There are numerous methods for implementing tomographic image reconstruction. Commercially, the technique which is most commonly used is filtered backprojection (FBP), due to its simplicity in computation. However, this technique is not the most appropriate in cases where the data set is incomplete or the signal-to-noise ratio is low. Therefore, methods based on statistics have been introduced, such as maximum a posteriori (MAP) estimation. These methods rely on modeling the process of projection data generation and modeling of the image a priori. Since the reconstruction problem posed in this manner usually reduces to an optimization problem, a variety of iterative numerical methods for solving this problem has been explored.
This dissertation will explore techniques for tomographic image reconstruction that use non-standard two-dimensional filtering of the sinogram prior to conventional backprojection. These techniques take advantage of features of both the conventional and statistical methods. In the first of our techniques, we will develop optimal nonstationary linear sinogram filters based on sinogram statistics. The advantages of this method are the exploitation of correlation information among the projections of the sinogram data for the filter design, while maintaining a low computational cost for the reconstruction, comparable to that of FBP.
This document also introduces a second technique named nonlinear backprojection (NBP), which attempts to directly model the pseudo-optimal reconstruction operator through off-line training. This allows a more general, less explicit modeling of the data than the traditional statistical methods. The reconstruction of the image is non-iterative, reducing the cost of the method but achieving quality comparable to that of other statistical methods. Results with both methods introduced in this dissertation refute the commonly held belief that sinogram filtering should be one-dimensional along the radial variable.