Forward and back projectors for gradient-based rigid motion estimation in x-ray imaging

Accurate rigid motion estimation plays a crucial role in numerous X-ray imaging tasks, including 2D/3D registration, motion-compensated reconstruction, and geometric calibration. Despite its importance, gradient-based optimization for these tasks has been constrained by the absence of efficient and generalizable projectors that are differentiable with respect to motion. In this work, we introduce a framework for differentiable forward- and back-projectors that enables scalable, accurate, and memory-efficient gradient computation. Unlike prior approaches that depend on auto-differentiation or are limited to specific projector algorithms, our method derives a general analytical gradient formulation for both forward and backprojection in the continuous domain. The key insight is that the motion gradients of these operations can be expressed directly in terms of the original projection operators themselves, yielding a unified gradient computation scheme applicable across diverse projector types. Building on this analytical foundation, we implement a discretized version equipped with an acceleration strategy that effectively balances computational efficiency and memory consumption. Experimental evaluations demonstrate the capability of the proposed approach: in 2D/3D registration, our method achieves approximately 8× speedup over an existing differentiable forward projector with comparable accuracy, and in motion-compensated analytical reconstruction, it enhances image sharpness and structural fidelity on physical phantom data while offering substantial efficiency gains over existing gradient-based method.