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

@article{Averbuch2016DirectIO,
title={Direct Inversion of the Three-Dimensional Pseudo-polar Fourier Transform},
author={Amir Averbuch and Gil Shabat and Yoel Shkolnisky},
journal={SIAM J. Sci. Comput.},
year={2016},
volume={38}
}
• Published 14 April 2016
• Physics, Mathematics
• SIAM J. Sci. Comput.
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…

## Figures and Tables from this paper

### An Exact and Fast Computation of Discrete Fourier Transform for Polar and Spherical Grid

• Mathematics
IEEE Transactions on Signal Processing
• 2017
This paper develops exact algorithms to the above problem for 2-D and 3-D, which involve only 1-D equispaced fast Fourier transform with no interpolation or approximation at any stage and leads to a fast solution with very high accuracy.

### Direct inversion of the nonequispaced fast Fourier transform

• Computer Science, Mathematics
Linear Algebra and its Applications
• 2019

### Three-dimensional multiscale discrete Radon and John transforms

• Mathematics
• 2020
Abstract. Two algorithms are introduced for the computation of discrete integral transforms with a multiscale approach operating in discrete three-dimensional (3-D) volumes while considering its

### Signal processing with Fourier analysis, novel algorithms and applications

This thesis investigates novel techniques to solve a suite of four fundamental problems in signal processing that have a wide range of applications and conducts propositions and evaluations of certain applications relevant algorithms such as, frequency estimation algorithm using non-uniform sampling, polar and spherical polar Fourier transform.

### An Exact and Fast CBCT Reconstruction via Pseudo-Polar Fourier Transform-Based Discrete Grangeat’s Formula

• Mathematics
IEEE Transactions on Image Processing
• 2020
The proposed methodology is designed to provide 3D Radon space in linogram fashion to facilitate the use of FIRM with cone beam projections (CBP) for the reconstruction of 3D images in a sparse view angles Cone Beam CT (CBCT).

## References

SHOWING 1-10 OF 46 REFERENCES

### A Framework for Discrete Integral Transformations I-The Pseudopolar Fourier Transform

• Computer Science
SIAM J. Sci. Comput.
• 2008
This paper proves that the ppFT is invertible and develops two algorithms for its inversion: iterative and direct, both with complexity $O(n^{2}\log{n})$, where $n \times n$ is the size of the reconstructed image.

### Pseudo-log-polar Fourier transform for image registration

• Physics
IEEE Signal Processing Letters
• 2006
A new registration algorithm based on pseudo-log-polar Fourier transform (PLPFT) for estimating large translations, rotations, and scalings in images is developed and the robustness and high accuracy of this algorithm is verified.

### Accelerating the Nonuniform Fast Fourier Transform

• Computer Science
SIAM Rev.
• 2004
This paper observes that one of the standard interpolation or "gridding" schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights, of particular value in two- and three- dimensional settings.

### On the Fast Fourier Transform of Functions with Singularities

An explicit approximation of the Fourier Transform of generalized functions of functions with singularities based on projecting such functions on a subspace of Multiresolution Analysis is obtained and a fast algorithm based on its evaluation is developed.

### Nonuniform fast Fourier transforms using min-max interpolation

• Engineering, Computer Science
IEEE Trans. Signal Process.
• 2003
This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm and indicates that the proposed method easily generalizes to multidimensional signals.

### 3-D Symmetry Detection and Analysis Using the Pseudo-polar Fourier Transform

• Computer Science
International Journal of Computer Vision
• 2010
This work presents a computational approach to 3D symmetry detection and analysis conducted in the Fourier domain using the pseudo-polar Fourier transform and derives a novel rigorous analysis of the inherent constraints of 3D symmetries via groups-theory based analysis.

### Algebraically accurate volume registration using Euler's theorem and the 3D pseudo-polar FFT

• Physics
2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05)
• 2005
An algorithm for the registration of rotated and translated volumes, which operates in the frequency domain, and a variant of the angular difference function registration algorithm is derived for the estimation of the planar rotation around the axis.

### ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm

• Computer Science
SIAM J. Imaging Sci.
• 2012
A digital shearlet theory is developed which is rationally designed in the sense that it is the digitization of the existing shearlett theory for continuous data, which implies that shearLET theory provides a unified treatment of both the continuum and digital realms.