A short scan helical FDK cone beam algorithm based on surfaces satisfying the Tuy's condition

FDK method is the most popular cone beam algorithm to date. Traditionally, short scan helical FDK algorithms have been implemented based on horizontal transaxial slices. However, not even, point on the horizontal transaxial slice satisfies Tuy's condition for the corresponding (pi+fan angle) segment of helix, which means that some points on the horizontal slices are incompletely sampled and are impossible to be exactly reconstructed In this paper, we propose and implement an improved short scan helical cone beam FDK algorithm based on nutating curved surfaces satisfying the Tuy's condition. This surface is defined by averaging PI surfaces emanating the initial and final source points of a (pi+fan angle) segment of helix. One of the key characteristics of the surface is that every point on it satisfies the Tuy's condition for the corresponding (pi+fan angle) segment of helix, which means that we can potentially reconstruct every point on the surface exactly. This difference makes the proposed algorithm deliver a better-reconstructed image quality while requiring a smaller detector area than that of traditional FDK methods based on horizontal transaxial slices. Another characteristics of the proposed surface is that every point in the object space belongs to one and only one such surface. Therefore, the location of the short scan segment for reconstruction of a point in Cartesian coordinate can be pre-calculated and stored in a look up table. This enables us to perform reconstruction directly on rectangular grids. We compare the performance of the improved FDK algorithm with that of a quasi-exact algorithm based on data combination technique. The simulation results show that the reconstructed image quality of these two methods is about the same. We also provide a qualitative analysis of the link between the improved FDK and exact methods. The computational requirement of the proposed algorithm is the same as that of the traditional FDK method. We validate the proposed algorithm with a disc phantom