X-ray fluorescence tomography: Jacobin matrix and confidence of the reconstructed images

The goal of the X-ray Fluorescence Computed Tomography (XFCT) is to give the quantitative description of an object under investigation (sample) in terms of the element composition. However, light and heavy elements inside the object give different contribution to the attenuation of the X-ray probe and of the fluorescence. It leads to the elements got in the shadow area do not give any contribution to the registered spectrum. Iterative reconstruction procedures will try to set to zero the variables describing the element content in composition of corresponding unit volumes as these variables do not change system's condition number. Inversion of the XFCT Radon transform gives random values in these areas. To evaluate the confidence of the reconstructed images we first propose, in addition to the reconstructed images, to calculate a generalized image based on Jacobian matrix. This image highlights the areas of doubt in case if there are exist. In the work we have attempted to prove the advisability of such an approach. For this purpose, we analyzed in detail the process of tomographic projection formation.