Computed Tomography

When a two-dimensional function f(x, y) is line-integrated in the y-direction or, as we say, is projected onto the x-axis, the resulting one-dimensional function ∫ f(x, y) dy does not contain as much information as the original. But other projections, such as the projection ∫ f(x, y) dy′ onto the x′-axis, where (x′, y′) is a rotated coordinate system, contain different information. From a set of such projections one might hope to be able to reconstruct the original function. The best-known example comes from x-ray computed tomography, where Hounsfield’s brain scanner has had a dramatic effect in radiology and dependent fields such as neurology, but reconstruction of projections was already known in more than one branch of astronomy and has continued to arise in many different contexts. Some indispensable preliminary theory for following the reconstruction techniques is given in the preceding chapter. In this chapter the theory is developed in terms of the x-ray application in order to get the benefit of being able to interpret the equations in physical terms.