Fourier transform resampling: theory and application

One of the most challenging problems in medical imaging is the development of reconstruction algorithms for nonstandard geometries. The standard geometry is the parallel ray geometry of the conventional Radon transform. This work focuses on the resampling of a nonstandard geometry to obtain a data set in standard geometry. The approach is guided by the application of Fourier analysis to resampling. Fourier Transform Resampling (FTRS) offers potential improvement because the Modulation Transfer Function (MTF) of the process behaves like an ideal low pass filter. Simulated MTFs were obtained by projecting point sources at different transverse positions in the flat fan beam detector geometry. These MTFs were compared to the closed form expression for FTRS. Excellent agreement was obtained for frequencies at or below the estimated cutoff frequency. The resulting FTRS algorithm is applied to simulations with a symmetric fan beam geometry, an elliptical orbit and uniform attenuation, with a normalized root mean square error (NRME) of 0.036. FTRS is also compared to sine interpolation, and it is shown empirically that the two methods are not equivalent. General expressions are obtained for the transfer function, the MTF, the frequency map, and the resampled autocovariance function. A closed form expression is found for the frequency map associated with the circular arc fan beam geometry.