Direct Inversion of the 3D Pseudo-polar Fourier Transform

TL;DR

This paper introduces a direct, fast, and stable algorithm for inverting the 3D pseudo-polar Fourier transform, improving over iterative methods by using only one-dimensional resampling operations.

Contribution

A novel direct inversion algorithm for the 3D pseudo-polar Fourier transform that is faster and more stable than existing iterative methods.

Findings

The algorithm is significantly faster than iterative methods.

It is based solely on one-dimensional resampling operations.

The method provides a stable inversion of the 3D pseudo-polar Fourier transform.

Abstract

The pseudo-polar Fourier transform is a specialized non-equally 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 non-uniform 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 non-equally spaced Fourier transform can be inverted, the only known algorithms are iterative. The convergence speed of these algorithms as well as their accuracy are difficult to control, as they depend both on the sampling geometry as well as on the unknown reconstructed object. In this paper, we present a direct inversion algorithm for the three-dimensional pseudo-polar Fourier transform. The algorithm is based only on one-dimensional resampling operations, and is shown to be significantly faster than existing iterative inversion algorithms.