Solution existence, uniqueness, and stability of discrete basis
  sinograms in multispectral CT
Yu Gao Xiaochuan Pan Chong Chen
This work investigates conditions for quantitative image reconstruction in
multispectral computed tomography (MSCT), which remains a topic of active
research. In MSCT, one seeks to obtain from data the spatial distribution of
linear attenuation coefficient, referred to as a virtual monochromatic image
(VMI), at a given X-ray energy, within the subject imaged. As a VMI is
decomposed often into a linear combination of basis images with known
decomposition coefficients, the reconstruction of a VMI is thus tantamount to
that of the basis images. An empirical, but highly effective, two-step
data-domain-decomposition (DDD) method has been developed and used widely for
quantitative image reconstruction in MSCT. In the two-step DDD method, step (1)
estimates the so-called basis sinogram from data through solving a nonlinear
transform, whereas step (2) reconstructs basis images from their basis
sinograms estimated. Subsequently, a VMI can readily be obtained from the
linear combination of basis images reconstructed. As step (2) involves the
inversion of a straightforward linear system, step (1) is the key component of
the DDD method in which a nonlinear system needs to be inverted for estimating
the basis sinograms from data. In this work, we consider a {\it discrete} form
of the nonlinear system in step (1), and then carry out theoretical and
numerical analyses of conditions on the existence, uniqueness, and stability of
a solution to the discrete nonlinear system for accurately estimating the
discrete basis sinograms, leading to quantitative reconstruction of VMIs in
MSCT.