TABLE OF CONTENTS Tomography with diffracting energy requires an entirely different approach to the manner in which projections are mathematically modeled. Acoustic and electromagnetic waves don t travel along straight rays and the projections aren t line integrals, so we will describe the flow of energy with a wave equation.
We will first consider the propagation of waves in homogeneous media, although our ultimate interest lies in imaging the inhomogeneities within an object. The propagation of waves in a homogeneous object is described by a wave equation, which is a second-order linear differential equation. Given such an equation and the source fields in an aperture, we can determine the fields everywhere else in the homogeneous medium.
There are no direct methods for solving the problem of wave propagation in an inhomogeneous medium; in practice, approximate formalisms are used that allow the theory of homogeneous medium wave propagation to be used for generating solutions in the presence of weak inhomogeneities. The better known among these approximate methods go under the names of Born and Rytov approximations.
Although in most cases we are interested in reconstructing three-dimensional objects, the diffraction tomography theory presented in this chapter will deal mostly with the two-dimensional case. Note that when a three-dimensional object can be assumed to vary only slowly along one of the dimensions, a two-dimensional theory can be readily applied to such an object. This assumption, for example, is often made in conventional computerized tomography where images are made of single slices of the object. In...
