  We study the problem of spectrum estimation from transmission data of a known
phantom. The goal is to reconstruct an x-ray spectrum that can accurately model
the x-ray transmission curves and reflects a realistic shape of the typical
energy spectra of the CT system. To this end, spectrum estimation is posed as
an optimization problem with x-ray spectrum as unknown variables, and a
Kullback-Leibler (KL) divergence constraint is employed to incorporate prior
knowledge of the spectrum and enhance numerical stability of the estimation
process. The formulated constrained optimization problem is convex and can be
solved efficiently by use of the exponentiated-gradient (EG) algorithm. We
demonstrate the effectiveness of the proposed approach on the simulated and
experimental data. The comparison to the expectation-maximization (EM) method
is also discussed. In simulations, the proposed algorithm is seen to yield
x-ray spectra that closely match the ground truth and represent the attenuation
process of x-ray photons in materials, both included and not included in the
estimation process. In experiments, the calculated transmission curve is in
good agreement with the measured transmission curve, and the estimated spectra
exhibits physically realistic looking shapes. The results further show the
comparable performance between the proposed optimization-based approach and EM.
In conclusion, our formulation of a constrained optimization provides an
interpretable and flexible framework for spectrum estimation. Moreover, a
KL-divergence constraint can include a prior spectrum and appears to capture
important features of x-ray spectrum, allowing accurate and robust estimation
of x-ray spectrum in CT imaging.
