Direct Inversion of the Three-Dimensional Pseudo-polar Fourier Transform

The pseudo-polar Fourier transform is a specialized nonequally spaced Fourier transform, which evaluates the Fourier transform on a near-polar grid known as the pseudo-polar grid. The advantage of the pseudo-polar grid over other nonuniform sampling geometries is that the transformation, which samples the Fourier transform on the pseudo-polar grid, can be inverted using a fast and stable algorithm. For other sampling geometries, even if the nonequally spaced Fourier transform can be inverted, the only known algorithms are iterative. The convergence speed of these algorithms and their accuracy are difficult to control, as they depend both on the sampling geometry and on the unknown reconstructed object. In this paper, a direct inversion algorithm for the three-dimensional pseudo-polar Fourier transform is presented. The algorithm is based only on one-dimensional resampling operations and is shown to be significantly faster than existing iterative inversion algorithms.