Overview

An entirely different approach for tomographic imaging consists of assuming that the cross section consists of an array of unknowns, and then setting up algebraic equations for the unknowns in terms of the measured projection data. Although conceptually this approach is much simpler than the transform-based methods discussed in previous sections, for medical applications it lacks the accuracy and the speed of implementation. However, there are situations where it is not possible to measure a large number of projections, or the projections are not uniformly distributed over 180 or 360 , both these conditions being necessary requirements for the transformbased techniques to produce results with the accuracy desired in medical imaging. An example of such a situation is earth resources imaging using cross-borehole measurements discussed in Chapter 4. Problems of this type are sometimes more amenable to solution by algebraic techniques. Algebraic techniques are also useful when the energy propagation paths between the source and receiver positions are subject to ray bending on account of refraction, or when the energy propagation undergoes attenuation along ray paths as in emission CT. [Unfortunately, many imaging problems where refraction is encountered also suffer from diffraction effects (see Chap. 4).] As will be obvious from the discussion to follow, in algebraic methods it is essential to know ray paths that connect the corresponding transmitter and receiver positions. When refraction and diffraction effects are substantial (medium inhomogeneities exceed 10% of the average background value and the correlation length of these inhomogeneities is comparable to a...