Data consistency conditions for truncated fanbeam and parallel projections.

PURPOSE In image reconstruction from projections, data consistency conditions (DCCs) are mathematical relationships that express the overlap of information between ideal projections. DCCs have been incorporated in image reconstruction procedures for positron emission tomography, single photon emission computed tomography, and x-ray computed tomography (CT). Building on published fanbeam DCCs for nontruncated projections along a line, the authors recently announced new DCCs that can be applied to truncated parallel projections in classical (two-dimensional) image reconstruction. These DCCs take the form of polynomial expressions for a weighted backprojection of the projections. The purpose of this work was to present the new DCCs for truncated parallel projections, to extend these conditions to truncated fanbeam projections on a circular trajectory, to verify the conditions with numerical examples, and to present a model of how DCCs could be applied with a toy problem in patient motion estimation with truncated projections. METHODS A mathematical derivation of the new parallel DCCs was performed by substituting the underlying imaging equation into the mathematical expression for the weighted backprojection and demonstrating the resulting polynomial form. This DCC result was extended to fanbeam projections by a substitution of parallel to fanbeam variables. Ideal fanbeam projections of a simple mathematical phantom were simulated and the DCCs for these projections were evaluated by fitting polynomials to the weighted backprojection. For the motion estimation problem, a parametrized motion was simulated using a dynamic version of the mathematical phantom, and both noiseless and noisy fanbeam projections were simulated for a full circular trajectory. The fanbeam DCCs were applied to extract the motion parameters, which allowed the motion contamination to be removed from the projections. A reconstruction was performed from the corrected projections. RESULTS The mathematical derivation revealed the anticipated polynomial behavior. The conversion to fanbeam variables led to a straight-forward weighted fanbeam backprojection which yielded the same function and therefore the same polynomial behavior as occurred in the parallel case. Plots of the numerically calculated DCCs showed polynomial behavior visually indistinguishable from the fitted polynomials. For the motion estimation problem, the motion parameters were satisfactorily recovered and ten times more accurately for the noise-free case. The reconstructed images showed that only a faint trace of the motion blur was still visible after correction from the noisy motion-contaminated projections. CONCLUSIONS New DCCs have been established for fanbeam and parallel projections, and these conditions have been validated using numerical experiments with truncated projections. It has been shown how these DCCs could be applied to extract parameters of unwanted physical effects in tomographic imaging, even with truncated projections.