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

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.

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.

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 $\sim$8× 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.

The proposed differentiable projectors enable effective and efficient gradient-based solutions for X-ray imaging tasks requiring rigid motion estimation.