Nonstationary and Nonlinear Sinogram Filtering for Tomographic Image Reconstruction

There are numerous methods for implementing tomographic imagereconstruction. Commercially, the technique which is most commonlyused is filtered backprojection (FBP), due to its simplicityin computation. However, this technique is not the most appropriatein cases where the data set is incomplete or the signal-to-noise ratiois low. Therefore, methods based on statistics have been introduced,such as maximum a posteriori (MAP) estimation. These methods rely onmodeling the process of projection data generation and modeling of theimage a priori. Since the reconstruction problem posed in thismanner usually reduces to an optimization problem, a variety of iterativenumerical methods for solving this problem has been explored.This dissertation will explore techniques for tomographic imagereconstruction that use non-standard two-dimensional filtering of thesinogram prior to conventional backprojection. These techniques take advantageof features of both the conventional and statistical methods. In the firstof our techniques, we will developoptimal nonstationary linear sinogram filters based on sinogramstatistics. The advantages of this method are the exploitation ofcorrelation information among the projections of the sinogram data forthe filter design, while maintaining a low computational cost for thereconstruction, comparable to that of FBP.

This document also introduces a second technique named nonlinearbackprojection (NBP), which attempts to directly model thepseudo-optimal reconstruction operator through off-line training.This allows a more general, less explicit modeling of the data thanthe traditional statistical methods. The reconstruction of the imageis non-iterative, reducing the cost of the method but achievingquality comparable to that of other statistical methods. Results withboth methods introduced in this dissertation refute the commonly heldbelief that sinogram filtering should be one-dimensional alongthe radial variable.