Tomographic Imaging with Non-Diffracting Sources

Image reconstruction is an important topic of tomographic imaging because spatial information is encoded into the measured data during the data acquisition step. Depending on how spatial information is encoded into the measured data, the image reconstruction technique can vary considerably. In this chapter we deal with the mathematical basis of tomography with non-diffracting sources. We outline two fundamental image reconstruction problems for detailed discussion: (i) reconstruction from Fourier transform samples, and (ii) reconstruction from Radon transform samples. Many practical MRI data acquisition schemes lend themselves naturally to one of these two reconstruction problems while computed tomography (CT) produces data exclusively as a series of projections.

This chapter is organized as follows. First, some general issues in image reconstruction are discussed. Then the algorithms of Fourier reconstruction are described. Finally, image reconstruction from Radon transform data is discussed, starting with a description of the inverse Radon transform, which is followed by an exposition of the practical algorithms.