Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Derivative-Free iterative One-Step Reconstruction for Multispectral CT

Thomas Prohaszka, Lukas Neumann, Markus Haltmeier

TL;DR

The paper addresses the nonlinear MSCT reconstruction problem by introducing a derivative-free, channel-preconditioned one-step framework. It leverages the full nonlinear forward model for the forward update while employing a derivative-free adjoint based on a linearization at zero, using channel preconditioning to mitigate ill-conditioning. The two algorithms CP-full and CP-fast deliver fast convergence and robust performance, outperforming published one-step methods and two-step approaches in simulations with multiple materials and energy bins. The approach is extendable to advanced regularization, Poisson noise models, plug-and-play priors, and learned components, offering a practical pathway for MSCT reconstruction.

Abstract

Image reconstruction in Multispectral Computed Tomography (MSCT) requires solving a challenging nonlinear inverse problem, commonly tackled via iterative optimization algorithms. Existing methods necessitate computing the derivative of the forward map and potentially its regularized inverse. In this work, we present a simple yet highly effective algorithm for MSCT image reconstruction, utilizing iterative update mechanisms that leverage the full forward model in the forward step and a derivative-free adjoint problem. Our approach demonstrates both fast convergence and superior performance compared to existing algorithms, making it an interesting candidate for future work. We also discuss further generalizations of our method and its combination with additional regularization and other data discrepancy terms.

Derivative-Free iterative One-Step Reconstruction for Multispectral CT

TL;DR

The paper addresses the nonlinear MSCT reconstruction problem by introducing a derivative-free, channel-preconditioned one-step framework. It leverages the full nonlinear forward model for the forward update while employing a derivative-free adjoint based on a linearization at zero, using channel preconditioning to mitigate ill-conditioning. The two algorithms CP-full and CP-fast deliver fast convergence and robust performance, outperforming published one-step methods and two-step approaches in simulations with multiple materials and energy bins. The approach is extendable to advanced regularization, Poisson noise models, plug-and-play priors, and learned components, offering a practical pathway for MSCT reconstruction.

Abstract

Image reconstruction in Multispectral Computed Tomography (MSCT) requires solving a challenging nonlinear inverse problem, commonly tackled via iterative optimization algorithms. Existing methods necessitate computing the derivative of the forward map and potentially its regularized inverse. In this work, we present a simple yet highly effective algorithm for MSCT image reconstruction, utilizing iterative update mechanisms that leverage the full forward model in the forward step and a derivative-free adjoint problem. Our approach demonstrates both fast convergence and superior performance compared to existing algorithms, making it an interesting candidate for future work. We also discuss further generalizations of our method and its combination with additional regularization and other data discrepancy terms.
Paper Structure (16 sections, 1 theorem, 18 equations, 4 figures)

This paper contains 16 sections, 1 theorem, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Multispectral CT
  3. Two-step and one-step algorithms
  4. Our contributions
  5. Mathematical modelling of MSCT
  6. Continuous model
  7. Discretisation
  8. Algorithm development
  9. Derivatives computation
  10. Gradient and Newton-type one-step algorithms
  11. Proposed algorithms
  12. Numerical simulations
  13. Comparison methods
  14. Numerical implementation
  15. Results
  16. ...and 1 more sections

Key Result

Theorem 3.3

Let $\mathbf{F}, \mathbf{H} \colon \mathbb R^{N_x \times M} \to \mathbb R^{B \times L}$ and $\mathcal{D}$ be defined by eq:fwd, eq:fwd-log, eq:lsq. The the derivative of $\mathbf{F}$, $\mathbf{H}$, the adjoint and the gradient of $\mathcal{D}$ are given by where $Q_X \triangleq \boldsymbol{\exp}_{E \times N_y}(- {\mathbf M} (\mathbf{A} X)^\intercal)$ denote virtual spectrally resolved data.

Figures (4)

  • Figure 2.1: Illustration of the forward model in MSCT for $m=3$ materials and $b=5$ energy bins, using $E=150$ energy discretizations: First, the Radon transform is applied separately to each of the given material densities $X_1$, $X_2$, and $X_3$, resulting in three material sinograms, which can be seen as a three-channel sinogram. Next, the matrix ${\mathbf M}$ is applied to each pixel, resulting in 150 energy sinograms. To each of these sinograms, $x \mapsto \exp(-x)$ is applied, resulting in 150 virtual energy data maps. By applying the matrix ${\mathbf S}$ pixel by pixel, one obtains the final data consisting of data maps. The continuous forward model can be visualized in a similar way by replacing the material images with continuous counterparts and the 150 energy channels with a function-valued channel.
  • Figure 4.1: Physical parameters determining the forward model. Top Left: Attenuation functions. Top right: Incident spectrum. Bottom left: Spectral response of the detectors. Bottom right: effective spectra.
  • Figure 4.2: Ground truth phantom (top row), and reconstructions using the CP-fast (second row), the CP-full (third row), Mechlem2018 (fourth row), and the two-step algorithm (bottom).
  • Figure 4.3: Relative reconstruction error using proposed CP-fast (top), proposed CP-full (middle) and Mechlem2018 (bootom) as a function of the iteration index.

Theorems & Definitions (9)

  • Remark 2.2: Recalibration
  • Remark 2.4: Noise modelling
  • Remark 2.5: Regularization
  • Remark 3.1: Gradients, inner products and preconditioning
  • Remark 3.2: Some calculus rules
  • Theorem 3.3: Derivatives computation
  • proof
  • Remark 3.4: Derivative at zero
  • Remark 3.5: Composite structure of derivatives