  Advances in multi-spectral detectors are causing a paradigm shift in X-ray
Computed Tomography (CT). Spectral information acquired from these detectors
can be used to extract volumetric material composition maps of the object of
interest. If the materials and their spectral responses are known a priori, the
image reconstruction step is rather straightforward. If they are not known,
however, the maps as well as the responses need to be estimated jointly. A
conventional workflow in spectral CT involves performing volume reconstruction
followed by material decomposition, or vice versa. However, these methods
inherently suffer from the ill-posedness of the joint reconstruction problem.
To resolve this issue, we propose 'A Dictionary-based Joint reconstruction and
Unmixing method for Spectral Tomography' (ADJUST). Our formulation relies on
forming a dictionary of spectral signatures of materials common in CT and prior
knowledge of the number of materials present in an object. In particular, we
decompose the spectral volume linearly in terms of spatial material maps, a
spectral dictionary, and the indicator of materials for the dictionary
elements. We propose a memory-efficient accelerated alternating proximal
gradient method to find an approximate solution to the resulting bi-convex
problem. From numerical demonstrations on several synthetic phantoms, we
observe that ADJUST performs exceedingly well compared to other
state-of-the-art methods. Additionally, we address the robustness of ADJUST
against limited and noisy measurement patterns. The demonstration of the
proposed approach on a spectral micro-CT dataset shows its potential for
real-world applications. Code is available at
https://github.com/mzeegers/ADJUST.
