Image reconstruction in cone beam computed tomography using controlled gradient sparsity

Alexander Meaney; Mikael A. K. Brix; Miika T. Nieminen; Samuli Siltanen

doi:10.3934/ammc.2025003

Article Contents

March 2025, Volume 3, 64-89. Doi: 10.3934/ammc.2025003

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Image reconstruction in cone beam computed tomography using controlled gradient sparsity

1.
University of Helsinki, Helsinki, Finland
2.
Research Unit of Health Sciences and Technology, University of Oulu, Oulu, Finland
3.
Department of Diagnostic Radiology, Oulu University Hospital, Oulu, Finland

^*Corresponding author: Alexander Meaney
^*Corresponding author: Alexander Meaney

Received: December 2024

Revised: February 2025

Early access: March 2025

Published: March 2025

All authors are supported by the FAME Flagship (Research Council of Finland decision 359182) and Business Finland (decision 8132/31/2022).

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Abstract

Total variation (TV) regularization is a popular reconstruction method for ill-posed imaging problems, and particularly useful for applications with piecewise constant targets. However, using TV for cone beam computed X-ray tomography (CBCT) has been limited so far, mainly due to heavy computational loads at practically relevant 3D resolutions and the difficulty in choosing the regularization parameter. Here an efficient minimization algorithm is presented, combined with a dynamic parameter adjustment based on control theory. The result is a fully automatic 3D reconstruction method. The input on top of projection data and system geometry is the desired degree of gradient magnitude sparsity of the reconstruction. This can be determined from an atlas of CT scans, or alternatively used as an easily adjustable parameter with straightforward interpretation.

Keywords:

Mathematics Subject Classification: Primary: 15A29; Secondary: 49N45.

Citation:

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Figure 1. Convergence of the relative distance between consecutive TV-CGS iterations for the 3D Shepp-Logan phantom when $ I_0 = 1000 $

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Figure 2. Convergence of the relative error for the 3D Shepp-Logan phantom when $ I_0 = 1000 $

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Figure 3. Convergence of a) $ \mathcal{C}_{\nabla f} $, and b) $ \alpha $ for the 3D Shepp-Logan phantom when $ I_0 = 1000 $

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Figure 4. Center slices in the axial, coronal, and sagittal directions of the ground truth, the FDK reconstruction, and the TV-CGS reconstruction with $ I_0 = 1000 $ and $ \mathcal{C}_\mathrm{pr} = 0.15 $

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Figure 5. Convergence of the relative error for the 3D Shepp-Logan phantom when $ \mathcal{C}_\mathrm{pr} = 0.15 $

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Figure 6. Convergence of a) $ \mathcal{C}_{\nabla f} $, and b) $ \alpha $ for the 3D Shepp-Logan phantom when $ \mathcal{C}_\mathrm{pr} = 0.15 $

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Figure 7. Convergence of the relative distance between consecutive TV-CGS iterations for the 10% dose experimental dataset

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Figure 8. Convergence of a) $ \mathcal{C}_{\nabla f} $, and b) $ \alpha $ for the 10% dose experimental dataset

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Figure 9. Center slices in the axial, coronal, and sagittal directions of the experimental data set for FDK (dose levels 10% and 100%) and TV-CGS (dose level 10%, $ \mathcal{C}_\mathrm{pr} = \{0.50, 0.56, 0.60\} $)

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Figure 10. ROI closeups of the center slices in the axial, coronal, and sagittal directions of the experimental dataset for FDK (dose levels 10% and 100%) and TV-CGS (dose level 10%, $ \mathcal{C}_\mathrm{pr} = \{0.50, 0.56, 0.60\} $)

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Figure 11. Locations of the ROI closeups of the center slices in the axial, coronal, and sagittal directions of the experimental data, indicated using the dose level 100% FDK reconstruction

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Figure 12. Convergence of the relative distance between consecutive TV-CGS iterations for the 10% dose experimental dataset for constant sparsity $ \mathcal{C}_\mathrm{pr} = 0.56 $ and varying dose levels

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Figure 13. Convergence of a) $ \mathcal{C}_{\nabla f} $, and b) $ \alpha $ for constant sparsity $ \mathcal{C}_\mathrm{pr} = 0.56 $ and varying dose levels of the experimental data

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Figure 14. Center slices in the axial plane of the experimental data set for FDK and TV-CGS ($ \mathcal{C}_\mathrm{pr} = 0.56 $) at all dose levels

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Figure 15. ROI closeups of the center slices in the axial plane of the experimental data set for FDK and TV-CGS ($ \mathcal{C}_\mathrm{pr} = 0.56 $) at all dose levels

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Table 1. Settings for the micro-CT scans of the walnut target

Relative dose	100%	50%	25%	10%
X-ray filter	0.5 mm Al	0.5 mm Al	0.5 mm Al	0.5 mm Al
U (kV)	40	40	40	40
I (mA)	1.0	1.0	0.5	0.2
Exposure time (ms)	2000	1000	1000	1000
Angular range	$ 360^\circ $	$ 360^\circ $	$ 360^\circ $	$ 360^\circ $
# projections	360	360	360	360
Voxel size (mm)	0.0725	0.0725	0.0725	0.0725

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