# Solution existence, uniqueness, and stability of discrete basis sinograms in multispectral CT

arXiv: https://arxiv.org/abs/2305.03330 · HTML: https://mlsanity.com/paper/2305.03330/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 37 references that are in the map.
- First posted on arXiv 5 May 2023.

## Builds on (most cited first, 1 of 1)

- [An Extended Primal-Dual Algorithm Framework for Nonconvex Problems with Application to Nonlinear Imaging](https://mlsanity.com/paper/2109.07174/) (2021)

## Built on by (most cited first, 1 of 1)

- [A Dual-domain Refinement Network with FBP-based Jacobian Learning for Sparse-view Dual-Energy CT Material Decomposition](https://mlsanity.com/paper/2606.30159/) (2026)

## Abstract (arXiv)

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.