(compare to section 3.1.2). In the simplest case the ray L may be thought of as a straight line. Modeling L as a strip or cone, possibly with a weight factor to account for detector inhomogeneities, may be more appropriate. Equation 14.1 neglects the dependence of a with the energy (beam hardening effect) and other nonlinear phenomena (e.g., partial volume effect); see section 3.2.3.
The mathematical problem in transmission tomography is to determine a from measurements of I for a large set of rays L. If L is simply the straight line connecting the source x0 with the detector x1, equation 14.1 gives rise to the integral
where dx is the restriction to L of the Lebesgue measure in Rn. The task is to compute a in a domain.W Í Rn.from the values of equation 14.2 where x0 and x1 run through certain subjects of ¶W.
For n = 2, equation 14.2 is simply a reparametrization of the Radon transform R. The operator R is defined to be
where is a unit vector in Rn ; i.e., q Î Sn-1 , and s Î R. Thus a is in principle found through Radon's inversion formula for R,
R* is given explicitly by
and the operator K is given by
where H is the Hilbert transform. In fact the numerical implementation of equation 14.4 leads to the filtered backprojection algorithm, which is the standard algorithm in commercial CT scanners.