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

Yu Gao, Xiaochuan Pan, Chong Chen

TL;DR

Results of the numerical study confirm that unique truth basis sinograms can numerically accurately be recovered from noise-free data and that the solution stability is demonstrated using noisy data.

Abstract

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.

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

TL;DR

Results of the numerical study confirm that unique truth basis sinograms can numerically accurately be recovered from noise-free data and that the solution stability is demonstrated using noisy data.

Abstract

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.
Paper Structure (16 sections, 16 theorems, 64 equations, 12 figures)

This paper contains 16 sections, 16 theorems, 64 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Data models in multispectral CT
  3. Continuous-to-continuous (CC)-data model
  4. Discrete-to-discrete (DD)-data model
  5. Existence, uniqueness and stability of solution to \ref{['eq:nonlinear_systems_map']}
  6. Mathematical preliminaries
  7. Mathematical analysis
  8. Existence
  9. Case I:
  10. Uniqueness
  11. Stability
  12. Numerical studies
  13. Numerical study with the Forbild head phantom
  14. Numerical study with the patient-torso phantom
  15. Discussion and Conclusion
  16. ...and 1 more sections

Key Result

Proposition 1

Parthasarathy1983 Mapping $\boldsymbol{H}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a homeomorphism if and only if $\boldsymbol{H}$ is a proper mapping and a local homeomorphism.

Figures (12)

  • Figure 1: Truth basis images of water (column 1) and bone (column 2) and truth VMIs at energies of 60 keV (column 3) and 100 keV (column 4), respectively, of the Forbild head phantom (row 1) and the patient-torso phantom (row 2).
  • Figure 2: (a) Spectra I: low-kV ( red, solid) and high-kV ( blue, dashed) spectra; and (b) Spectra II: low-kV spectrum ( red, solid) identical to the low-kV spectrum in (a) and filtered high-kV spectrum ( blue, dashed dot).
  • Figure 3: Truth (column 1) and estimated (columns 2 and 3) basis sinograms of water (row 1) and bone (row 2) of the Forbild head phantom from noiseless data. Columns 2 and 3 are obtained with spectral pairs in \ref{['fig:normalized_spectra']}a and \ref{['fig:normalized_spectra']}b, respectively.
  • Figure 4: Metrics $\text{RE}_{\boldsymbol{x}}^{n}$ plotted in a log-log scale as functions of the iteration number of the ordinary Newton method obtained with the spectral pairs in \ref{['fig:normalized_spectra']}a (solid) and \ref{['fig:normalized_spectra']}b (dashed), respectively, from noiseless data (a) and noisy data (b) of the Forbild head phantom.
  • Figure 5: Basis images of water (column 1) and bone (column 2) and VMIs at energies 60 keV (column 3) and 100 keV (column 4) reconstructed from noiseless data, respectively, with the spectral pairs in \ref{['fig:normalized_spectra']}a (row 1) and \ref{['fig:normalized_spectra']}b (row 2) of the Forbild head phantom, respectively.
  • ...and 7 more figures

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Lemma 4.1
  • ...and 24 more