Parabolic Radon transform, sampling and efficiency

A good choice of the sampling in the transform domain is essential for a successful application of the parabolic Radon transform. The parabolic Radon transform is computed for each temporal frequency and is essentially equivalent to the nonuniform Fourier transform. This leads to new and useful insights in the parabolic Radon transform. Using nonuniform Fourier theory, we derive a minimum sampling interval for the curvature parameter and a maximum curvature range for which stability is guaranteed for general (irregular) sampling. A significantly smaller sampling interval requires stabilization. If diagonal stabilization is used, no gain in resolution is obtained. In contrast to conventional implementations, the curvature sampling interval is proposed to be inversely proportional to the temporal frequency. This results in improved quality of the transform and yields significant savings in computation time.