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Automatic and Computationally Efficient Alignment in Fan- and Cone-beam Tomography

Patricio Guerrero, Simon Bellens, Ricardo Santander, Wim Dewulf

TL;DR

This work tackles geometry estimation for fan- and cone-beam CT with circular trajectories, a prerequisite for high-quality reconstructions. It introduces fast, automatic methods for 2D fan-beam alignment based on symmetry (Yang-like, 2D registration, and fixed-point approaches) and extends to 3D cone-beam alignment via a variable-projection framework that decouples the inner shift $h$ from the outer rotation $η$. The fan-beam methods (2DR and FP) outperform prior low-cost approaches, especially under low-contrast or noisy conditions, while the cone-beam VP approach demonstrates competitive accuracy against reference-object calibration, with rapid convergence and robust performance. The results on simulated and industrial data, together with open-source code, highlight the practical utility for accelerating and stabilizing CT geometry estimation prior to reconstruction.

Abstract

This work is concerned with fan- and cone-beam computed tomography with circular source trajectory, where the reconstruction inverse problem requires an accurate knowledge of source, detector and rotational axis relative positions and orientations. We address this additional inverse problem as a preceding step of the reconstruction process directly from the acquired projections. In the cone-beam case, we present a method that estimates both the detector shift (orthogonal to both focal and rotational axes) and the in-plane detector rotation (over the focal axis) based on the variable projection optimization approach. In addition and for the fan-beam case, two new strategies with low computational cost are presented to estimate the detector shift based on a fan-beam symmetry condition. The methods are validated with simulated and experimental industrial tomographic data with code examples available for both fan- and cone-beam geometries.

Automatic and Computationally Efficient Alignment in Fan- and Cone-beam Tomography

TL;DR

This work tackles geometry estimation for fan- and cone-beam CT with circular trajectories, a prerequisite for high-quality reconstructions. It introduces fast, automatic methods for 2D fan-beam alignment based on symmetry (Yang-like, 2D registration, and fixed-point approaches) and extends to 3D cone-beam alignment via a variable-projection framework that decouples the inner shift from the outer rotation . The fan-beam methods (2DR and FP) outperform prior low-cost approaches, especially under low-contrast or noisy conditions, while the cone-beam VP approach demonstrates competitive accuracy against reference-object calibration, with rapid convergence and robust performance. The results on simulated and industrial data, together with open-source code, highlight the practical utility for accelerating and stabilizing CT geometry estimation prior to reconstruction.

Abstract

This work is concerned with fan- and cone-beam computed tomography with circular source trajectory, where the reconstruction inverse problem requires an accurate knowledge of source, detector and rotational axis relative positions and orientations. We address this additional inverse problem as a preceding step of the reconstruction process directly from the acquired projections. In the cone-beam case, we present a method that estimates both the detector shift (orthogonal to both focal and rotational axes) and the in-plane detector rotation (over the focal axis) based on the variable projection optimization approach. In addition and for the fan-beam case, two new strategies with low computational cost are presented to estimate the detector shift based on a fan-beam symmetry condition. The methods are validated with simulated and experimental industrial tomographic data with code examples available for both fan- and cone-beam geometries.
Paper Structure (19 sections, 6 theorems, 52 equations, 10 figures, 3 tables)

This paper contains 19 sections, 6 theorems, 52 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. 2D problem: alignment in fan-beam tomography
  3. Yang's method as a nearest neighbour interpolation
  4. Higher degree interpolation refinement
  5. Main approaches
  6. 2D sinogram registration
  7. A fixed point method
  8. Discussion and computational cost
  9. 3D problemcase: alignment in cone-beam tomography
  10. A variable projection method
  11. Discussion
  12. Cone-beam Numerical results
  13. Fan-beam data
  14. Simulated experiment
  15. Industrial CT experimentdata
  16. ...and 4 more sections

Key Result

Proposition 1

For $\tilde{g}$ given as in (eq.shifted), the signals $\tilde{p}$ and $w$operators $\Phi_0 \tilde{g}$ and $\Psi_0\tilde{g}$ are related by

Figures (10)

  • Figure 1: Cone-beam parameterization. The alignment variables to estimate are $\{h,\eta\}$.
  • Figure 2: Fan-beam aligned tomographic acquisitions.
  • Figure 3: Phantoms, denoted p$_1$ (left), p$_2$ (middle). Right: Detail of FBP reconstructions of p$_1$ (top) and p$_2$ (bottom) with uncorrected $h=1$ (pixel) illustrating misalignment artifacts.
  • Figure 4: Shift estimation results for phantoms p$_1$ (left) and p$_2$ (right), showing better performances of 2DR and FP when the beam instability increases.
  • Figure 5: low noise fan-beam sinogram of the manufactured object (left), followed by its FBP reconstruction without any detector shift estimation and its detail where double edge artifacts due to misalignments are visible.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • Proposition 5
  • Proposition 6