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SAMCIRT: A Simultaneous Reconstruction and Affine Motion Compensation Technique for Four Dimensional Computed Tomography (4DCT)

Anh-Tuan Nguyen, Jens Renders, Khoi-Nguyen Nguyen, Tat-Dat To, Domenico Iuso, Yves Maris

TL;DR

SAMCIRT introduces a gradient-based framework for simultaneous reconstruction and affine motion estimation in 4DCT by formulating the forward model as $W\mathcal{M}(p)x=b$ with affine warps $\mathcal{M}(p)$ and providing analytic derivatives $\nabla_x g$ and $\nabla_p g$. The authors prove convergence through separated Lipschitz continuity of the forward, gradient, and objective mappings within compact domains, enabling independent step sizes for $x$ and $p$ without nested iterations. Empirical results on simulated and real datasets, including a moving diamond, show substantial reductions in projection distance and improved computational feasibility compared with state-of-the-art affine-motion methods. The work enables accurate 4DCT reconstructions under motion and highlights a novel application area for 4DCT in nonstationary materials, while outlining avenues for extending to more complex motion models and stopping criteria.

Abstract

The majority of the recent iterative approaches in 4DCT not only rely on nested iterations, thereby increasing computational complexity and constraining potential acceleration, but also fail to provide a theoretical proof of convergence for their proposed iterative schemes. On the other hand, the latest MATLAB and Python image processing toolboxes lack the implementation of analytic adjoints of affine motion operators for 3D object volumes, which does not allow gradient methods using exact derivatives towards affine motion parameters. In this work, we propose the Simultaneous Affine Motion-Compensated Image Reconstruction Technique (SAMCIRT)- an efficient iterative reconstruction scheme that combines image reconstruction and affine motion estimation in a single update step, based on the analytic adjoints of the motion operators then exact partial derivatives with respect to both the reconstruction and the affine motion parameters. Moreover, we prove the separated Lipschitz continuity of the objective function and its associated functions, including the gradient, which supports the convergence of our proposed iterative scheme, despite the non-convexity of the objective function with respect to the affine motion parameters. Results from simulation and real experiments show that our method outperforms the state-of-the-art CT reconstruction with affine motion correction methods in computational feasibility and projection distance. In particular, this allows accurate reconstruction for a real, nonstationary diamond, showing a novel application of 4DCT.

SAMCIRT: A Simultaneous Reconstruction and Affine Motion Compensation Technique for Four Dimensional Computed Tomography (4DCT)

TL;DR

SAMCIRT introduces a gradient-based framework for simultaneous reconstruction and affine motion estimation in 4DCT by formulating the forward model as with affine warps and providing analytic derivatives and . The authors prove convergence through separated Lipschitz continuity of the forward, gradient, and objective mappings within compact domains, enabling independent step sizes for and without nested iterations. Empirical results on simulated and real datasets, including a moving diamond, show substantial reductions in projection distance and improved computational feasibility compared with state-of-the-art affine-motion methods. The work enables accurate 4DCT reconstructions under motion and highlights a novel application area for 4DCT in nonstationary materials, while outlining avenues for extending to more complex motion models and stopping criteria.

Abstract

The majority of the recent iterative approaches in 4DCT not only rely on nested iterations, thereby increasing computational complexity and constraining potential acceleration, but also fail to provide a theoretical proof of convergence for their proposed iterative schemes. On the other hand, the latest MATLAB and Python image processing toolboxes lack the implementation of analytic adjoints of affine motion operators for 3D object volumes, which does not allow gradient methods using exact derivatives towards affine motion parameters. In this work, we propose the Simultaneous Affine Motion-Compensated Image Reconstruction Technique (SAMCIRT)- an efficient iterative reconstruction scheme that combines image reconstruction and affine motion estimation in a single update step, based on the analytic adjoints of the motion operators then exact partial derivatives with respect to both the reconstruction and the affine motion parameters. Moreover, we prove the separated Lipschitz continuity of the objective function and its associated functions, including the gradient, which supports the convergence of our proposed iterative scheme, despite the non-convexity of the objective function with respect to the affine motion parameters. Results from simulation and real experiments show that our method outperforms the state-of-the-art CT reconstruction with affine motion correction methods in computational feasibility and projection distance. In particular, this allows accurate reconstruction for a real, nonstationary diamond, showing a novel application of 4DCT.
Paper Structure (21 sections, 3 theorems, 66 equations, 7 figures, 1 table)

This paper contains 21 sections, 3 theorems, 66 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proposed method
  3. Forward model
  4. Analytic derivation of general affine motion operator with respect to affine motion parameters
  5. Dynamic CT model
  6. Optimization strategies
  7. The compactness of the domains, the Lipschitz continuity of the relevant functions and the convergence of the schemes \ref{['eq:GDBB_x']} and \ref{['eq:GDBB_p']}
  8. Initialization
  9. Initial stepsizes
  10. Initial reconstruction
  11. Optimization criteria
  12. Datasets
  13. Experiments
  14. Rigid motion estimation
  15. On the homogeneous diamond phantom: simulation experiment 1
  16. ...and 6 more sections

Key Result

Proposition 2.2

\newlabelprop:Mpx0 The projection functionNot be confused with the projection data $\bm{b}$.$\bm{b}\left(\bm{x}, \bm{p} \right) = \bm{W}\bm{\mathcal{M}\left(\bm{p}\right)}\bm{x}$ is separated Lipschitz continuous in $\bm{x}$ and $\bm{p}$, then so are the norm function $\left \| \bm{b}\left(\bm{x},

Figures (7)

  • Figure 1: central cross-sections of the 3D homogeneous diamond phantom.
  • Figure 2: projection difference and projection distance (PD) results on the first projection of the homogeneous diamond phantom.
  • Figure 3: the real diamond in front of the X-ray source of the CT scanner.
  • Figure 4: difference of the projections at the projection angle $0$ and the projection angle $2 \pi$ the real diamond.
  • Figure 5: parameter estimation result of the translation (top) and the rotation (bottom) on the real diamond dataset (horizontal axis: $i^{th}$ subscan).
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 2.1
  • Proposition 2.2
  • Proof 1
  • Proposition 2.3
  • Proof 2
  • Theorem 2.4
  • Proof 3
  • Definition 2.5