Analytic Image Reconstruction Methods

This chapter describes analytic image reconstruction methods, which includes the standard two-dimensional filtered backprojection (FBP) method and three-dimensional image reconstruction. It also reviews theoretical developments in three-dimensional reconstruction methods. The chapter begins with description of a simple model of the data acquisition process, followed by a discussion on how the Fourier transforms of the data and the original object are related through the central section theorem— also known as the central slice theorem. The central section theorem is used to explain two-dimensional image reconstruction and three-dimensional image reconstruction. It first derives the two-dimensional version, which is then easily extended to the three-dimensional X-ray transform. This is followed by discussion on the version for the three-dimensional Radon transform and generalizations for other transforms. This chapter also highlights the literature that has evolved in the 1990s regarding more esoteric aspects of three-dimensional FBP reconstruction filters, which have a rich complexity not found in two-dimensional image reconstruction, and for which other uses may be found in the future.