Radon transform and x - squared fk - transform - aliasing and efficiency

Per temporal frequency component, the (least squares) Parabo-lic Radon Transform (PRT) is equivalent to the (least squares) Nonuniform Discrete Fourier Transform (NDFT). This can be used in two ways. Firstly, the efficiency of the PRT can be improved by using very fast algorithms for the calculation of the direct nonuniform Fourier transform. Secondly, comparing the PRT to the NDFT provides useful insight into the effects of sampling in the transform domain (the q-range and q). Although a frequency independent q is characteristic for the PRT, it is shown that a q that is inversely proportional to ! yields better stability of the least squares inversion. Using this q / 1 ! relationship yields the x 2-f k-transform. For this x 2-f k-transform even more efficient algorithms are possible.