3. Algorithms for Reconstruction with Nondiffracting Sources

Abstract

In this chapter we will deal with the mathematical basis of tomography with nondiffracting sources. We will show how one can go about recovering the image of the cross section of an object from the projection data. In ideal situations, projections are a set of measurements of the integrated values of some parameter of the object—integrations being along straight lines through the object and being referred to as line integrals. We will show that the key to tomographic imaging is the Fourier Slice Theorem which relates the measured projection data to the two-dimensional Fourier transform of the object cross section.This chapter will start with the definition of line integrals and how they are combined to form projections of an object. By finding the Fourier transform of a projection taken along parallel lines, we will then derive the Fourier Slice Theorem. The reconstruction algorithm used depends on the type of projection data measured; we will discuss algorithms based on parallel beam projection data and two types of fan beam data.3.1 Line Integrals and ProjectionsA line integral, as the name implies, represents the integral of some parameter of the object along a line. In this chapter we will not concern ourselves with the physical phenomena that generate line integrals, but a typical example is the attenuation of x-rays as they propagate through biological tissue. In this case the object is modeled as a two-dimensional (or three-dimensional) distribution of the x-ray attenuation constant and a line integral represents the total attenuation suffered by a beam of x-rays as it travels in a straight line through the object. More details of this process and other examples will be presented in Chapter 4.We will use the coordinate system defined in Fig. 3.1 to describe line integrals and projections. In this example the object is represented by a two-dimensional function ƒ (x, y) and each line integral by the (θ, t) parameters.