SCBP is of limited applicability for XDT, as the scatter signal from an arbitrary site within a specimen will generally vary in a non-linear way, when viewed at different projection angles. This is due to the attenuation of both the incident beam from source, S, to a point P with the sample, and the scattered X-ray beam from P to the detector, D.

If the experimental XDT condition is such that the X-ray photon mean free path is far greater than the specimen dimensions, the average count rate, N(E, D), at the detector may be expressed approximately as

where N(E, 0) is the incident count rate, d _p(E)/sindis the total differential cross-section at the point P in the sample, and k is a geometrical constant that has to be empirically determined. Expressed this way, XDT is a linear inverse scattering problem and can be solved in a fashion similar to TCT. In general, however, the beam of X-rays are attenuated in travelling from the source, S, through the specimen to the point P, and again from the point P to the specimen boundary and on to the detector, D. This renders the problem non-linear and not amenable to image reconstruction using the conventional SCBP methods.

In this paper, we discuss the reconstruction problem and present a reconstruction strategy for XDT. This discussion is supplemented with examples of experimental XDT arrangements, and some typical results on simple test objects. The efficacy of the reconstruction algorithm is demonstrated utilising Monte Carlo simulations of photon transport through simple phantom objects.