  The work seeks to develop an algorithm for image reconstruction by directly
inverting the non-linear data model in spectral CT. Using the non-linear data
model, we formulate the image-reconstruction problem as a non-convex
optimization program, and develop a non-convex primal-dual (NCPD) algorithm to
solve the program. We devise multiple convergence conditions and perform
verification studies numerically to demonstrate that the NCPD algorithm can
solve the non-convex optimization program and under appropriate data condition,
can invert the non-linear data model. Using the NCPD algorithm, we then
reconstruct monochromatic images from simulated and real data of numerical and
physical phantoms acquired with a standard, full-scan dual-energy
configuration. The result of the reconstruction studies shows that the NCPD
algorithm can correct accurately for the non-linear beam-hardening effect.
Furthermore, we apply the NCPD algorithm to simulated and real data of the
numerical and physical phantoms collected with non-standard, short-scan
dual-energy configurations, and obtain monochromatic images comparable to those
of the standard, full-scan study, thus revealing the potential of the NCPD
algorithm for enabling non-standard scanning configurations in spectral CT,
where the existing indirect methods are limited.
