Bayesian Methods for XFI

Abstract

X-ray fluorescence imaging (XFI) has the potential to become a highly advantageous medical imaging technique. It offers functional imaging over arbitrary time intervals into significant tissue depth and with high spatial resolution. There are two key steps in processing XFI data, the evaluation of the energy spectra at a given point and the reconstruction of the image. Both of these tasks can be interpreted as inference tasks in the statistical sense. One powerful statistical framework for inference is given by the field of Bayesian statistics which revolves around updating beliefs using new information. This work attempts to extend the established XFI processing pipeline using various Bayesian techniques.

Firstly, this work discusses how x-ray fluorescence (XRF) spectra can be modeled as Poisson variables over a linear combination of base curves. This model then allows simple predictions for measurements at different tracer concentrations and dose levels, which is particularly useful for planning measurements using simulation data. This model is also validated against simulation data.

Next, linear curve fitting of spectral models is reexamined, with least squares, weighted least squares and the first order correction to the Poisson likelihood being studied. Additionally this work shows that projecting the uncertainty of the energy-bin-wise posterior into the parameter space provides a robust and useful uncertainty estimate. The error estimation for signal isolation via spectral subtraction is also examined. Here, the Bayesian posterior leads to a correction of the uncertainty estimate that effectively removes singular behavior in the low count regime. It is shown that both of these revised uncertainty estimates produce strictly more powerful $z$-tests for signal detection than the established methods. Finally a new method for incorporating data from multiple detectors into spectral processing is proposed.

To handle quantitative 3d image reconstruction several methods are evaluated. It is shown that the reconstruction of depth information using multiple detectors, but only one projection, is generally not feasible. However, hierarchical generative Bayesian models are capable of reconstructing 3d images from very few (e.g.\ eleven) projections at reasonable resolution with additional detectors being able to reduce the required dose at equal image quality. The uncertainty estimate this model provides are also shown to be usable for quantitative work. The resolution of the reconstruction is shown to be at the expected limits induced by the data dimensionality itself.