Standard computed tomography (CT) cannot reproduce spectral information
of an object. Hardware solutions include dual-energy CT which scans the object
twice in different x-ray energy levels, and energy-discriminative detectors
which can separate lower and higher energy levels from a single x-ray scan. In
this paper, we propose a software solution and give an iterative algorithm that
reconstructs an image with spectral information from just one scan with a
standard energy-integrating detector. The spectral information obtained can be
used to produce color CT images, spectral curves of the attenuation coefficient
μ(r, E)at points
inside the object, and photoelectric images, which are all valuable imaging
tools in cancerous diagnosis. Our software solution requires no change on
hardware of a CT machine. With the Shepp–Logan phantom, we have found
that although the photoelectric and Compton components were not perfectly
reconstructed, their composite effect was very accurately reconstructed as
compared to the ground truth and the dual-energy CT counterpart. This means that
our proposed method has an intrinsic benefit in beam hardening correction and
metal artifact reduction. The algorithm is based on a nonlinear polychromatic
acquisition model for x-ray CT. The key technique is a sparse representation of
iterations in a framelet system. Convergence of the algorithm is studied. This
is believed to be the first application of framelet imaging tools to a nonlinear
inverse problem.