Relation to Computerized Tomography

Most major hospitals have a machine called a CAT (Computerized Axial Tomography) scanner (or CT scanner) that provides images of a 2-dimensional section of a patient, to be used by medical staff for diagnosis or surgery. Computerized tomography (also referred to as computer tomography or computed tomography) is the combination of mathematics, computer science, physics, and engineering involved in the image reconstruction, typically from several hundred X-rays taken in coplanar directions. Both parallel and point (or fan-beam) X-rays have been used, though modern scanners usually employ the latter. Two pioneers, A. M. Cormack, a physicist, and G. N. Hounsfield, an engineer, were awarded the 1979 Nobel prize in medicine for the "development of computer assisted tomography."

Mathematically, the problem is to reconstruct an unknown density function \(f(x,y)\) describing a 2-dimensional section of an object from its integrals along lines.The main mathematical tool behind computerized tomography is the Fourier transform. It is known that \(f\) can be reconstructed if its integrals along all lines parallel to an infinite set of directions are known, while if the set of directions is finite, then the reconstruction problem does not generally have a unique solution. Worse, in practise only a finite number of line integrals are available and consequently the Fourier inversion process must be discretized. The utility of the entire process rests on the fact that despite the lack of uniqueness, a sufficiently large number of X-rays permits a reconstruction to any desired degree of accuracy. This very short summary is inadequate, since there are a huge number of variations on this theme, necessitated by the many various applications. In addition, there are quite different algorithms based, for example, on linear algebra. The literature is vast. The reader who wishes to learn more could begin with the texts of Epstein [7] and Kak and Slaney [24], both of which are accessible to advanced undergraduate students; Natterer and Wübbeling [29] is set at a slightly higher level.

Shepp-Logan Image and Reconstruction
Figure 12. Shepp-Logan image (left) and a reconstruction (right) using Murrell’s Mathematica program (see [27])

The relation of geometric tomography to computer tomography can be summed up as follows. While geometric tomography focuses entirely on obtaining information about sets (equivalently, density functions taking only 0 or 1 as values), computerized tomography attempts to reconstruct general density functions. On the other hand, while computerized tomography deals only with X-rays, geometric tomography employs other forms of data such as those listed above.