  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.
