Fast backprojection algorithms for divergent beam tomography

Tomographic imaging involves forming an image of an object from projections or line integrals through the object. The method of choice for performing this reconstruction is filtered backprojection, which consist of a filter step and a backprojection step. Algorithms of this type are preferable, due to their simple implementation and good image quality. However, the computational cost of filtered backprojection algorithms is dominated by the backprojection step, which is O(N3) for 2-D and O(N4) for 3-D reconstruction. Fast backprojection algorithms reduce this complexity to O( N2logN) and O( N3logN), respectively.; Conventional imaging systems use divergent beam geometries for 2-D fan beam and 3-D cone beam tomography, as opposed to the well studied nondivergent parallel beam geometry. The divergent nature of the beams makes it challenging to extend fast parallel beam backprojection algorithms to these geometries.; This thesis covers the application of hierarchical backprojection algorithms to divergent beam geometries. An analysis of the hierarchical algorithm for 2-D fan beam and 3-D circular cone beam is provided, justifying the use of the method in divergent beam scenarios. Efficient implementations of the algorithm are also considered, along with methods for preserving image reconstruction accuracy. The algorithm is also extended to the case of 3-D helical cone beam tomography.; The use of hierarchical backprojection algorithms in divergent beam tomography is found to have good image quality while providing a speedup of 10--80 times over the conventional backprojection algorithm. These fast algorithms will be important when moving to higher resolution images or utilizing iterative reconstruction algorithms, which use multiple backprojection operations.