On the computational implementation of forward and back-projection operations for cone-beam computed tomography

Forward- and back-projection operations are the main computational burden in iterative image reconstruction in computed tomography. In addition, their implementation has to be accurate to ensure stable convergence to a high-quality image. This paper reviews and compares some of the variations in the implementation of these operations in cone-beam computed tomography. We compare four algorithms for computing the system matrix, including a distance-driven algorithm, an algorithm based on cubic basis functions, another based on spherically symmetric basis functions, and a voxel-driven algorithm. The focus of our study is on understanding how the choice of the implementation of the system matrix will influence the performance of iterative image reconstruction algorithms, including such factors as the noise strength and spatial resolution in the reconstructed image. Our experiments with simulated and real cone-beam data reveal the significance of the speed-accuracy trade-off in the implementation of the system matrix. Our results suggest that fast convergence of iterative image reconstruction methods requires accurate implementation of forward- and back-projection operations, involving a direct estimation of the convolution of the footprint of the voxel basis function with the surface of the detectors. The required accuracy decreases by increasing the resolution of the projection measurements beyond the resolution of the reconstructed image. Moreover, reconstruction of low-contrast objects needs more accurate implementation of these operations. Our results also show that, compared with regularized reconstruction methods, the behavior of iterative reconstruction algorithms that do not use a proper regularization is influenced more significantly by the implementation of the forward- and back-projection operations.