Differentiable Forward and Back Projectors for Gradient-Based Deformable Motion Estimation

Accurate estimation and compensation of patient motion are critical for many X-ray imaging applications, including image-guided interventions, cone-beam CT (CBCT) reconstruction, and multi-phase CT imaging. While gradient-based optimization may offer an effective framework for motion estimation, its practicality has been limited by the difficulty of computing analytical gradients of forward and back-projection operators with respect to motion. To address this issue, our recent work proposed a scalable, accurate, and efficient differentiable projector with respect to rigid patient motion. In this work, we extend this method from rigid motion to deformable motion by deriving analytical gradients of forward and back-projectors with respect to a spatially varying displacement vector field (DVF) in the continuous domain. We show that, similar to the rigid case, these gradients preserve the same integration structure as the original projection operators, enabling effective gradient computation using standard projector and backprojector implementations. The proposed method is evaluated in two X-ray imaging tasks: deformable 2D/3D registration using simulated biplane radiography and motion-compensated filtered backprojection (FBP) reconstruction. Simulation results demonstrate effective deformable motion estimation for both 2D/3D registration and motion-compensated FBP tasks, where both qualitative assessment and quantitative analysis reveal substantially reduced projection mismatch and motion-induced artifacts. These results indicate that the proposed differentiable projector provides a practical solution for gradient-based deformable motion estimation and compensation.