Optical EngineeringPixel-pyramid model for divergent projection geometry
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Divergent projection refers to fan-beam or cone-beam projection originating from a point source, which is commonly seen in x-ray imaging. Conventionally, the divergent projection calculation is carried out using a ray model, in which each detector pixel (cell) receives radiations emanating from a point source, in a form of straight-line ray. The finite dimension of the detector pixel develops a zero-width ray into a nonzero-width strip, with its cross section assuming the effective aperture of a detector pixel facet. In the pursuit of accuracy, we portray the divergent projection from the point source radiation on a detector cell by a pyramidal element (pyrel), which is formed by a pyramidal base (detector pixel facet) and an apex (point source). A pyrel intersects the object and circumscribes a trapezoidal zone for fan-beam projection (a frustum volume for cone-beam projection), wherein the object voxels are accumulated (appropriately weighted) to produce the detector cell's signal. In addition to the forward divergent projection calculation, the pyrel model can be used for iterative object reconstruction by repeating the backprojection and reprojection. Since the pyrel model portrays more naturally the divergent projection geometry, it maximizes the calculation accuracy, though at the cost of more computations. This model may be rewarding in certain situations: (1) when the detector resolution is unacceptably low, (2) when the projection data set is severely incomplete (for example, the number of projections <100), and (3) when the projection-beam angle is too large (say, >40 deg). Simulations on an iterative fan-beam tomographic reconstruction, using the pyrel model, the ray model, and the voxel-splatting model, are demonstrated.