Image reconstruction from truncated cone-beam data

In this dissertation, new approaches to cone-beam tomography are developed that allow reconstruction to be achieved from truncated projections.; A wide class of x-ray source trajectories called saddles is defined and a mathematical analysis of the number of intersections between a saddle and an arbitrary plane is given. This analysis demonstrates that axially truncated cone-beam projections acquired along a saddle can be used for exact reconstruction at any point in a large volume. The shape of the reconstructed volume and the properties of saddles make saddles attractive for cardiac imaging.; In addition, a flexible new methodology is described for accurate cone-beam reconstruction using general vertex paths. The inversion formulas employed by this methodology are based on first backprojecting a derivative in the projection space and then applying a Hilbert transform inversion in the image space. The local nature of the projection space filtering distinguishes this approach from conventional filtered-backprojection methods. This characteristic together with a degree of flexibility in choosing the direction of the Hilbert transform used for inversion offers two important features for the design of data acquisition geometries and reconstruction algorithms. First, the size of the detector necessary to acquire sufficient data for accurate reconstruction of a given region is often smaller than that required by previously documented approaches. In other words, more data truncation is allowed. Second, redundant data can be incorporated for the purpose of noise reduction.; Finally, three additional exact formulas are derived for cone-beam reconstruction using general vertex paths. For reconstruction at a single point, these formulas operate by applying a filtration step followed by a backprojection step to cone-beam data. The filtering is performed along the intersection of the detector surface with a filtering plane. Two of these formulas allow flexibility in choosing the filtering direction. In some cases, this flexibility allows the efficiency of volume reconstruction to be improved. Alternatively, the flexibility can be used to reduce the detector size necessary to avoid truncation artifacts in the reconstruction or to change the noise properties of the reconstruction.; The validity of all inversion formulas are illustrated with reconstructions from computer simulated data.