Penalized-likelihood image reconstruction for x-ray fluorescence computed tomography with unknown fluorescence attenuation maps

X-ray fluorescence computed tomography (XFCT) 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. This is challenging because the attenuation map is, in,general, known only at the stimulating beam energy and not at the various fluorescence energies of interest. We make use of empirically fit analytic expressions for x-ray attenuation coefficients to express the unknown attenuation maps as linear combinations of known quantities and the unknown elemental concentrations of interest. We develop an alternating-update iterative reconstruction algorithm based on maximizing a penalized Poisson likelihood objective function. Studies with numerical phantoms indicate that the approach is able to produce qualitatively and quantitatively accurate reconstructed images of numerical phantoms even in the face of severe attenuation.