Accurate 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 qualitatively and quantitatively accurate images are to be obtained. Attenuation correction has generally been addressed either by use of simple, somewhat inaccurate, analytic methods or by use of computationally intensive iterative methods. In this work, we show that in the case of uniform attenuation, a fairly reasonable approximation allows the XFCT image reconstruction problem to be converted into an exponential Radon transform (ERT) problem. At this point any one of a number of techniques for inverting the well-studied ERT can be brought to bear on the XFCT problem. We demonstrate the validity of our approximation and provide reconstructions from simulated data showing the approach's qualitative and quantitative accuracy.