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.

翻译：本研究探讨多谱计算机断层扫描（MSCT）中定量图像重建的条件，该领域仍是活跃的研究课题。在MSCT中，目标是从数据中获取受试对象内部给定X射线能量下的线性衰减系数空间分布，即虚拟单色图像（VMI）。由于VMI通常被分解为具有已知分解系数的基图像的线性组合，VMI的重建因此等价于基图像的重建。一种经验性但高效的两步数据域分解（DDD）方法已被开发并广泛用于MSCT中的定量图像重建。在两步DDD方法中，步骤（1）通过求解非线性变换从数据中估计所谓的基正弦图，而步骤（2）则从估计的基正弦图重建基图像。随后，可通过基图像重建的线性组合直接获得VMI。由于步骤（2）涉及简单线性系统的反演，步骤（1）是DDD方法的关键组成部分，其中需要反演非线性系统以从数据中估计基正弦图。在本工作中，我们考虑步骤（1）中非线性系统的离散形式，随后对离散非线性系统解的存在性、唯一性与稳定性条件进行理论与数值分析，以精确估计离散基正弦图，从而实现MSCT中VMI的定量重建。