Abstract
Expressions for the fan-beam projection and back-projection operators are given along with an evaluation of the impulse response for the composite of the projection and back-projection operations. The fan-beam geometries for both curved and flat detectors have back-projection operators such that the impulse response for the composite of the projection and back-projection operations is equal to 1/| r − r 0| if the data are taken over 360°, and thus two-dimensional Fourier filter techniques can be used to reconstruct transverse sections from fan-beam projection data. It is shown that the impulse response for the composite of the projection and back-projection operators for fan-beam geometry is not spatially invariant if the data are taken over 180° as is true for parallel-beam projection data. The convolution algorithm is approximately four times faster and requires 40% less memory than the filter of the back-projection algorithm. However, the filter of the back-projection algorithm has the advantage of easily implementing different frequency space filters which lends itself to easily changing the noise propagation vs resolution properties of the convolution kernel.
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