Approximate analytic reconstruction in x-ray fluorescence computed tomography.

X-ray fluorescence computed tomography (XFCT) is an emerging imaging modality that allows for the reconstruction of the distribution of nonradioactive elements within a sample from measurements of fluorescence x-rays produced by irradiation of the sample with monochromatic synchrotron radiation. XFCT is not a transmission tomography modality, but rather a stimulated emission tomography modality and thus correction for attenuation of the incident and fluorescence photons is essential if accurate images are to be obtained. In this work, we develop and characterize an approximate analytic approach to image reconstruction with attenuation correction in XFCT that is applicable when the incident beam attenuation is uniform and when a factor involving fluorescence attenuation and solid angle effects satisfies a certain approximation. When these conditions hold, we demonstrate that the XFCT imaging equation reduces to the exponential Radon transform, which can be readily inverted. The necessary approximation worsens as the total fluorescence attenuation in the sample grows, but the approach is found to be relatively robust as the approximation breaks down. In a long-axis, small solid angle geometry the proposed approach performs comparably to a previously proposed, more computationally expensive approximate method across a range of attenuation levels. In a short-axis, large solid angle geometry, the proposed approach is found to outperform this previous method.