IMAGE-RECONSTRUCTION FROM TRUNCATED, 2-DIMENSIONAL, PARALLEL PROJECTIONS

Full three-dimensional scanning allows a significant improvement in image quality in X-ray transmission computerized tomography (CT), in single-photon emission computerized tomography (SPECT) and in positron emission tomography (PET). Increased detection efficiency is obtained by increasing the solid angle seen by the detectors and by detecting photons which are no longer confined to a set of parallel slices as in the standard 2D scanning mode. Consequently, 3D image reconstruction cannot be factored as usual into a set of independent 2D reconstructions, and hence one has to invert the 3D X-ray transform with limited data. Assuming a basic knowledge of standard 2D tomography, this paper presents a review of analytic methods for the reconstruction of a 3D image from a set of 2D parallel projections along some limited set of directions. This inverse problem is overdetermined, i.e., the projection data are redundant, and the consequences of this property are analysed. Redundancy is used to generate classes of exact filters for 3D filtered-backprojection, thereby allowing considerable versatility in the design of inversion algorithms tailored to specific applications. The review also covers the inversion of the 3D X-ray transform when the 2D parallel projections are incompletely measured (truncated), a situation which arises for example in PET, In view of data redundancy, it is possible to build convolution kernels for filtered- backprojection which have a limited support and are not, therefore, affected by truncation. A similar analysis and utilization of data redundancy could be proposed in any application where, instead of trying to define the smallest data set from which the problem can be solved, one attempts to optimize the signal-to-noise ratio by measuring and by incorporating into the reconstruction as much data as practically feasible.