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The following HTML text is provided to enhance online readability. Many aspects of typography translate only awkwardly to HTML. Please use the page image as the authoritative form to ensure accuracy. Page 200
object, then
(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.
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