Joint cone-beam CT reconstruction and rigid motion compensation using a differentiable projector

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Purpose: To develop and assess the feasibility of integrating model-based CBCT reconstruction with rigid motion estimation, using only a projection-domain data fidelity objective coupled with a differentiable forward projector. Methods: We propose a joint image reconstruction and motion estimation framework that integrates motion compensation directly into a model-based iterative reconstruction (MBIR) scheme. Rigid motion with six degrees of freedom (DoF) is modeled as geometric perturbations that alter the projection geometry and thereby affect the estimated projections. To enable gradient-based optimization, a differentiable forward projector with an analytical gradient formulation is employed, allowing efficient backpropagation of the loss between the estimated and measured projections. Within this differentiable framework, the motion parameters and attenuation volume are optimized simultaneously. To evaluate the proposed method, we conducted both simulation and physical phantom studies. In the simulation study, we assessed performance under jerk motion with progressively increasing motion amplitudes, ranging from 1 mm to 10 mm in translation and from 0.02 rad to 0.2 rad in rotation. Reconstruction accuracy was quantified using structural similarity index (SSIM) relative to motion-free ground truth. In the physical phantom study, a jerk motion of 5 mm translation and 0.1 rad rotation was applied during scanning. Motion-free, motion-corrupted, and motion-compensated reconstructions were displayed side-by-side for qualitative comparison, enabling visual assessment of motion artifact reduction. Results: The experimental results demonstrate that the proposed motion compensation method substantially enhances image quality across a range of motion amplitudes. Qualitative evaluation reveal that even under significant translational and rotational motion, the method effectively recovers anatomical structures with minimal residual errors. In simulation studies, SSIM values improved by 18%, 37%, and 48% for motion amplitudes (3 mm, 0.06 rad), (6 mm, 0.12 rad), and (9 mm, 0.18 rad), respectively. Reconstructions from physical bench data further confirm efficacy, showing restored sharpness and anatomical continuity across coronal and sagittal planes. Conclusion: The discrepancy between the estimated and acquired projections serves as a sufficient objective for jointly estimating both the image volume and motion parameters, potentially avoiding the need for image-based sharpness criteria.

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