Differentiable Forward and Back-Projector for Deformable Motion Estimation In X-Ray Imaging
Abstract
Purpose
Patient motion estimation is essential for various X-ray imaging applications, and is typically formulated as an optimization problem. Solution using gradient-free methods often suffer from limited accuracy and high computational cost. Gradient-based approaches offer improved efficiency and accuracy but are hindered by the difficulty of analytically computing gradients of projection operators with respect to motion parameters. This work proposes a general differentiable projector framework that enables scalable, accurate, and efficient gradient computation for both rigid and deformable motion estimation.
Methods
We derive analytical gradients of forward and back-projection operators with respect to motion parameters in the continuous domain. A key insight is that these gradients can be expressed in terms of the forward and back-projection operations themselves, allowing gradient computation via standard projection algorithms without relying on auto-differentiation. We implement variants for both rigid motion, parameterized using six degrees of freedom, and deformable motion using downsampled displacement vector fields. Closed-form gradients with respect to both rigid and deformable motion parameters are derived and implemented using forward and back-projection. We illustrate these operators in 2D/3D registration and motion compensated CT reconstruction.
Results
For rigid motion, the proposed forward projector achieves high registration accuracy and effective motion compensation, while demonstrating improved computational efficiency over both gradient-free approaches and existing gradient-based methods. For deformable motion, the differentiable forward projector accurately aligns preoperative volumes to intraoperative projections, substantially reducing projection mismatch in both anteroposterior and lateral views. The differentiable backprojector mitigates motion-induced artifacts across a range of motion speeds, recovering subtle vascular structures and improving quantitative image quality metrics.
Conclusion
The proposed differentiable projector framework enables effective computation of the gradient with respect to motion. This tool has widespread potential application in motion estimation algorithms.