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Learning-based approaches for reconstructions with inexact operators in nanoCT applications

Tom Lütjen, Fabian Schönfeld, Alice Oberacker, Johannes Leuschner, Maximilian Schmidt, Anne Wald, Tobias Kluth

TL;DR

This work addresses the problem where the forward model is inexact due to stochastic or deterministic deviations during the measurement process, and investigates the performance of non-learned iterative reconstruction methods dealing with inexactness and learned reconstruction schemes, which are based on U-Nets and conditional invertible neural networks.

Abstract

Imaging problems such as the one in nanoCT require the solution of an inverse problem, where it is often taken for granted that the forward operator, i.e., the underlying physical model, is properly known. In the present work we address the problem where the forward model is inexact due to stochastic or deterministic deviations during the measurement process. We particularly investigate the performance of non-learned iterative reconstruction methods dealing with inexactness and learned reconstruction schemes, which are based on U-Nets and conditional invertible neural networks. The latter also provide the opportunity for uncertainty quantification. A synthetic large data set in line with a typical nanoCT setting is provided and extensive numerical experiments are conducted evaluating the proposed methods.

Learning-based approaches for reconstructions with inexact operators in nanoCT applications

TL;DR

This work addresses the problem where the forward model is inexact due to stochastic or deterministic deviations during the measurement process, and investigates the performance of non-learned iterative reconstruction methods dealing with inexactness and learned reconstruction schemes, which are based on U-Nets and conditional invertible neural networks.

Abstract

Imaging problems such as the one in nanoCT require the solution of an inverse problem, where it is often taken for granted that the forward operator, i.e., the underlying physical model, is properly known. In the present work we address the problem where the forward model is inexact due to stochastic or deterministic deviations during the measurement process. We particularly investigate the performance of non-learned iterative reconstruction methods dealing with inexactness and learned reconstruction schemes, which are based on U-Nets and conditional invertible neural networks. The latter also provide the opportunity for uncertainty quantification. A synthetic large data set in line with a typical nanoCT setting is provided and extensive numerical experiments are conducted evaluating the proposed methods.
Paper Structure (24 sections, 22 equations, 21 figures, 9 tables, 2 algorithms)

This paper contains 24 sections, 22 equations, 21 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem description and setting
  3. Materials and methods
  4. Sequential algorithms explicitly taking into account inexactness
  5. Dremel
  6. RESESOP-Kaczmarz
  7. Post-processing via neural networks
  8. Conditional invertible neural networks
  9. Data set generation
  10. Numerical experiments and results
  11. Architecture specifications
  12. Non-perturbed phantom case
  13. Perturbed phantom case
  14. Discussion and Conclusion
  15. Pseudocode Dremel
  16. ...and 9 more sections

Figures (21)

  • Figure 1: Three sample phantoms with $255\times 255$ pixels included in the generated test data set.
  • Figure 2: Randomly generated perturbations for the data set generation with respect to scanner positions $\phi$.
  • Figure 3: Non-perturbed (top) and perturbed (bottom) sinograms of the phantoms in \ref{['phantoms']} resulting from the perturbations like illustrated in \ref{['perturbances']}. Scanner positions $\phi$ are on the horizontal axis and detector positions are on the vertical axis.
  • Figure 4: Image reconstructions using the non-trained reconstruction methods $\mathcal{T}$ on perturbed parallel beam data for the phantoms in \ref{['phantoms']}.
  • Figure 5: Image reconstructions $x_\mathrm{reco}$ and differences to ground truth using the non-trained reconstruction methods $\mathcal{T}$ combined with UNet post-processing on perturbed parallel beam data for the phantoms in \ref{['phantoms']}.
  • ...and 16 more figures