended to three-dimensional image reconstruction for two types of cone beam geometries. In one case a cone-beam source is rotated about an object of interest in a manner similar to commonly used fan beam systems. In the second case a small number of randomly placed sources (or receivers) is considered. As in the two-dimensional cases, savings is achieved by considering symmetries of the measurement geometry and by reordering or factoring the measurement covariance matrix so that large blocks of zero elements corresponding to uncorrelated mesurements are used to advantage. Two benefits of using an optimal estimator are that the best use is made of the information in the measurements and that the expected performance of a given measurement geometry can be evaluated.