In spectral computed tomography (spectral CT), the additional information about the energy dependence of
attenuation coefficients can be exploited to generate material selective images. These images have found applications
in various areas such as artifact reduction, quantitative imaging or clinical diagnosis. However, significant
noise amplification on material decomposed images remains a fundamental problem of spectral CT. Most spectral
CT algorithms separate the process of material decomposition and image reconstruction. Separating these
steps is suboptimal because the full statistical information contained in the spectral tomographic measurements
cannot be exploited. Statistical iterative reconstruction (SIR) techniques provide an alternative, mathematically
elegant approach to obtaining material selective images with improved tradeoffs between noise and resolution.
Furthermore, image reconstruction and material decomposition can be performed jointly. This is accomplished
by a forward model which directly connects the (expected) spectral projection measurements and the material
selective images. To obtain this forward model, detailed knowledge of the different photon energy spectra and
the detector response was assumed in previous work. However, accurately determining the spectrum is often
difficult in practice. In this work, a new algorithm for statistical iterative material decomposition is presented.
It uses a semi-empirical forward model which relies on simple calibration measurements. Furthermore, an efficient optimization algorithm based on separable surrogate functions is employed. This partially negates one
of the major shortcomings of SIR, namely high computational cost and long reconstruction times. Numerical
simulations and real experiments show strongly improved image quality and reduced statistical bias compared
to projection-based material decomposition.