A Fast Implementation of a Minimum Variance Estimator for Computerized Tomography Image Reconstruction

Most present day computerized tomography (CT) systems are based on reconstruction algorithms that produce only approximate deterministic solutions of the image reconstruction problem. These algorithms yield reasonable results in cases of low measurement noise and regular measurement geometry, and are considered acceptable because they require far less computation and storage than more powerful algorithms that can yield near optimal results. However, the special geometry of the CT image reconstruction problem can be used to reduce by orders of magnitude the computation required for optimal reconstruction methods, such as the minimum variance estimator. These simplifications can make the minimum variance technique very competitive with well-known approximate techniques such as the algebraic reconstruction technique (ART) and convolution-back projection. The general minimum variance estimator for CT is first presented, and then a fast algorithm is described that uses Fourier transform techniques to implement the estimator for either fan beam or parallel beam geometries. The computational requirements of these estimators are examined and compared to other techniques. To allow further comparison with the commonly used convolution-back projection method, a representation of the fast algorithm is derived which allows its equivalent convolving function to be examined. Several examples are presented.