FOURIER-TRANSFORM RESAMPLING - THEORY AND APPLICATION

One of the most challenging problems in medical imaging is the develop ment of reconstruction algorithms for nonstandard geometries. The stan dard geometry is the parallel ray geometry of the conventional Radon t ransform. This work focuses on the resampling of a nonstandard geometr y 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 MTF's were obtained by projecting point sources at d ifferent transverse positions in the flat fan beam detector geometry. These MTF's were compared to the closed form expression for FTRS. Exce llent agreement was obtained for frequencies at or below the estimated cutoff frequency.The resulting FTRS algorithm is applied to simulatio ns with symmetric fan beam geometry an elliptical orbit and uniform at tenuation, with a normalized root mean square error (NRME) of 0.036.. FTRS is also compared to sine interpolation, and it is shown empirical ly that the two methods are not equivalent. General expressions are ob tained for the transfer function, the MTF, the frequency map, and the resampled autocovariance function. A closed form expression is found f or the frequency map associated with the circular are fan beam geometr y.