3D Forward and Back-Projection for X-Ray CT Using Separable Footprints with Trapezoid Functions

The greatest impediment to practical adoption of iterative methods for X-ray CT is the computation burden of cone-beam forward and back-projectors. Moreover, forward and back-projector accuracy is also crucial to iterative reconstruction methods. We previously described a computationally efficient projector that approximates the voxel footprint functions by the 2D separable products of trapezoid functions in the transaxial plane and rectangular functions in the axial direction [1], [2]. The separability of these footprint functions simplifies calculating their integrals over rectangular detector cells. We showed that this separable footprint (SF-TR) method was more accurate than the distance-driven (DD) method but with comparable computation time. This paper describes a new extension of that projector, called the SF-TT projector, that uses trapezoid functions in both directions. We show that using a trapezoid along the axial direction improves projector accuracy for voxels associated with larger cone angles. However, this improved accuracy requires increased computation compared to the rectangular approximation. Having both options available facilitates evaluation of the trade offs between accuracy and computation for different cone-beam geometries.