Differentiable Forward and Back-Projector for Rigid Motion Estimation in X-ray Imaging

TL;DR

The paper introduces differentiable forward and back-projectors for rigid motion estimation in X-ray imaging by deriving analytical gradients in the continuous domain, showing that the motion derivatives of forward and back-projection share the same integration structure as the original operators. It then provides a discretized, GPU-friendly implementation with efficiency strategies (on-the-fly gradient computation and patch-based buffering) to balance speed and memory usage, enabling large-scale 2D/3D registration and motion-aware reconstruction. Numerical validations show near-perfect gradient accuracy against finite differences (cosine similarities near 1) and significant memory and time advantages over previous differentiable methods and gradient-free baselines. The framework proves effective across several X-ray tasks, including 2D/3D registration, motion-compensated analytical reconstruction, and online calibration of noncircular gantry orbits, and offers pathways to deformable motion modeling and integration with learning-based losses for broader clinical impact.

Abstract

Objective: In this work, we propose a framework for differentiable forward and back-projector that enables scalable, accurate, and memory-efficient gradient computation for rigid motion estimation tasks. Methods: Unlike existing approaches that rely on auto-differentiation or that are restricted to specific projector types, our method is based on a general analytical gradient formulation for forward/backprojection in the continuous domain. A key insight is that the gradients of both forward and back-projection can be expressed directly in terms of the forward and back-projection operations themselves, providing a unified gradient computation scheme across different projector types. Leveraging this analytical formulation, we develop a discretized implementation with an acceleration strategy that balances computational speed and memory usage. Results: Simulation studies illustrate the numerical accuracy and computational efficiency of the proposed algorithm. Experiments demonstrates the effectiveness of this approach for multiple X-ray imaging tasks we conducted. In 2D/3D registration, the proposed method achieves ~8x speedup over an existing differentiable forward projector while maintaining comparable accuracy. In motion-compensated analytical reconstruction and cone-beam CT geometry calibration, the proposed method enhances image sharpness and structural fidelity on real phantom data while showing significant efficiency advantages over existing gradient-free and gradient-based solutions. Conclusion: The proposed differentiable projectors enable effective and efficient gradient-based solutions for X-ray imaging tasks requiring rigid motion estimation.

Figures (5)

• Figure 1: Geometric illustration of X-ray imaging projection model with rigid object motion.
• Figure 2: Geometric illustration of X-ray imaging projection model with rigid object motion.
• Figure 3: Mean absolute error (MAE) of the motion parameters estimated by CMA-ES, DiffDRR, and proposed approach.
• Figure 4: Motion compensated FBP reconstruction of a spine phantom. The full size volume is reconstructed with a standard resolution (0.5mm$^3$), and the zoom-in image containing a 0.127mm diameter tungsten wire, is reconstructed using the estimated motion with a high resolution (0.0625mm$^3$). Display window: full size/zoom-in: [0.01, 0.03]mm$^{-1}$/[0.015,0.1]mm$^{-1}$
• Figure 5: Reconstruction results using circular (initial) and calibrated geometries. The reference image was reconstructed using FBP from 1440 projections acquired with a circular scan. Structural Similarity Index (SSIM) and Normalized Cross-Correlation (NCC) are used to quatify the similarty between the reconstruction and reference image. The right panel shows zoomed-in views of bone ROIs to highlight the reconstruction quality of fine structural details and sharp features.